diff --git a/.drone.star b/.drone.star
index 3a6a4739d..2202c6017 100644
--- a/.drone.star
+++ b/.drone.star
@@ -32,10 +32,10 @@ def main(ctx):
     #
     # Sanitizers:
     #
-    result.append(linux_cxx("Ubuntu g++-12 C++20 ASAN" + " " + suite, "g++-12", packages="g++-12", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2204:1", environment={'TOOLSET': 'gcc', 'COMPILER': 'g++-12', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=address <linkflags>-fsanitize=address <cxxflags>-DBOOST_CI_SANITIZER_BUILD' }, globalenv=globalenv))
-    result.append(linux_cxx("Ubuntu g++-12 C++20 USAN" + " " + suite, "g++-12", packages="g++-12", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2204:1", environment={'TOOLSET': 'gcc', 'COMPILER': 'g++-12', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=undefined <linkflags>-fsanitize=undefined <cxxflags>-DBOOST_CI_SANITIZER_BUILD' }, globalenv=globalenv))
-    result.append(linux_cxx("Ubuntu g++-12 C++20 TSAN" + " " + suite, "g++-12", packages="g++-12", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2204:1", environment={'TOOLSET': 'gcc', 'COMPILER': 'g++-12', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=thread <linkflags>-fsanitize=thread <cxxflags>-DBOOST_CI_SANITIZER_BUILD' }, globalenv=globalenv))
-    result.append(linux_cxx("Ubuntu clang++-14 C++20 ISAN" + " " + suite, "clang++-14", packages="clang-14", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2204:1", environment={'TOOLSET': 'clang', 'COMPILER': 'clang++-14', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=integer <linkflags>-fsanitize=integer' }, globalenv=globalenv))
+    result.append(linux_cxx("Ubuntu Clang-18 C++20 ASAN" + " " + suite, "clang++-18", packages="clang-18", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2404:1", environment={'TOOLSET': 'clang', 'COMPILER': 'clang++-18', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=address <linkflags>-fsanitize=address <cxxflags>-DBOOST_CI_SANITIZER_BUILD' }, globalenv=globalenv))
+    result.append(linux_cxx("Ubuntu Clang-18 C++20 USAN" + " " + suite, "clang++-18", packages="clang-18", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2404:1", environment={'TOOLSET': 'clang', 'COMPILER': 'clang++-18', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=undefined <linkflags>-fsanitize=undefined <cxxflags>-DBOOST_CI_SANITIZER_BUILD' }, globalenv=globalenv))
+    result.append(linux_cxx("Ubuntu Clang-18 C++20 TSAN" + " " + suite, "clang++-18", packages="clang-18", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2404:1", environment={'TOOLSET': 'clang', 'COMPILER': 'clang++-18', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=thread <linkflags>-fsanitize=thread <cxxflags>-DBOOST_CI_SANITIZER_BUILD' }, globalenv=globalenv))
+    result.append(linux_cxx("Ubuntu Clang-18 C++20 ISAN" + " " + suite, "clang++-18", packages="clang-18", privileged=True, buildtype="boost", image="cppalliance/droneubuntu2404:1", environment={'TOOLSET': 'clang', 'COMPILER': 'clang++-18', 'CXXSTD': 'gnu++20', 'TEST_SUITE': suite, 'OPTIONS': '<cxxflags>-fsanitize=integer <linkflags>-fsanitize=integer' }, globalenv=globalenv))
 
   for suite in things_to_test:
     for cxx in gnu_5_stds:
@@ -63,7 +63,8 @@ def main(ctx):
       result.append(osx_cxx("M1 Clang " + cxx + " " + suite, "clang++", buildscript="drone", buildtype="boost", xcode_version="14.1", environment={'TOOLSET': 'clang', 'CXXSTD': cxx, 'TEST_SUITE': suite, 'DEFINE': 'BOOST_MATH_NO_REAL_CONCEPT_TESTS,BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS,BOOST_MATH_MULTI_ARCH_CI_RUN', }, globalenv=globalenv))
   for suite in gcc13_things_to_test:
     for cxx in gcc13_stds:
-      result.append(linux_cxx("Ubuntu g++-13 " + cxx + " " + suite, "g++-13", packages="g++-13", buildtype="boost", image="cppalliance/droneubuntu2304:1", environment={'TOOLSET': 'gcc', 'COMPILER': 'g++-13', 'CXXSTD': cxx, 'TEST_SUITE': suite, }, globalenv=globalenv))
+      result.append(linux_cxx("Ubuntu g++-13 " + cxx + " " + suite, "g++-13", packages="g++-13", buildtype="boost", image="cppalliance/droneubuntu2404:1", environment={'TOOLSET': 'gcc', 'COMPILER': 'g++-13', 'CXXSTD': cxx, 'TEST_SUITE': suite, }, globalenv=globalenv))
+      result.append(linux_cxx("Ubuntu g++-14 " + cxx + " " + suite, "g++-14", packages="g++-14", buildtype="boost", image="cppalliance/droneubuntu2404:1", environment={'TOOLSET': 'gcc', 'COMPILER': 'g++-14', 'CXXSTD': cxx, 'TEST_SUITE': suite, }, globalenv=globalenv))
 
   return result
 
diff --git a/.drone/drone.sh b/.drone/drone.sh
index 4ccf92eb7..86dbf129f 100644
--- a/.drone/drone.sh
+++ b/.drone/drone.sh
@@ -40,6 +40,10 @@ python tools/boostdep/depinst/depinst.py math
 ./bootstrap.sh
 ./b2 headers
 
+if [[ $(uname) == "Linux" ]]; then
+    echo 0 | sudo tee /proc/sys/kernel/randomize_va_space
+fi
+
 echo '==================================> BEFORE_SCRIPT'
 
 . $DRONE_BUILD_DIR/.drone/before-script.sh
diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 4ab36ec74..cdd5404fc 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -19,12 +19,72 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
+  ubuntu-noble:
+    runs-on: ubuntu-24.04
+    strategy:
+      fail-fast: false
+      matrix:
+        compiler: [ g++-14, clang++-16, clang++-17, clang++-18 ]
+        standard: [ c++20 ]
+        suite: [ github_ci_block_1, github_ci_block_2 ]
+    steps:
+      - uses: actions/checkout@v4
+        with:
+          fetch-depth: '0'
+      - name: Set TOOLSET
+        run: echo ${{ matrix.compiler }} | awk '/^g/ { print "TOOLSET=gcc" } /^clang/ { print "TOOLSET=clang" }' >> $GITHUB_ENV
+      - name: Add repository
+        continue-on-error: true
+        id: addrepo
+        run: sudo apt-add-repository -y "ppa:ubuntu-toolchain-r/test"
+      - name: Retry Add Repo
+        continue-on-error: true
+        id: retry1
+        if: steps.addrepo.outcome=='failure'
+        run: sudo apt-add-repository -y "ppa:ubuntu-toolchain-r/test"
+      - name: Retry Add Repo 2
+        continue-on-error: true
+        id: retry2
+        if: steps.retry1.outcome=='failure'
+        run: sudo apt-add-repository -y "ppa:ubuntu-toolchain-r/test"
+      - name: Install packages
+        run: sudo apt-get install -y g++-13 g++-14 clang-16 clang-17 clang-18 libgmp-dev libmpfr-dev libfftw3-dev
+      - name: Checkout main boost
+        run: git clone -b develop --depth 1 https://github.com/boostorg/boost.git ../boost-root
+      - name: Update tools/boostdep
+        run: git submodule update --init tools/boostdep
+        working-directory: ../boost-root
+      - name: Copy files
+        run: cp -r $GITHUB_WORKSPACE/* libs/math
+        working-directory: ../boost-root
+      - name: Install deps
+        run: python tools/boostdep/depinst/depinst.py math -I example -I tools
+        working-directory: ../boost-root
+      - name: Bootstrap
+        run: ./bootstrap.sh
+        working-directory: ../boost-root
+      - name: Generate headers
+        run: ./b2 headers
+        working-directory: ../boost-root
+      - name: Generate user config
+        run: 'echo "using $TOOLSET : : ${{ matrix.compiler }} : <cxxflags>-std=${{ matrix.standard }} ;" > ~/user-config.jam'
+        working-directory: ../boost-root
+      - name: Config info install
+        run: ../../../b2 config_info_travis_install toolset=$TOOLSET
+        working-directory: ../boost-root/libs/config/test
+      - name: Config info
+        run: ./config_info_travis
+        working-directory: ../boost-root/libs/config/test
+      - name: Test
+        run: ../../../b2 toolset=$TOOLSET ${{ matrix.suite }} define=CI_SUPPRESS_KNOWN_ISSUES define=SLOW_COMPILER define=BOOST_MATH_RUN_MP_TESTS
+        working-directory: ../boost-root/libs/math/test
+
   ubuntu-jammy:
     runs-on: ubuntu-22.04
     strategy:
       fail-fast: false
       matrix:
-        compiler: [ g++-12, clang++-14 ]
+        compiler: [ g++-12, clang++-14, clang++-15 ]
         standard: [ c++14, c++17 ]
         suite: [ github_ci_block_1, github_ci_block_2 ]
     steps:
@@ -48,7 +108,7 @@ jobs:
         if: steps.retry1.outcome=='failure'
         run: sudo apt-add-repository -y "ppa:ubuntu-toolchain-r/test"
       - name: Install packages
-        run: sudo apt-get install -y g++-12 clang-14 libgmp-dev libmpfr-dev libfftw3-dev
+        run: sudo apt-get install -y g++-12 clang-14 clang-15 libgmp-dev libmpfr-dev libfftw3-dev
       - name: Checkout main boost
         run: git clone -b develop --depth 1 https://github.com/boostorg/boost.git ../boost-root
       - name: Update tools/boostdep
@@ -185,7 +245,7 @@ jobs:
     strategy:
       fail-fast: false
       matrix:
-        toolset: [ msvc-14.0, msvc-14.2 ]
+        toolset: [ msvc-14.2 ]
         standard: [ 14, 17 ]
         suite: [ github_ci_block_1, github_ci_block_2 ]
     steps:
diff --git a/.github/workflows/codecov.yml b/.github/workflows/codecov.yml
index 7978200e4..763b5e1d3 100644
--- a/.github/workflows/codecov.yml
+++ b/.github/workflows/codecov.yml
@@ -195,3 +195,15 @@ jobs:
       - name: Upload coverage
         if: matrix.coverage
         run: ci/codecov.sh "upload"
+        env:
+          BOOST_CI_CODECOV_IO_UPLOAD: skip
+
+      - name: Upload coverage
+        if: matrix.coverage
+        uses: codecov/codecov-action@v4
+        with:
+          disable_search: true
+          file: coverage.info
+          name: Github Actions
+          token: ${{secrets.CODECOV_TOKEN}}
+          verbose: true
diff --git a/include/boost/math/ccmath/fmin.hpp b/include/boost/math/ccmath/fmin.hpp
index 4d0dc229e..1c113e0d6 100644
--- a/include/boost/math/ccmath/fmin.hpp
+++ b/include/boost/math/ccmath/fmin.hpp
@@ -72,13 +72,13 @@ constexpr auto fmin(T1 x, T2 y) noexcept
     }
 }
 
-float fminf(float x, float y) noexcept
+constexpr float fminf(float x, float y) noexcept
 {
     return boost::math::ccmath::fmin(x, y);
 }
 
 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
-long double fminl(long double x, long double y) noexcept
+constexpr long double fminl(long double x, long double y) noexcept
 {
     return boost::math::ccmath::fmin(x, y);
 }
diff --git a/include/boost/math/concepts/distributions.hpp b/include/boost/math/concepts/distributions.hpp
index 538bbe702..7f9af2dd5 100644
--- a/include/boost/math/concepts/distributions.hpp
+++ b/include/boost/math/concepts/distributions.hpp
@@ -343,7 +343,7 @@ struct DistributionConcept
    template <class R, class P>
    static void test_extra_members(const boost::math::hypergeometric_distribution<R, P>& d)
    {
-      unsigned u = d.defective();
+      std::uint64_t u = d.defective();
       u = d.sample_count();
       u = d.total();
       suppress_unused_variable_warning(u);
diff --git a/include/boost/math/concepts/std_real_concept.hpp b/include/boost/math/concepts/std_real_concept.hpp
index 277c35caf..f77935c7f 100644
--- a/include/boost/math/concepts/std_real_concept.hpp
+++ b/include/boost/math/concepts/std_real_concept.hpp
@@ -90,6 +90,9 @@ class std_real_concept
    std_real_concept& operator=(float c) { m_value = c; return *this; }
    std_real_concept& operator=(double c) { m_value = c; return *this; }
    std_real_concept& operator=(long double c) { m_value = c; return *this; }
+#ifdef BOOST_MATH_USE_FLOAT128
+   std_real_concept& operator=(BOOST_MATH_FLOAT128_TYPE c) { m_value = c; return *this; }
+#endif
 
    // Access:
    std_real_concept_base_type value()const{ return m_value; }
diff --git a/include/boost/math/distributions/detail/hypergeometric_pdf.hpp b/include/boost/math/distributions/detail/hypergeometric_pdf.hpp
index 9eeef270d..0ec3065eb 100644
--- a/include/boost/math/distributions/detail/hypergeometric_pdf.hpp
+++ b/include/boost/math/distributions/detail/hypergeometric_pdf.hpp
@@ -297,7 +297,7 @@ T hypergeometric_pdf_prime_loop_imp(hypergeometric_pdf_prime_loop_data& data, hy
    while(data.current_prime <= data.N)
    {
       std::uint64_t base = data.current_prime;
-      std::int64_t prime_powers = 0;
+      std::uint64_t prime_powers = 0;
       while(base <= data.N)
       {
          prime_powers += data.n / base;
@@ -313,7 +313,7 @@ T hypergeometric_pdf_prime_loop_imp(hypergeometric_pdf_prime_loop_data& data, hy
       }
       if(prime_powers)
       {
-         T p = integer_power<T>(static_cast<T>(data.current_prime), prime_powers);
+         T p = integer_power<T>(static_cast<T>(data.current_prime), static_cast<int>(prime_powers));
          if((p > 1) && (tools::max_value<T>() / p < result.value))
          {
             //
@@ -321,7 +321,7 @@ T hypergeometric_pdf_prime_loop_imp(hypergeometric_pdf_prime_loop_data& data, hy
             // to sidestep the issue:
             //
             hypergeometric_pdf_prime_loop_result_entry<T> t = { p, &result };
-            data.current_prime = prime(++data.prime_index);
+            data.current_prime = prime(static_cast<unsigned>(++data.prime_index));
             return hypergeometric_pdf_prime_loop_imp<T>(data, t);
          }
          if((p < 1) && (tools::min_value<T>() / p > result.value))
@@ -331,12 +331,12 @@ T hypergeometric_pdf_prime_loop_imp(hypergeometric_pdf_prime_loop_data& data, hy
             // to sidestep the issue:
             //
             hypergeometric_pdf_prime_loop_result_entry<T> t = { p, &result };
-            data.current_prime = prime(++data.prime_index);
+            data.current_prime = prime(static_cast<unsigned>(++data.prime_index));
             return hypergeometric_pdf_prime_loop_imp<T>(data, t);
          }
          result.value *= p;
       }
-      data.current_prime = prime(++data.prime_index);
+      data.current_prime = prime(static_cast<unsigned>(++data.prime_index));
    }
    //
    // When we get to here we have run out of prime factors,
@@ -395,18 +395,18 @@ T hypergeometric_pdf_factorial_imp(std::uint64_t x, std::uint64_t r, std::uint64
 {
    BOOST_MATH_STD_USING
    BOOST_MATH_ASSERT(N <= boost::math::max_factorial<T>::value);
-   T result = boost::math::unchecked_factorial<T>(n);
+   T result = boost::math::unchecked_factorial<T>(static_cast<unsigned>(n));
    T num[3] = {
-      boost::math::unchecked_factorial<T>(r),
-      boost::math::unchecked_factorial<T>(N - n),
-      boost::math::unchecked_factorial<T>(N - r)
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(r)),
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(N - n)),
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(N - r))
    };
    T denom[5] = {
-      boost::math::unchecked_factorial<T>(N),
-      boost::math::unchecked_factorial<T>(x),
-      boost::math::unchecked_factorial<T>(n - x),
-      boost::math::unchecked_factorial<T>(r - x),
-      boost::math::unchecked_factorial<T>(N - n - r + x)
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(N)),
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(x)),
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(n - x)),
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(r - x)),
+      boost::math::unchecked_factorial<T>(static_cast<unsigned>(N - n - r + x))
    };
    std::size_t i = 0;
    std::size_t j = 0;
diff --git a/include/boost/math/distributions/detail/hypergeometric_quantile.hpp b/include/boost/math/distributions/detail/hypergeometric_quantile.hpp
index 7ab85fb14..2441a7330 100644
--- a/include/boost/math/distributions/detail/hypergeometric_quantile.hpp
+++ b/include/boost/math/distributions/detail/hypergeometric_quantile.hpp
@@ -14,7 +14,7 @@
 namespace boost{ namespace math{ namespace detail{
 
 template <class T>
-inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned lbound, unsigned /*ubound*/, const policies::discrete_quantile<policies::integer_round_down>&)
+inline std::uint64_t round_x_from_p(std::uint64_t x, T p, T cum, T fudge_factor, std::uint64_t lbound, std::uint64_t /*ubound*/, const policies::discrete_quantile<policies::integer_round_down>&)
 {
    if((p < cum * fudge_factor) && (x != lbound))
    {
@@ -25,7 +25,7 @@ inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned /*lbound*/, unsigned ubound, const policies::discrete_quantile<policies::integer_round_up>&)
+inline std::uint64_t round_x_from_p(std::uint64_t x, T p, T cum, T fudge_factor, std::uint64_t /*lbound*/, std::uint64_t ubound, const policies::discrete_quantile<policies::integer_round_up>&)
 {
    if((cum < p * fudge_factor) && (x != ubound))
    {
@@ -36,7 +36,7 @@ inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned lbound, unsigned ubound, const policies::discrete_quantile<policies::integer_round_inwards>&)
+inline std::uint64_t round_x_from_p(std::uint64_t x, T p, T cum, T fudge_factor, std::uint64_t lbound, std::uint64_t ubound, const policies::discrete_quantile<policies::integer_round_inwards>&)
 {
    if(p >= 0.5)
       return round_x_from_p(x, p, cum, fudge_factor, lbound, ubound, policies::discrete_quantile<policies::integer_round_down>());
@@ -44,7 +44,7 @@ inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned lbound, unsigned ubound, const policies::discrete_quantile<policies::integer_round_outwards>&)
+inline std::uint64_t round_x_from_p(std::uint64_t x, T p, T cum, T fudge_factor, std::uint64_t lbound, std::uint64_t ubound, const policies::discrete_quantile<policies::integer_round_outwards>&)
 {
    if(p >= 0.5)
       return round_x_from_p(x, p, cum, fudge_factor, lbound, ubound, policies::discrete_quantile<policies::integer_round_up>());
@@ -52,13 +52,13 @@ inline unsigned round_x_from_p(unsigned x, T p, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_p(unsigned x, T /*p*/, T /*cum*/, T /*fudge_factor*/, unsigned /*lbound*/, unsigned /*ubound*/, const policies::discrete_quantile<policies::integer_round_nearest>&)
+inline std::uint64_t round_x_from_p(std::uint64_t x, T /*p*/, T /*cum*/, T /*fudge_factor*/, std::uint64_t /*lbound*/, std::uint64_t /*ubound*/, const policies::discrete_quantile<policies::integer_round_nearest>&)
 {
    return x;
 }
 
 template <class T>
-inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned lbound, unsigned /*ubound*/, const policies::discrete_quantile<policies::integer_round_down>&)
+inline std::uint64_t round_x_from_q(std::uint64_t x, T q, T cum, T fudge_factor, std::uint64_t lbound, std::uint64_t /*ubound*/, const policies::discrete_quantile<policies::integer_round_down>&)
 {
    if((q * fudge_factor > cum) && (x != lbound))
    {
@@ -69,7 +69,7 @@ inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned /*lbound*/, unsigned ubound, const policies::discrete_quantile<policies::integer_round_up>&)
+inline std::uint64_t round_x_from_q(std::uint64_t x, T q, T cum, T fudge_factor, std::uint64_t /*lbound*/, std::uint64_t ubound, const policies::discrete_quantile<policies::integer_round_up>&)
 {
    if((q < cum * fudge_factor) && (x != ubound))
    {
@@ -80,7 +80,7 @@ inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned lbound, unsigned ubound, const policies::discrete_quantile<policies::integer_round_inwards>&)
+inline std::uint64_t round_x_from_q(std::uint64_t x, T q, T cum, T fudge_factor, std::uint64_t lbound, std::uint64_t ubound, const policies::discrete_quantile<policies::integer_round_inwards>&)
 {
    if(q < 0.5)
       return round_x_from_q(x, q, cum, fudge_factor, lbound, ubound, policies::discrete_quantile<policies::integer_round_down>());
@@ -88,7 +88,7 @@ inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned lbound, unsigned ubound, const policies::discrete_quantile<policies::integer_round_outwards>&)
+inline std::uint64_t round_x_from_q(std::uint64_t x, T q, T cum, T fudge_factor, std::uint64_t lbound, std::uint64_t ubound, const policies::discrete_quantile<policies::integer_round_outwards>&)
 {
    if(q >= 0.5)
       return round_x_from_q(x, q, cum, fudge_factor, lbound, ubound, policies::discrete_quantile<policies::integer_round_down>());
@@ -96,13 +96,13 @@ inline unsigned round_x_from_q(unsigned x, T q, T cum, T fudge_factor, unsigned
 }
 
 template <class T>
-inline unsigned round_x_from_q(unsigned x, T /*q*/, T /*cum*/, T /*fudge_factor*/, unsigned /*lbound*/, unsigned /*ubound*/, const policies::discrete_quantile<policies::integer_round_nearest>&)
+inline std::uint64_t round_x_from_q(std::uint64_t x, T /*q*/, T /*cum*/, T /*fudge_factor*/, std::uint64_t /*lbound*/, std::uint64_t /*ubound*/, const policies::discrete_quantile<policies::integer_round_nearest>&)
 {
    return x;
 }
 
 template <class T, class Policy>
-unsigned hypergeometric_quantile_imp(T p, T q, unsigned r, unsigned n, unsigned N, const Policy& pol)
+std::uint64_t hypergeometric_quantile_imp(T p, T q, std::uint64_t r, std::uint64_t n, std::uint64_t N, const Policy& pol)
 {
 #ifdef _MSC_VER
 #  pragma warning(push)
@@ -113,8 +113,8 @@ unsigned hypergeometric_quantile_imp(T p, T q, unsigned r, unsigned n, unsigned
    BOOST_FPU_EXCEPTION_GUARD
    T result;
    T fudge_factor = 1 + tools::epsilon<T>() * ((N <= boost::math::prime(boost::math::max_prime - 1)) ? 50 : 2 * N);
-   unsigned base = static_cast<unsigned>((std::max)(0, static_cast<int>(n + r) - static_cast<int>(N)));
-   unsigned lim = (std::min)(r, n);
+   std::uint64_t base = static_cast<std::uint64_t>((std::max)(0, static_cast<int>(n + r) - static_cast<int>(N)));
+   std::uint64_t lim = (std::min)(r, n);
 
    BOOST_MATH_INSTRUMENT_VARIABLE(p);
    BOOST_MATH_INSTRUMENT_VARIABLE(q);
@@ -127,7 +127,7 @@ unsigned hypergeometric_quantile_imp(T p, T q, unsigned r, unsigned n, unsigned
 
    if(p <= 0.5)
    {
-      unsigned x = base;
+      std::uint64_t x = base;
       result = hypergeometric_pdf<T>(x, r, n, N, pol);
       T diff = result;
       if (diff == 0)
@@ -175,7 +175,7 @@ unsigned hypergeometric_quantile_imp(T p, T q, unsigned r, unsigned n, unsigned
    }
    else
    {
-      unsigned x = lim;
+      std::uint64_t x = lim;
       result = 0;
       T diff = hypergeometric_pdf<T>(x, r, n, N, pol);
       if (diff == 0)
@@ -225,7 +225,7 @@ unsigned hypergeometric_quantile_imp(T p, T q, unsigned r, unsigned n, unsigned
 }
 
 template <class T, class Policy>
-inline unsigned hypergeometric_quantile(T p, T q, unsigned r, unsigned n, unsigned N, const Policy&)
+inline std::uint64_t hypergeometric_quantile(T p, T q, std::uint64_t r, std::uint64_t n, std::uint64_t N, const Policy&)
 {
    BOOST_FPU_EXCEPTION_GUARD
    typedef typename tools::promote_args<T>::type result_type;
diff --git a/include/boost/math/distributions/geometric.hpp b/include/boost/math/distributions/geometric.hpp
index baff0e249..7c511ef2d 100644
--- a/include/boost/math/distributions/geometric.hpp
+++ b/include/boost/math/distributions/geometric.hpp
@@ -384,6 +384,8 @@ namespace boost
     inline RealType logcdf(const geometric_distribution<RealType, Policy>& dist, const RealType& k)
     { // Cumulative Distribution Function of geometric.
       using std::pow;
+      using std::log;
+      using std::exp;
       static const char* function = "boost::math::logcdf(const geometric_distribution<%1%>&, %1%)";
 
       // k argument may be integral, signed, or unsigned, or floating point.
diff --git a/include/boost/math/distributions/non_central_beta.hpp b/include/boost/math/distributions/non_central_beta.hpp
index b32a605f2..66b12e870 100644
--- a/include/boost/math/distributions/non_central_beta.hpp
+++ b/include/boost/math/distributions/non_central_beta.hpp
@@ -63,6 +63,17 @@ namespace boost
                ? detail::ibeta_imp(T(a + k), b, x, pol, false, true, &xterm)
                : detail::ibeta_imp(b, T(a + k), y, pol, true, true, &xterm);
 
+            while (fabs(beta * pois) < tools::min_value<T>())
+            {
+               if ((k == 0) || (pois == 0))
+                  return init_val;
+               k /= 2;
+               pois = gamma_p_derivative(T(k + 1), l2, pol);
+               beta = x < y
+                  ? detail::ibeta_imp(T(a + k), b, x, pol, false, true, &xterm)
+                  : detail::ibeta_imp(b, T(a + k), y, pol, true, true, &xterm);
+            }
+
             xterm *= y / (a + b + k - 1);
             T poisf(pois), betaf(beta), xtermf(xterm);
             T sum = init_val;
@@ -80,7 +91,7 @@ namespace boost
             {
                T term = beta * pois;
                sum += term;
-               if(((fabs(term/sum) < errtol) && (last_term >= term)) || (term == 0))
+               if(((fabs(term/sum) < errtol) && (fabs(last_term) >= fabs(term))) || (term == 0))
                {
                   count = k - i;
                   break;
@@ -95,6 +106,7 @@ namespace boost
                
                last_term = term;
             }
+            last_term = 0;
             for(auto i = k + 1; ; ++i)
             {
                poisf *= l2 / i;
@@ -103,10 +115,11 @@ namespace boost
 
                T term = poisf * betaf;
                sum += term;
-               if((fabs(term/sum) < errtol) || (term == 0))
+               if(((fabs(term/sum) < errtol) && (fabs(last_term) >= fabs(term))) || (term == 0))
                {
                   break;
                }
+               last_term = term;
                if(static_cast<std::uintmax_t>(count + i - k) > max_iter)
                {
                   return policies::raise_evaluation_error("cdf(non_central_beta_distribution<%1%>, %1%)", "Series did not converge, closest value was %1%", sum, pol); // LCOV_EXCL_LINE
@@ -542,6 +555,24 @@ namespace boost
             T beta = x < y ?
                ibeta_derivative(a + k, b, x, pol)
                : ibeta_derivative(b, a + k, y, pol);
+
+            while (fabs(beta * pois) < tools::min_value<T>())
+            {
+               if ((k == 0) || (pois == 0))
+                  return 0;  // Nothing else we can do!
+               //
+               // We only get here when a+k and b are large and x is small,
+               // in that case reduce k (bisect) until both terms are finite:
+               //
+               k /= 2;
+               pois = gamma_p_derivative(T(k + 1), l2, pol);
+               // Starting beta term:
+               beta = x < y ?
+                  ibeta_derivative(a + k, b, x, pol)
+                  : ibeta_derivative(b, a + k, y, pol);
+            }
+
+
             T sum = 0;
             T poisf(pois);
             T betaf(beta);
@@ -550,15 +581,19 @@ namespace boost
             // Stable backwards recursion first:
             //
             std::uintmax_t count = k;
+            T ratio = 0;
+            T old_ratio = 0;
             for(auto i = k; i >= 0; --i)
             {
                T term = beta * pois;
                sum += term;
-               if((fabs(term/sum) < errtol) || (term == 0))
+               ratio = fabs(term / sum);
+               if(((ratio < errtol) && (ratio < old_ratio)) || (term == 0))
                {
                   count = k - i;
                   break;
                }
+               ratio = old_ratio;
                pois *= i / l2;
 
                if (a + b + i != 1)
@@ -566,6 +601,7 @@ namespace boost
                   beta *= (a + i - 1) / (x * (a + i + b - 1));
                }
             }
+            old_ratio = 0;
             for(auto i = k + 1; ; ++i)
             {
                poisf *= l2 / i;
@@ -573,10 +609,12 @@ namespace boost
 
                T term = poisf * betaf;
                sum += term;
-               if((fabs(term/sum) < errtol) || (term == 0))
+               ratio = fabs(term / sum);
+               if(((ratio < errtol) && (ratio < old_ratio)) || (term == 0))
                {
                   break;
                }
+               old_ratio = ratio;
                if(static_cast<std::uintmax_t>(count + i - k) > max_iter)
                {
                   return policies::raise_evaluation_error("pdf(non_central_beta_distribution<%1%>, %1%)", "Series did not converge, closest value was %1%", sum, pol); // LCOV_EXCL_LINE
diff --git a/include/boost/math/distributions/non_central_t.hpp b/include/boost/math/distributions/non_central_t.hpp
index 4651268d5..a6c988d75 100644
--- a/include/boost/math/distributions/non_central_t.hpp
+++ b/include/boost/math/distributions/non_central_t.hpp
@@ -16,6 +16,8 @@
 #include <boost/math/distributions/students_t.hpp>
 #include <boost/math/distributions/detail/generic_quantile.hpp> // quantile
 #include <boost/math/special_functions/trunc.hpp>
+#include <boost/math/special_functions/detail/hypergeometric_series.hpp>
+#include <boost/math/quadrature/exp_sinh.hpp>
 
 namespace boost
 {
@@ -58,6 +60,20 @@ namespace boost
             beta = x < y
                ? detail::ibeta_imp(T(k + 1), T(v / 2), x, pol, false, true, &xterm)
                : detail::ibeta_imp(T(v / 2), T(k + 1), y, pol, true, true, &xterm);
+
+            while (fabs(beta * pois) < tools::min_value<T>())
+            {
+               if ((k == 0) || (pois == 0))
+                  return init_val;
+               k /= 2;
+               pois = gamma_p_derivative(T(k + 1), d2, pol)
+                  * tgamma_delta_ratio(T(k + 1), T(0.5f))
+                  * delta / constants::root_two<T>();
+               beta = x < y
+                  ? detail::ibeta_imp(T(k + 1), T(v / 2), x, pol, false, true, &xterm)
+                  : detail::ibeta_imp(T(v / 2), T(k + 1), y, pol, true, true, &xterm);
+            }
+
             xterm *= y / (v / 2 + k);
             T poisf(pois), betaf(beta), xtermf(xterm);
             T sum = init_val;
@@ -262,7 +278,13 @@ namespace boost
                }
                else
                   result = 0;
-               result += cdf(boost::math::normal_distribution<T, Policy>(), -delta);
+               if (invert)
+               {
+                  result = cdf(complement(boost::math::normal_distribution<T, Policy>(), -delta)) - result;
+                  invert = false;
+               }
+               else
+                  result += cdf(boost::math::normal_distribution<T, Policy>(), -delta);
             }
             else
             {
@@ -370,6 +392,60 @@ namespace boost
                function);
          }
 
+         template <class T, class Policy>
+         T non_central_t_pdf_integral(T x, T v, T mu, const Policy& pol)
+         {
+            BOOST_MATH_STD_USING
+            boost::math::quadrature::exp_sinh<T, Policy> integrator;
+            T integral = pow(v, v / 2) * exp(-v * mu * mu / (2 * (x * x + v)));
+            if (integral != 0)
+            {
+               integral *= integrator.integrate([&x, v, mu](T y) 
+                  {
+                     T p;
+                     if (v * log(y) < tools::log_max_value<T>())
+                        p = pow(y, v) * exp(boost::math::pow<2>((y - mu * x / sqrt(x * x + v))) / -2);
+                     else
+                        p = exp(log(y) * v + boost::math::pow<2>((y - mu * x / sqrt(x * x + v))) / -2);
+                     return p; 
+                  });
+            }
+            integral /= boost::math::constants::root_pi<T>() * boost::math::tgamma(v / 2, pol) * pow(T(2), (v - 1) / 2) * pow(x * x + v, (v + 1) / 2);
+            return integral;
+         }
+
+         template <class T, class Policy>
+         T non_central_t_pdf_hypergeometric(T x, T v, T mu, const Policy& pol)
+         {
+            BOOST_MATH_STD_USING
+            long long scale = 0;
+            const char* function = "non central T PDF";
+            //
+            // We only call this routine when we know that the series form of 1F1 is cheap to evaluate,
+            // so no need to call the whole 1F1 function, just the series will do:
+            //
+            T Av = hypergeometric_1F1_generic_series(static_cast<T>((v + 1) / 2), boost::math::constants::half<T>(), static_cast<T>(mu * mu * x * x / (2 * (x * x + v))), pol, scale, function);
+            Av = ldexp(Av, static_cast<int>(scale));
+            scale = 0;
+            T Bv = hypergeometric_1F1_generic_series(static_cast<T>(v / 2 + T(1)), static_cast<T>(T(3) / 2), static_cast<T>(mu * mu * x * x / (2 * (x * x + v))), pol, scale, function);
+            Bv = ldexp(Bv, static_cast<int>(scale));
+            Bv *= boost::math::tgamma_delta_ratio(v / 2 + T(1), -constants::half<T>(), pol);
+            Bv *= boost::math::constants::root_two<T>() * mu * x / sqrt(x * x + v);
+
+            T tolerance = tools::root_epsilon<T>() * Av * 4;
+            Av += Bv;
+
+            if (Av < tolerance)
+            {
+               // More than half the digits have cancelled, fall back to the integral method:
+               return non_central_t_pdf_integral(x, v, mu, pol);
+            }
+
+            Av *= exp(-mu * mu / 2) * pow(1 + x * x / v, -(v + 1) / 2) * boost::math::tgamma_delta_ratio(v / 2 + constants::half<T>(), -constants::half<T>(), pol);
+            Av /= sqrt(v) * boost::math::constants::root_pi<T>();
+            return Av;
+         }
+
          template <class T, class Policy>
          T non_central_t2_pdf(T n, T delta, T x, T y, const Policy& pol, T init_val)
          {
@@ -398,10 +474,25 @@ namespace boost
                ? ibeta_derivative(T(k + 1), n / 2, x, pol)
                : ibeta_derivative(n / 2, T(k + 1), y, pol);
             BOOST_MATH_INSTRUMENT_VARIABLE(xterm);
+
+            while (fabs(xterm * pois) < tools::min_value<T>())
+            {
+               if (k == 0)
+                  return init_val;
+               k /= 2;
+               pois = gamma_p_derivative(T(k + 1), d2, pol)
+                  * tgamma_delta_ratio(T(k + 1), T(0.5f))
+                  * delta / constants::root_two<T>();
+               BOOST_MATH_INSTRUMENT_VARIABLE(pois);
+               // Starting beta term:
+               xterm = x < y
+                  ? ibeta_derivative(T(k + 1), n / 2, x, pol)
+                  : ibeta_derivative(n / 2, T(k + 1), y, pol);
+               BOOST_MATH_INSTRUMENT_VARIABLE(xterm);
+            }
+
             T poisf(pois), xtermf(xterm);
             T sum = init_val;
-            if((pois == 0) || (xterm == 0))
-               return init_val;
 
             //
             // Backwards recursion first, this is the stable
@@ -427,15 +518,18 @@ namespace boost
                }
             }
             BOOST_MATH_INSTRUMENT_VARIABLE(sum);
+            old_ratio = 0;
             for(auto i = k + 1; ; ++i)
             {
                poisf *= d2 / (i + 0.5f);
                xtermf *= (x * (n / 2 + i)) / (i);
                T term = poisf * xtermf;
                sum += term;
-               if((fabs(term/sum) < errtol) || (term == 0))
+               T ratio = fabs(term / sum);
+               if(((ratio < errtol) && (ratio < old_ratio)) || (term == 0))
                   break;
                ++count;
+               old_ratio = ratio;
                if(count > max_iter)
                {
                   return policies::raise_evaluation_error("pdf(non_central_t_distribution<%1%>, %1%)", "Series did not converge, closest value was %1%", sum, pol); // LCOV_EXCL_LINE
@@ -454,14 +548,7 @@ namespace boost
                normal_distribution<T, Policy> norm(delta, 1);
                return pdf(norm, t);
             }
-            //
-            // Otherwise, for t < 0 we have to use the reflection formula:
-            if(t < 0)
-            {
-               t = -t;
-               delta = -delta;
-            }
-            if(t == 0)
+            if(t * t < tools::epsilon<T>())
             {
                //
                // Handle this as a special case, using the formula
@@ -492,13 +579,30 @@ namespace boost
                return pdf(students_t_distribution<T, Policy>(n), t - delta);
             }
             //
+            // Figure out if the hypergeometric formula will be efficient or not, 
+            // based on where the summit of the series.
+            //
+            T a = (n + 1) / 2;
+            T x = delta * delta * t * t / (2 * (t * t + n));
+            T summit = (sqrt(x * (4 * a + x)) + x) / 2;
+            if (summit < 40)
+               return non_central_t_pdf_hypergeometric(t, n, delta, pol);
+            // 
+            // Otherwise, for t < 0 we have to use the reflection formula:
+            //
+            if (t < 0)
+            {
+               t = -t;
+               delta = -delta;
+            }
+            //
             // x and y are the corresponding random
             // variables for the noncentral beta distribution,
             // with y = 1 - x:
             //
-            T x = t * t / (n + t * t);
+            x = t * t / (n + t * t);
             T y = n / (n + t * t);
-            T a = 0.5f;
+            a = 0.5f;
             T b = n / 2;
             T d2 = delta * delta;
             //
@@ -508,12 +612,22 @@ namespace boost
             BOOST_MATH_INSTRUMENT_VARIABLE(dt);
             T result = non_central_beta_pdf(a, b, d2, x, y, pol);
             BOOST_MATH_INSTRUMENT_VARIABLE(result);
-            T tol = tools::epsilon<T>() * result * 500;
+            T tol = tools::root_epsilon<T>() * result;
             result = non_central_t2_pdf(n, delta, x, y, pol, result);
             BOOST_MATH_INSTRUMENT_VARIABLE(result);
-            if(result <= tol)
-               result = 0;
             result *= dt;
+            if (result <= tol)
+            {
+               // More than half the digits in the result have cancelled, 
+               // Try direct integration... super slow but reliable as far as we can tell...
+               if (delta < 0)
+               {
+                  // reflect back:
+                  delta = -delta;
+                  t = -t;
+               }
+               result = non_central_t_pdf_integral(t, n, delta, pol);
+            }
             return result;
          }
 
diff --git a/include/boost/math/distributions/skew_normal.hpp b/include/boost/math/distributions/skew_normal.hpp
index f3cc5a6b5..7df81b6f5 100644
--- a/include/boost/math/distributions/skew_normal.hpp
+++ b/include/boost/math/distributions/skew_normal.hpp
@@ -27,6 +27,10 @@
 #include <utility>
 #include <algorithm> // std::lower_bound, std::distance
 
+#ifdef BOOST_MATH_INSTRUMENT_SKEW_NORMAL_ITERATIONS
+extern std::uintmax_t global_iter_count;
+#endif
+
 namespace boost{ namespace math{
 
   namespace detail
@@ -614,32 +618,6 @@ namespace boost{ namespace math{
     return factor * y*y / (x*x);
   }
 
-  namespace detail
-  {
-
-    template <class RealType, class Policy>
-    struct skew_normal_quantile_functor
-    {
-      skew_normal_quantile_functor(const boost::math::skew_normal_distribution<RealType, Policy> dist, RealType const& p)
-        : distribution(dist), prob(p)
-      {
-      }
-
-      boost::math::tuple<RealType, RealType> operator()(RealType const& x)
-      {
-        RealType c = cdf(distribution, x);
-        RealType fx = c - prob;  // Difference cdf - value - to minimize.
-        RealType dx = pdf(distribution, x); // pdf is 1st derivative.
-        // return both function evaluation difference f(x) and 1st derivative f'(x).
-        return boost::math::make_tuple(fx, dx);
-      }
-    private:
-      const boost::math::skew_normal_distribution<RealType, Policy> distribution;
-      RealType prob;
-    };
-
-  } // namespace detail
-
   template <class RealType, class Policy>
   inline RealType quantile(const skew_normal_distribution<RealType, Policy>& dist, const RealType& p)
   {
@@ -681,14 +659,58 @@ namespace boost{ namespace math{
 
     // refine the result by numerically searching the root of (p-cdf)
 
-    const RealType search_min = support(dist).first;
-    const RealType search_max = support(dist).second;
-
     const int get_digits = policies::digits<RealType, Policy>();// get digits from policy,
     std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); // and max iterations.
 
-    result = tools::newton_raphson_iterate(detail::skew_normal_quantile_functor<RealType, Policy>(dist, p), result,
-      search_min, search_max, get_digits, max_iter);
+    if (result == 0)
+       result = tools::min_value<RealType>(); // we need to be one side of zero or the other for the root finder to work.
+
+    auto fun = [&, dist, p](const RealType& x)->RealType { return cdf(dist, x) - p; };
+
+    RealType f_result = fun(result);
+
+    if (f_result == 0)
+       return result;
+
+    if (f_result * result > 0)
+    {
+       // If the root is in the direction of zero, we need to check that we're the correct side of it:
+       RealType f_zero = fun(static_cast<RealType>(0));
+       if (f_zero * f_result > 0)
+       {
+          // we're the wrong side of zero:
+          result = -result;
+          f_result = fun(result);
+       }
+    }
+
+    RealType scaling_factor = 1.25;
+    if (f_result * result > 0)
+    {
+       // We're heading towards zero... it's a long way down so use a larger scaling factor:
+       scaling_factor = 16;
+    }
+
+    auto p_result = tools::bracket_and_solve_root(fun, result, scaling_factor, true, tools::eps_tolerance<RealType>(get_digits), max_iter, Policy());
+
+#ifdef BOOST_MATH_INSTRUMENT_SKEW_NORMAL_ITERATIONS
+    global_iter_count += max_iter;
+#endif
+
+    result = (p_result.first + p_result.second) / 2;
+
+    //
+    // Try one last Newton step, just to close up the interval:
+    //
+    RealType step = fun(result) / pdf(dist, result);
+
+    if (result - step <= p_result.first)
+       result = p_result.first;
+    else if (result - step >= p_result.second)
+       result = p_result.second;
+    else
+       result -= step;
+
     if (max_iter >= policies::get_max_root_iterations<Policy>())
     {
        return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time: either there is no answer to quantile" // LCOV_EXCL_LINE
diff --git a/include/boost/math/quadrature/detail/exp_sinh_detail.hpp b/include/boost/math/quadrature/detail/exp_sinh_detail.hpp
index e829fa328..2df07b6ec 100644
--- a/include/boost/math/quadrature/detail/exp_sinh_detail.hpp
+++ b/include/boost/math/quadrature/detail/exp_sinh_detail.hpp
@@ -198,13 +198,29 @@ auto exp_sinh_detail<Real, Policy>::integrate(const F& f, Real* error, Real* L1,
         if (!have_first_j && (I1_last == I1))
         {
            // No change to the sum, disregard these values on the LHS:
-           min_abscissa = m_abscissas[1][i];
-           first_j = i;
+           if ((i < m_abscissas[1].size() - 1) && (m_abscissas[1][i + 1] > max_abscissa))
+           {
+              // The summit is so high, that we found nothing in this row which added to the integral!!
+              have_first_j = true;
+           }
+           else
+           {
+              min_abscissa = m_abscissas[1][i];
+              first_j = i;
+           }
         }
         else
            have_first_j = true;
     }
 
+    if (I0 == static_cast<Real>(0))
+    {
+       // We failed to find anything, is the integral zero, or have we just not found it yet?
+       // We'll try one more level, if that still finds nothing then it'll terminate.
+       min_abscissa = 0;
+       max_abscissa = boost::math::tools::max_value<Real>();
+    }
+
     I1 *= half<Real>();
     L1_I1 *= half<Real>();
     Real err = abs(I0 - I1);
@@ -243,7 +259,7 @@ auto exp_sinh_detail<Real, Policy>::integrate(const F& f, Real* error, Real* L1,
         err = abs(I0 - I1);
         //std::cout << "Estimate:        " << I1 << " Error estimate at level " << i  << " = " << err << std::endl;
         // Use L1_I1 here to make it work with both complex and real valued integrands:
-        if (!isfinite(L1_I1))
+        if (!(boost::math::isfinite)(L1_I1))
         {
             return static_cast<K>(policies::raise_evaluation_error(function, "The exp_sinh quadrature evaluated your function at a singular point and returned %1%. Please ensure your function evaluates to a finite number over its entire domain.", I1, Policy()));
         }
diff --git a/include/boost/math/quadrature/exp_sinh.hpp b/include/boost/math/quadrature/exp_sinh.hpp
index 14970bfdb..f28493737 100644
--- a/include/boost/math/quadrature/exp_sinh.hpp
+++ b/include/boost/math/quadrature/exp_sinh.hpp
@@ -91,8 +91,7 @@ auto exp_sinh<Real, Policy>::integrate(const F& f, Real tolerance, Real* error,
     static const char* function = "boost::math::quadrature::exp_sinh<%1%>::integrate";
     using std::abs;
     if (abs(tolerance) > 1) {
-        std::string msg = std::string(__FILE__) + ":" + std::to_string(__LINE__) + ":" + std::string(function) + ": The tolerance provided is unusually large; did you confuse it with a domain bound?";
-        throw std::domain_error(msg);
+        return policies::raise_domain_error(function, "The tolerance provided (%1%) is unusually large; did you confuse it with a domain bound?", tolerance, Policy());
     }
     return m_imp->integrate(f, error, L1, function, tolerance, levels);
 }
diff --git a/include/boost/math/special_functions/bessel.hpp b/include/boost/math/special_functions/bessel.hpp
index 82bed6e49..e9677d3c7 100644
--- a/include/boost/math/special_functions/bessel.hpp
+++ b/include/boost/math/special_functions/bessel.hpp
@@ -151,7 +151,7 @@ inline T sph_bessel_j_imp(unsigned n, T x, const Policy& pol)
    // Special case, n == 0 resolves down to the sinus cardinal of x:
    //
    if(n == 0)
-      return boost::math::sinc_pi(x, pol);
+      return (boost::math::isinf)(x) ? T(0) : boost::math::sinc_pi(x, pol);
    //
    // Special case for x == 0:
    //
diff --git a/include/boost/math/special_functions/detail/bessel_ik.hpp b/include/boost/math/special_functions/detail/bessel_ik.hpp
index da2310866..0c653b475 100644
--- a/include/boost/math/special_functions/detail/bessel_ik.hpp
+++ b/include/boost/math/special_functions/detail/bessel_ik.hpp
@@ -322,83 +322,96 @@ int bessel_ik(T v, T x, T* result_I, T* result_K, int kind, const Policy& pol)
         v = -v;                             // v is non-negative from here
         kind |= need_k;
     }
-    n = iround(v, pol);
-    u = v - n;                              // -1/2 <= u < 1/2
-    BOOST_MATH_INSTRUMENT_VARIABLE(n);
-    BOOST_MATH_INSTRUMENT_VARIABLE(u);
 
-    BOOST_MATH_ASSERT(x > 0); // Error handling for x <= 0 handled in cyl_bessel_i and cyl_bessel_k
-
-    // x is positive until reflection
-    W = 1 / x;                                 // Wronskian
-    if (x <= 2)                                // x in (0, 2]
-    {
-        temme_ik(u, x, &Ku, &Ku1, pol);             // Temme series
-    }
-    else                                       // x in (2, \infty)
-    {
-        CF2_ik(u, x, &Ku, &Ku1, pol);               // continued fraction CF2_ik
-    }
-    BOOST_MATH_INSTRUMENT_VARIABLE(Ku);
-    BOOST_MATH_INSTRUMENT_VARIABLE(Ku1);
-    prev = Ku;
-    current = Ku1;
     T scale = 1;
     T scale_sign = 1;
-    for (k = 1; k <= n; k++)                   // forward recurrence for K
+
+    if (((kind & need_i) == 0) && (fabs(4 * v * v - 25) / (8 * x) < tools::forth_root_epsilon<T>()))
     {
-        T fact = 2 * (u + k) / x;
-        // Check for overflow: if (max - |prev|) / fact > max, then overflow
-        // (max - |prev|) / fact > max
-        // max * (1 - fact) > |prev|
-        // if fact < 1: safe to compute overflow check
-        // if fact >= 1:  won't overflow
-        const bool will_overflow = (fact < 1)
-          ? tools::max_value<T>() * (1 - fact) > fabs(prev)
-          : false;
-        if(!will_overflow && ((tools::max_value<T>() - fabs(prev)) / fact < fabs(current)))
-        {
-           prev /= current;
-           scale /= current;
-           scale_sign *= ((boost::math::signbit)(current) ? -1 : 1);
-           current = 1;
-        }
-        next = fact * current + prev;
-        prev = current;
-        current = next;
+       // A&S 9.7.2
+       Iv = std::numeric_limits<T>::quiet_NaN(); // any value will do
+       T mu = 4 * v * v;
+       T eight_z = 8 * x;
+       Kv = 1 + (mu - 1) / eight_z + (mu - 1) * (mu - 9) / (2 * eight_z * eight_z) + (mu - 1) * (mu - 9) * (mu - 25) / (6 * eight_z * eight_z * eight_z);
+       Kv *= exp(-x) * constants::root_pi<T>() / sqrt(2 * x);
     }
-    Kv = prev;
-    Kv1 = current;
-    BOOST_MATH_INSTRUMENT_VARIABLE(Kv);
-    BOOST_MATH_INSTRUMENT_VARIABLE(Kv1);
-    if(kind & need_i)
+    else
     {
-       T lim = (4 * v * v + 10) / (8 * x);
-       lim *= lim;
-       lim *= lim;
-       lim /= 24;
-       if((lim < tools::epsilon<T>() * 10) && (x > 100))
+       n = iround(v, pol);
+       u = v - n;                              // -1/2 <= u < 1/2
+       BOOST_MATH_INSTRUMENT_VARIABLE(n);
+       BOOST_MATH_INSTRUMENT_VARIABLE(u);
+
+       BOOST_MATH_ASSERT(x > 0); // Error handling for x <= 0 handled in cyl_bessel_i and cyl_bessel_k
+
+       // x is positive until reflection
+       W = 1 / x;                                 // Wronskian
+       if (x <= 2)                                // x in (0, 2]
        {
-          // x is huge compared to v, CF1 may be very slow
-          // to converge so use asymptotic expansion for large
-          // x case instead.  Note that the asymptotic expansion
-          // isn't very accurate - so it's deliberately very hard
-          // to get here - probably we're going to overflow:
-          Iv = asymptotic_bessel_i_large_x(v, x, pol);
+          temme_ik(u, x, &Ku, &Ku1, pol);             // Temme series
        }
-       else if((v > 0) && (x / v < 0.25))
+       else                                       // x in (2, \infty)
        {
-          Iv = bessel_i_small_z_series(v, x, pol);
+          CF2_ik(u, x, &Ku, &Ku1, pol);               // continued fraction CF2_ik
        }
-       else
+       BOOST_MATH_INSTRUMENT_VARIABLE(Ku);
+       BOOST_MATH_INSTRUMENT_VARIABLE(Ku1);
+       prev = Ku;
+       current = Ku1;
+       for (k = 1; k <= n; k++)                   // forward recurrence for K
+       {
+          T fact = 2 * (u + k) / x;
+          // Check for overflow: if (max - |prev|) / fact > max, then overflow
+          // (max - |prev|) / fact > max
+          // max * (1 - fact) > |prev|
+          // if fact < 1: safe to compute overflow check
+          // if fact >= 1:  won't overflow
+          const bool will_overflow = (fact < 1)
+             ? tools::max_value<T>() * (1 - fact) > fabs(prev)
+             : false;
+          if (!will_overflow && ((tools::max_value<T>() - fabs(prev)) / fact < fabs(current)))
+          {
+             prev /= current;
+             scale /= current;
+             scale_sign *= ((boost::math::signbit)(current) ? -1 : 1);
+             current = 1;
+          }
+          next = fact * current + prev;
+          prev = current;
+          current = next;
+       }
+       Kv = prev;
+       Kv1 = current;
+       BOOST_MATH_INSTRUMENT_VARIABLE(Kv);
+       BOOST_MATH_INSTRUMENT_VARIABLE(Kv1);
+       if (kind & need_i)
        {
-          CF1_ik(v, x, &fv, pol);                         // continued fraction CF1_ik
-          Iv = scale * W / (Kv * fv + Kv1);                  // Wronskian relation
+          T lim = (4 * v * v + 10) / (8 * x);
+          lim *= lim;
+          lim *= lim;
+          lim /= 24;
+          if ((lim < tools::epsilon<T>() * 10) && (x > 100))
+          {
+             // x is huge compared to v, CF1 may be very slow
+             // to converge so use asymptotic expansion for large
+             // x case instead.  Note that the asymptotic expansion
+             // isn't very accurate - so it's deliberately very hard
+             // to get here - probably we're going to overflow:
+             Iv = asymptotic_bessel_i_large_x(v, x, pol);
+          }
+          else if ((v > 0) && (x / v < 0.25))
+          {
+             Iv = bessel_i_small_z_series(v, x, pol);
+          }
+          else
+          {
+             CF1_ik(v, x, &fv, pol);                         // continued fraction CF1_ik
+             Iv = scale * W / (Kv * fv + Kv1);                  // Wronskian relation
+          }
        }
+       else
+          Iv = std::numeric_limits<T>::quiet_NaN(); // any value will do
     }
-    else
-       Iv = std::numeric_limits<T>::quiet_NaN(); // any value will do
-
     if (reflect)
     {
         T z = (u + n % 2);
diff --git a/include/boost/math/special_functions/detail/bessel_jy_asym.hpp b/include/boost/math/special_functions/detail/bessel_jy_asym.hpp
index 16b6543f9..cb09b202d 100644
--- a/include/boost/math/special_functions/detail/bessel_jy_asym.hpp
+++ b/include/boost/math/special_functions/detail/bessel_jy_asym.hpp
@@ -69,6 +69,8 @@ inline T asymptotic_bessel_y_large_x_2(T v, T x, const Policy& pol)
    BOOST_MATH_STD_USING
    // Get the phase and amplitude:
    T ampl = asymptotic_bessel_amplitude(v, x);
+   if (0 == ampl)
+      return ampl;
    T phase = asymptotic_bessel_phase_mx(v, x);
    BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
    BOOST_MATH_INSTRUMENT_VARIABLE(phase);
@@ -97,6 +99,8 @@ inline T asymptotic_bessel_j_large_x_2(T v, T x, const Policy& pol)
    BOOST_MATH_STD_USING
    // Get the phase and amplitude:
    T ampl = asymptotic_bessel_amplitude(v, x);
+   if (0 == ampl) 
+      return ampl;  // shortcut.
    T phase = asymptotic_bessel_phase_mx(v, x);
    BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
    BOOST_MATH_INSTRUMENT_VARIABLE(phase);
diff --git a/include/boost/math/special_functions/ellint_1.hpp b/include/boost/math/special_functions/ellint_1.hpp
index 33d84c281..dfc1815f7 100644
--- a/include/boost/math/special_functions/ellint_1.hpp
+++ b/include/boost/math/special_functions/ellint_1.hpp
@@ -41,10 +41,12 @@ template <typename T, typename Policy>
 T ellint_k_imp(T k, const Policy& pol, std::integral_constant<int, 1> const&);
 template <typename T, typename Policy>
 T ellint_k_imp(T k, const Policy& pol, std::integral_constant<int, 2> const&);
+template <typename T, typename Policy>
+T ellint_k_imp(T k, const Policy& pol, T one_minus_k2);
 
 // Elliptic integral (Legendre form) of the first kind
 template <typename T, typename Policy>
-T ellint_f_imp(T phi, T k, const Policy& pol)
+T ellint_f_imp(T phi, T k, const Policy& pol, T one_minus_k2)
 {
     BOOST_MATH_STD_USING
     using namespace boost::math::tools;
@@ -75,12 +77,7 @@ T ellint_f_imp(T phi, T k, const Policy& pol)
     {
        // Phi is so large that phi%pi is necessarily zero (or garbage),
        // just return the second part of the duplication formula:
-       typedef std::integral_constant<int,
-          std::is_floating_point<T>::value&& std::numeric_limits<T>::digits && (std::numeric_limits<T>::digits <= 54) ? 0 :
-          std::is_floating_point<T>::value && std::numeric_limits<T>::digits && (std::numeric_limits<T>::digits <= 64) ? 1 : 2
-       > precision_tag_type;
-
-       result = 2 * phi * ellint_k_imp(k, pol, precision_tag_type()) / constants::pi<T>();
+       result = 2 * phi * ellint_k_imp(k, pol, one_minus_k2) / constants::pi<T>();
        BOOST_MATH_INSTRUMENT_VARIABLE(result);
     }
     else
@@ -121,31 +118,45 @@ T ellint_f_imp(T phi, T k, const Policy& pol)
           BOOST_MATH_ASSERT(rphi != 0); // precondition, can't be true if sin(rphi) != 0.
           //
           // Use http://dlmf.nist.gov/19.25#E5, note that
-          // c-1 simplifies to cot^2(rphi) which avoid cancellation:
+          // c-1 simplifies to cot^2(rphi) which avoids cancellation.
+          // Likewise c - k^2 is the same as (c - 1) + (1 - k^2).
           //
           T c = 1 / sinp;
-          result = static_cast<T>(s * ellint_rf_imp(T(cosp / sinp), T(c - k * k), c, pol));
+          T c_minus_one = cosp / sinp;
+          T arg2;
+          if (k != 0)
+          {
+             T cross = fabs(c / (k * k));
+             if ((cross > 0.9f) && (cross < 1.1f))
+                arg2 = c_minus_one + one_minus_k2;
+             else
+                arg2 = c - k * k;
+          }
+          else
+             arg2 = c;
+          result = static_cast<T>(s * ellint_rf_imp(c_minus_one, arg2, c, pol));
        }
        else
           result = s * sin(rphi);
        BOOST_MATH_INSTRUMENT_VARIABLE(result);
        if(m != 0)
        {
-          typedef std::integral_constant<int,
-             std::is_floating_point<T>::value&& std::numeric_limits<T>::digits && (std::numeric_limits<T>::digits <= 54) ? 0 :
-             std::is_floating_point<T>::value && std::numeric_limits<T>::digits && (std::numeric_limits<T>::digits <= 64) ? 1 : 2
-          > precision_tag_type;
-
-          result += m * ellint_k_imp(k, pol, precision_tag_type());
+          result += m * ellint_k_imp(k, pol, one_minus_k2);
           BOOST_MATH_INSTRUMENT_VARIABLE(result);
        }
     }
     return invert ? T(-result) : result;
 }
 
+template <typename T, typename Policy>
+inline T ellint_f_imp(T phi, T k, const Policy& pol)
+{
+   return ellint_f_imp(phi, k, pol, T(1 - k * k));
+}
+
 // Complete elliptic integral (Legendre form) of the first kind
 template <typename T, typename Policy>
-T ellint_k_imp(T k, const Policy& pol, std::integral_constant<int, 2> const&)
+T ellint_k_imp(T k, const Policy& pol, T one_minus_k2)
 {
     BOOST_MATH_STD_USING
     using namespace boost::math::tools;
@@ -162,12 +173,16 @@ T ellint_k_imp(T k, const Policy& pol, std::integral_constant<int, 2> const&)
     }
 
     T x = 0;
-    T y = 1 - k * k;
     T z = 1;
-    T value = ellint_rf_imp(x, y, z, pol);
+    T value = ellint_rf_imp(x, one_minus_k2, z, pol);
 
     return value;
 }
+template <typename T, typename Policy>
+inline T ellint_k_imp(T k, const Policy& pol, std::integral_constant<int, 2> const&)
+{
+   return ellint_k_imp(k, pol, T(1 - k * k));
+}
 
 //
 // Special versions for double and 80-bit long double precision,
diff --git a/include/boost/math/special_functions/heuman_lambda.hpp b/include/boost/math/special_functions/heuman_lambda.hpp
index c2d16d46e..0fbf4a980 100644
--- a/include/boost/math/special_functions/heuman_lambda.hpp
+++ b/include/boost/math/special_functions/heuman_lambda.hpp
@@ -63,8 +63,8 @@ T heuman_lambda_imp(T phi, T k, const Policy& pol)
           return policies::raise_domain_error<T>(function, "When 1-k^2 == 1 then phi must be < Pi/2, but got phi = %1%", phi, pol);
        }
        else
-          ratio = ellint_f_imp(phi, rkp, pol) / ellint_k_imp(rkp, pol, precision_tag_type());
-       result = ratio + ellint_k_imp(k, pol, precision_tag_type()) * jacobi_zeta_imp(phi, rkp, pol) / constants::half_pi<T>();
+          ratio = ellint_f_imp(phi, rkp, pol, k2) / ellint_k_imp(rkp, pol, k2);
+       result = ratio + ellint_k_imp(k, pol, precision_tag_type()) * jacobi_zeta_imp(phi, rkp, pol, k2) / constants::half_pi<T>();
     }
     return result;
 }
diff --git a/include/boost/math/special_functions/jacobi_zeta.hpp b/include/boost/math/special_functions/jacobi_zeta.hpp
index aca9ff6d4..c4ba7d23d 100644
--- a/include/boost/math/special_functions/jacobi_zeta.hpp
+++ b/include/boost/math/special_functions/jacobi_zeta.hpp
@@ -27,7 +27,7 @@ namespace detail{
 
 // Elliptic integral - Jacobi Zeta
 template <typename T, typename Policy>
-T jacobi_zeta_imp(T phi, T k, const Policy& pol)
+T jacobi_zeta_imp(T phi, T k, const Policy& pol, T kp)
 {
     BOOST_MATH_STD_USING
     using namespace boost::math::tools;
@@ -43,22 +43,22 @@ T jacobi_zeta_imp(T phi, T k, const Policy& pol)
     T result;
     T sinp = sin(phi);
     T cosp = cos(phi);
-    T s2 = sinp * sinp;
+    T c2 = cosp * cosp;
+    T one_minus_ks2 = kp + c2 - kp * c2;
     T k2 = k * k;
-    T kp = 1 - k2;
     if(k == 1)
        result = sinp * (boost::math::sign)(cosp);  // We get here by simplifying JacobiZeta[w, 1] in Mathematica, and the fact that 0 <= phi.
     else
     {
-       typedef std::integral_constant<int,
-          std::is_floating_point<T>::value&& std::numeric_limits<T>::digits && (std::numeric_limits<T>::digits <= 54) ? 0 :
-          std::is_floating_point<T>::value && std::numeric_limits<T>::digits && (std::numeric_limits<T>::digits <= 64) ? 1 : 2
-       > precision_tag_type;
-       result = k2 * sinp * cosp * sqrt(1 - k2 * s2) * ellint_rj_imp(T(0), kp, T(1), T(1 - k2 * s2), pol) / (3 * ellint_k_imp(k, pol, precision_tag_type()));
+       result = k2 * sinp * cosp * sqrt(one_minus_ks2) * ellint_rj_imp(T(0), kp, T(1), one_minus_ks2, pol) / (3 * ellint_k_imp(k, pol, kp));
     }
     return invert ? T(-result) : result;
 }
-
+template <typename T, typename Policy>
+inline T jacobi_zeta_imp(T phi, T k, const Policy& pol)
+{
+   return jacobi_zeta_imp(phi, k, pol, T(1 - k * k));
+}
 } // detail
 
 template <class T1, class T2, class Policy>
diff --git a/include/boost/math/special_functions/next.hpp b/include/boost/math/special_functions/next.hpp
index c696b97b5..02a208e4e 100644
--- a/include/boost/math/special_functions/next.hpp
+++ b/include/boost/math/special_functions/next.hpp
@@ -194,10 +194,14 @@ T float_next_imp(const T& val, const std::true_type&, const Policy& pol)
 
    int fpclass = (boost::math::fpclassify)(val);
 
-   if((fpclass == (int)FP_NAN) || (fpclass == (int)FP_INFINITE))
+   if (fpclass == (int)FP_INFINITE)
    {
-      if(val < 0)
+      if (val < 0)
          return -tools::max_value<T>();
+      return val;  // +INF
+   }
+   else if (fpclass == (int)FP_NAN)
+   {
       return policies::raise_domain_error<T>(
          function,
          "Argument must be finite, but got %1%", val, pol);
@@ -243,10 +247,14 @@ T float_next_imp(const T& val, const std::false_type&, const Policy& pol)
 
    int fpclass = (boost::math::fpclassify)(val);
 
-   if((fpclass == (int)FP_NAN) || (fpclass == (int)FP_INFINITE))
+   if (fpclass == (int)FP_INFINITE)
    {
-      if(val < 0)
+      if (val < 0)
          return -tools::max_value<T>();
+      return val;  // +INF
+   }
+   else if (fpclass == (int)FP_NAN)
+   {
       return policies::raise_domain_error<T>(
          function,
          "Argument must be finite, but got %1%", val, pol);
@@ -328,10 +336,14 @@ T float_prior_imp(const T& val, const std::true_type&, const Policy& pol)
 
    int fpclass = (boost::math::fpclassify)(val);
 
-   if((fpclass == (int)FP_NAN) || (fpclass == (int)FP_INFINITE))
+   if (fpclass == (int)FP_INFINITE)
    {
-      if(val > 0)
+      if (val > 0)
          return tools::max_value<T>();
+      return val; // -INF
+   }
+   else if (fpclass == (int)FP_NAN)
+   {
       return policies::raise_domain_error<T>(
          function,
          "Argument must be finite, but got %1%", val, pol);
@@ -378,10 +390,14 @@ T float_prior_imp(const T& val, const std::false_type&, const Policy& pol)
 
    int fpclass = (boost::math::fpclassify)(val);
 
-   if((fpclass == (int)FP_NAN) || (fpclass == (int)FP_INFINITE))
+   if (fpclass == (int)FP_INFINITE)
    {
-      if(val > 0)
+      if (val > 0)
          return tools::max_value<T>();
+      return val; // -INF
+   }
+   else if (fpclass == (int)FP_NAN)
+   {
       return policies::raise_domain_error<T>(
          function,
          "Argument must be finite, but got %1%", val, pol);
diff --git a/include/boost/math/special_functions/sinc.hpp b/include/boost/math/special_functions/sinc.hpp
index 7220d30b8..ff1b2e966 100644
--- a/include/boost/math/special_functions/sinc.hpp
+++ b/include/boost/math/special_functions/sinc.hpp
@@ -19,6 +19,7 @@
 #include <boost/math/tools/precision.hpp>
 #include <boost/math/policies/policy.hpp>
 #include <boost/math/special_functions/math_fwd.hpp>
+#include <boost/math/special_functions/fpclassify.hpp>
 #include <limits>
 #include <string>
 #include <stdexcept>
@@ -39,7 +40,11 @@ namespace boost
         {
             BOOST_MATH_STD_USING
 
-            if    (abs(x) >= 3.3 * tools::forth_root_epsilon<T>())
+            if ((boost::math::isinf)(x))
+            {
+               return 0;
+            }
+            else if (abs(x) >= 3.3 * tools::forth_root_epsilon<T>())
             {
                 return(sin(x)/x);
             }
diff --git a/include/boost/math/special_functions/sinhc.hpp b/include/boost/math/special_functions/sinhc.hpp
index 6606ad3ea..b8c5f9f2e 100644
--- a/include/boost/math/special_functions/sinhc.hpp
+++ b/include/boost/math/special_functions/sinhc.hpp
@@ -16,7 +16,9 @@
 #endif
 
 #include <boost/math/tools/precision.hpp>
+#include <boost/math/policies/error_handling.hpp>
 #include <boost/math/special_functions/math_fwd.hpp>
+#include <boost/math/special_functions/fpclassify.hpp>
 #include <limits>
 #include <string>
 #include <stdexcept>
@@ -32,8 +34,8 @@ namespace boost
        {
         // This is the "Hyperbolic Sinus Cardinal" of index Pi.
 
-        template<typename T>
-        inline T    sinhc_pi_imp(const T x)
+        template<typename T, typename Policy>
+        inline T    sinhc_pi_imp(const T x, const Policy&)
         {
             using    ::std::abs;
             using    ::std::sinh;
@@ -43,6 +45,10 @@ namespace boost
             static T const    taylor_2_bound = sqrt(taylor_0_bound);
             static T const    taylor_n_bound = sqrt(taylor_2_bound);
 
+            if((boost::math::isinf)(x))
+            {
+               return policies::raise_overflow_error<T>("sinhc(%1%)", nullptr, Policy());
+            }
             if    (abs(x) >= taylor_n_bound)
             {
                 return(sinh(x)/x);
@@ -72,17 +78,18 @@ namespace boost
 
        } // namespace detail
 
-       template <class T>
-       inline typename tools::promote_args<T>::type sinhc_pi(T x)
+       template <class T, class Policy>
+       inline typename tools::promote_args<T>::type sinhc_pi(T x, const Policy& pol)
        {
           typedef typename tools::promote_args<T>::type result_type;
-          return detail::sinhc_pi_imp(static_cast<result_type>(x));
+          return policies::checked_narrowing_cast<T, Policy>(detail::sinhc_pi_imp(static_cast<result_type>(x), pol), "sinhc(%1%)");
        }
 
-       template <class T, class Policy>
-       inline typename tools::promote_args<T>::type sinhc_pi(T x, const Policy&)
+       template <class T>
+       inline typename tools::promote_args<T>::type sinhc_pi(T x)
        {
-          return boost::math::sinhc_pi(x);
+          typedef typename tools::promote_args<T>::type result_type;
+          return sinhc_pi(static_cast<result_type>(x), policies::policy<>());
        }
 
         template<typename T, template<typename> class U>
diff --git a/include/boost/math/tools/config.hpp b/include/boost/math/tools/config.hpp
index 9e4b7d7ca..6d962a08a 100644
--- a/include/boost/math/tools/config.hpp
+++ b/include/boost/math/tools/config.hpp
@@ -365,42 +365,62 @@ struct non_type {};
 #endif
 
 //
-// Tune performance options for specific compilers:
+// Tune performance options for specific compilers,
+// but check at each step that nothing has been previously defined by the user first
 //
 #ifdef _MSC_VER
-#  define BOOST_MATH_POLY_METHOD 2
+#  ifndef BOOST_MATH_POLY_METHOD
+#    define BOOST_MATH_POLY_METHOD 2
+#  endif
 #if _MSC_VER <= 1900
-#  define BOOST_MATH_RATIONAL_METHOD 1
+#  ifndef BOOST_MATH_POLY_METHOD
+#    define BOOST_MATH_RATIONAL_METHOD 1
+#  endif
 #else
-#  define BOOST_MATH_RATIONAL_METHOD 2
+#  ifndef BOOST_MATH_RATIONAL_METHOD
+#    define BOOST_MATH_RATIONAL_METHOD 2
+#  endif
 #endif
 #if _MSC_VER > 1900
-#  define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
-#  define BOOST_MATH_INT_VALUE_SUFFIX(RV, SUF) RV##.0L
+#  ifndef BOOST_MATH_INT_TABLE_TYPE
+#    define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
+#  endif
+#  ifndef BOOST_MATH_INT_VALUE_SUFFIX
+#    define BOOST_MATH_INT_VALUE_SUFFIX(RV, SUF) RV##.0L
+#  endif
 #endif
 
 #elif defined(__INTEL_COMPILER)
-#  define BOOST_MATH_POLY_METHOD 2
-#  define BOOST_MATH_RATIONAL_METHOD 1
+#  ifndef BOOST_MATH_POLY_METHOD
+#    define BOOST_MATH_POLY_METHOD 2
+#  endif
+#  ifndef BOOST_MATH_RATIONAL_METHOD
+#    define BOOST_MATH_RATIONAL_METHOD 1
+#  endif
 
 #elif defined(__GNUC__)
-#if __GNUC__ < 4
-#  define BOOST_MATH_POLY_METHOD 3
-#  define BOOST_MATH_RATIONAL_METHOD 3
-#  define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
-#  define BOOST_MATH_INT_VALUE_SUFFIX(RV, SUF) RV##.0L
-#else
-#  define BOOST_MATH_POLY_METHOD 3
-#  define BOOST_MATH_RATIONAL_METHOD 3
-#endif
+#  ifndef BOOST_MATH_POLY_METHOD
+#    define BOOST_MATH_POLY_METHOD 3
+#  endif
+#  ifndef BOOST_MATH_RATIONAL_METHOD
+#    define BOOST_MATH_RATIONAL_METHOD 3
+#  endif
 
 #elif defined(__clang__)
 
 #if __clang__ > 6
-#  define BOOST_MATH_POLY_METHOD 3
-#  define BOOST_MATH_RATIONAL_METHOD 3
-#  define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
-#  define BOOST_MATH_INT_VALUE_SUFFIX(RV, SUF) RV##.0L
+#  ifndef BOOST_MATH_POLY_METHOD
+#    define BOOST_MATH_POLY_METHOD 3
+#  endif
+#  ifndef BOOST_MATH_RATIONAL_METHOD
+#    define BOOST_MATH_RATIONAL_METHOD 3
+#  endif
+#  ifndef BOOST_MATH_INT_TABLE_TYPE
+#    define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
+#  endif
+#  ifndef BOOST_MATH_INT_VALUE_SUFFIX
+#    define BOOST_MATH_INT_VALUE_SUFFIX(RV, SUF) RV##.0L
+#  endif
 #endif
 
 #endif
diff --git a/include/boost/math/tools/detail/rational_horner1_10.hpp b/include/boost/math/tools/detail/rational_horner1_10.hpp
index dd23e0d7b..6a04128ca 100644
--- a/include/boost/math/tools/detail/rational_horner1_10.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_10.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_11.hpp b/include/boost/math/tools/detail/rational_horner1_11.hpp
index 7ca35d39f..d43e53433 100644
--- a/include/boost/math/tools/detail/rational_horner1_11.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_11.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_12.hpp b/include/boost/math/tools/detail/rational_horner1_12.hpp
index 5ee313852..33d19eb38 100644
--- a/include/boost/math/tools/detail/rational_horner1_12.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_12.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_13.hpp b/include/boost/math/tools/detail/rational_horner1_13.hpp
index 6e675777a..2069aa515 100644
--- a/include/boost/math/tools/detail/rational_horner1_13.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_13.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_14.hpp b/include/boost/math/tools/detail/rational_horner1_14.hpp
index b8e5f7b2e..5ebcde626 100644
--- a/include/boost/math/tools/detail/rational_horner1_14.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_14.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -170,7 +170,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((a[13] * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((b[13] * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_15.hpp b/include/boost/math/tools/detail/rational_horner1_15.hpp
index 801f4439b..9da8e1b71 100644
--- a/include/boost/math/tools/detail/rational_horner1_15.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_15.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -170,7 +170,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((a[13] * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((b[13] * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((a[14] * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((b[14] * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_16.hpp b/include/boost/math/tools/detail/rational_horner1_16.hpp
index b0f8cbb03..203ba7819 100644
--- a/include/boost/math/tools/detail/rational_horner1_16.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_16.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -170,7 +170,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((a[13] * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((b[13] * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((a[14] * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((b[14] * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -194,7 +194,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((a[15] * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((b[15] * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_17.hpp b/include/boost/math/tools/detail/rational_horner1_17.hpp
index 32759b831..e382d2931 100644
--- a/include/boost/math/tools/detail/rational_horner1_17.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_17.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -170,7 +170,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((a[13] * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((b[13] * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((a[14] * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((b[14] * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -194,7 +194,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((a[15] * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((b[15] * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -206,7 +206,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((((a[16] * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((((b[16] * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_18.hpp b/include/boost/math/tools/detail/rational_horner1_18.hpp
index 6c91a930d..66f668ee3 100644
--- a/include/boost/math/tools/detail/rational_horner1_18.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_18.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -170,7 +170,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((a[13] * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((b[13] * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((a[14] * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((b[14] * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -194,7 +194,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((a[15] * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((b[15] * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -206,7 +206,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((((a[16] * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((((b[16] * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -218,7 +218,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((((a[17] * x + a[16]) * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((((b[17] * x + b[16]) * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_19.hpp b/include/boost/math/tools/detail/rational_horner1_19.hpp
index 8c9e08315..9cd139143 100644
--- a/include/boost/math/tools/detail/rational_horner1_19.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_19.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -170,7 +170,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((a[13] * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((b[13] * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((a[14] * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((b[14] * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -194,7 +194,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((a[15] * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((b[15] * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -206,7 +206,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((((a[16] * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((((b[16] * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -218,7 +218,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((((a[17] * x + a[16]) * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((((b[17] * x + b[16]) * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -230,7 +230,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 19>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((((((a[18] * x + a[17]) * x + a[16]) * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((((((b[18] * x + b[17]) * x + b[16]) * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_2.hpp b/include/boost/math/tools/detail/rational_horner1_2.hpp
index 605bb0424..f42cbfc64 100644
--- a/include/boost/math/tools/detail/rational_horner1_2.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_2.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_20.hpp b/include/boost/math/tools/detail/rational_horner1_20.hpp
index 2c29d6ba8..0a6c2a0f2 100644
--- a/include/boost/math/tools/detail/rational_horner1_20.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_20.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -122,7 +122,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((a[9] * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((b[9] * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -134,7 +134,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((a[10] * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((b[10] * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -146,7 +146,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((a[11] * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((b[11] * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -158,7 +158,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((a[12] * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((b[12] * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -170,7 +170,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((a[13] * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((b[13] * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((a[14] * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((b[14] * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -194,7 +194,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((a[15] * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((b[15] * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -206,7 +206,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((((a[16] * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((((b[16] * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -218,7 +218,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((((a[17] * x + a[16]) * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((((b[17] * x + b[16]) * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -230,7 +230,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 19>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((((((((((((a[18] * x + a[17]) * x + a[16]) * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((((((((((((b[18] * x + b[17]) * x + b[16]) * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -242,7 +242,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 20>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((((((((((((((a[19] * x + a[18]) * x + a[17]) * x + a[16]) * x + a[15]) * x + a[14]) * x + a[13]) * x + a[12]) * x + a[11]) * x + a[10]) * x + a[9]) * x + a[8]) * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((((((((((((((b[19] * x + b[18]) * x + b[17]) * x + b[16]) * x + b[15]) * x + b[14]) * x + b[13]) * x + b[12]) * x + b[11]) * x + b[10]) * x + b[9]) * x + b[8]) * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_3.hpp b/include/boost/math/tools/detail/rational_horner1_3.hpp
index 3f16fdf1d..d0ab213b3 100644
--- a/include/boost/math/tools/detail/rational_horner1_3.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_3.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_4.hpp b/include/boost/math/tools/detail/rational_horner1_4.hpp
index 778a692c8..44f40114a 100644
--- a/include/boost/math/tools/detail/rational_horner1_4.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_4.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_5.hpp b/include/boost/math/tools/detail/rational_horner1_5.hpp
index 0ae61fca0..db032f15e 100644
--- a/include/boost/math/tools/detail/rational_horner1_5.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_5.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_6.hpp b/include/boost/math/tools/detail/rational_horner1_6.hpp
index 9e2501696..4de5143ca 100644
--- a/include/boost/math/tools/detail/rational_horner1_6.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_6.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_7.hpp b/include/boost/math/tools/detail/rational_horner1_7.hpp
index 7dc78d1f9..7d4ef69e9 100644
--- a/include/boost/math/tools/detail/rational_horner1_7.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_7.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_8.hpp b/include/boost/math/tools/detail/rational_horner1_8.hpp
index 8f5df2c22..bf4d7f57e 100644
--- a/include/boost/math/tools/detail/rational_horner1_8.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_8.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner1_9.hpp b/include/boost/math/tools/detail/rational_horner1_9.hpp
index a46c9d7aa..cf3be7f82 100644
--- a/include/boost/math/tools/detail/rational_horner1_9.hpp
+++ b/include/boost/math/tools/detail/rational_horner1_9.hpp
@@ -26,7 +26,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V&, const std::in
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 2>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((a[1] * x + a[0]) / (b[1] * x + b[0]));
    else
    {
@@ -38,7 +38,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 3>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((a[2] * x + a[1]) * x + a[0]) / ((b[2] * x + b[1]) * x + b[0]));
    else
    {
@@ -50,7 +50,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 4>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((a[3] * x + a[2]) * x + a[1]) * x + a[0]) / (((b[3] * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -62,7 +62,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((a[4] * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((b[4] * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -74,7 +74,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((a[5] * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((b[5] * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -86,7 +86,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((a[6] * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((b[6] * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -98,7 +98,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>((((((((a[7] * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / (((((((b[7] * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
@@ -110,7 +110,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
      return static_cast<V>(((((((((a[8] * x + a[7]) * x + a[6]) * x + a[5]) * x + a[4]) * x + a[3]) * x + a[2]) * x + a[1]) * x + a[0]) / ((((((((b[8] * x + b[7]) * x + b[6]) * x + b[5]) * x + b[4]) * x + b[3]) * x + b[2]) * x + b[1]) * x + b[0]));
    else
    {
diff --git a/include/boost/math/tools/detail/rational_horner2_10.hpp b/include/boost/math/tools/detail/rational_horner2_10.hpp
index f3ff07535..1a59aa334 100644
--- a/include/boost/math/tools/detail/rational_horner2_10.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_10.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_11.hpp b/include/boost/math/tools/detail/rational_horner2_11.hpp
index 250b4ee29..1333a40bc 100644
--- a/include/boost/math/tools/detail/rational_horner2_11.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_11.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner2_12.hpp b/include/boost/math/tools/detail/rational_horner2_12.hpp
index df2b2771f..a37cf5a05 100644
--- a/include/boost/math/tools/detail/rational_horner2_12.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_12.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_13.hpp b/include/boost/math/tools/detail/rational_horner2_13.hpp
index 2bbc97e4d..648f3079c 100644
--- a/include/boost/math/tools/detail/rational_horner2_13.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_13.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner2_14.hpp b/include/boost/math/tools/detail/rational_horner2_14.hpp
index 157a844ce..7771c3da9 100644
--- a/include/boost/math/tools/detail/rational_horner2_14.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_14.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -188,7 +188,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_15.hpp b/include/boost/math/tools/detail/rational_horner2_15.hpp
index 9b392bef2..03fae0d94 100644
--- a/include/boost/math/tools/detail/rational_horner2_15.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_15.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -188,7 +188,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -204,7 +204,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner2_16.hpp b/include/boost/math/tools/detail/rational_horner2_16.hpp
index d884131ad..d8565e104 100644
--- a/include/boost/math/tools/detail/rational_horner2_16.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_16.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -188,7 +188,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -204,7 +204,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -220,7 +220,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_17.hpp b/include/boost/math/tools/detail/rational_horner2_17.hpp
index 87efe07f8..bd8990e0c 100644
--- a/include/boost/math/tools/detail/rational_horner2_17.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_17.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -188,7 +188,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -204,7 +204,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -220,7 +220,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -236,7 +236,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[16] * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((((b[16] * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner2_18.hpp b/include/boost/math/tools/detail/rational_horner2_18.hpp
index 6989c6dba..38b99ecf1 100644
--- a/include/boost/math/tools/detail/rational_horner2_18.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_18.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -188,7 +188,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -204,7 +204,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -220,7 +220,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -236,7 +236,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[16] * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((((b[16] * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -252,7 +252,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((((a[17] * x2 + a[15]) * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((((a[16] * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((((b[17] * x2 + b[15]) * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((((b[16] * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_19.hpp b/include/boost/math/tools/detail/rational_horner2_19.hpp
index 11d382a5c..b77d2eb0b 100644
--- a/include/boost/math/tools/detail/rational_horner2_19.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_19.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -188,7 +188,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -204,7 +204,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -220,7 +220,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -236,7 +236,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[16] * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((((b[16] * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -252,7 +252,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((((a[17] * x2 + a[15]) * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((((a[16] * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((((b[17] * x2 + b[15]) * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((((b[16] * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -268,7 +268,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 19>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((((a[18] * x2 + a[16]) * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((((a[17] * x2 + a[15]) * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((((b[18] * x2 + b[16]) * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((((b[17] * x2 + b[15]) * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner2_20.hpp b/include/boost/math/tools/detail/rational_horner2_20.hpp
index 469cbb7ee..485639dce 100644
--- a/include/boost/math/tools/detail/rational_horner2_20.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_20.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -124,7 +124,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -140,7 +140,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((a[9] * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((b[9] * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -156,7 +156,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((a[10] * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((b[10] * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -172,7 +172,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((a[11] * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((b[11] * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -188,7 +188,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((a[12] * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((b[12] * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -204,7 +204,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((a[13] * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((b[13] * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -220,7 +220,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((((a[14] * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((((b[14] * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -236,7 +236,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((a[16] * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((((((a[15] * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((((((b[16] * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((((((b[15] * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -252,7 +252,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((((a[17] * x2 + a[15]) * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + (((((((a[16] * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / (((((((((b[17] * x2 + b[15]) * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + (((((((b[16] * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -268,7 +268,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 19>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((((((((a[18] * x2 + a[16]) * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((((((((a[17] * x2 + a[15]) * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / (((((((((b[18] * x2 + b[16]) * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((((((((b[17] * x2 + b[15]) * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
@@ -284,7 +284,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 20>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((((((((a[19] * x2 + a[17]) * x2 + a[15]) * x2 + a[13]) * x2 + a[11]) * x2 + a[9]) * x2 + a[7]) * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((((((((a[18] * x2 + a[16]) * x2 + a[14]) * x2 + a[12]) * x2 + a[10]) * x2 + a[8]) * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((((((((b[19] * x2 + b[17]) * x2 + b[15]) * x2 + b[13]) * x2 + b[11]) * x2 + b[9]) * x2 + b[7]) * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((((((((b[18] * x2 + b[16]) * x2 + b[14]) * x2 + b[12]) * x2 + b[10]) * x2 + b[8]) * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_5.hpp b/include/boost/math/tools/detail/rational_horner2_5.hpp
index e1cf56671..8633d5dc1 100644
--- a/include/boost/math/tools/detail/rational_horner2_5.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_5.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner2_6.hpp b/include/boost/math/tools/detail/rational_horner2_6.hpp
index 480fcb471..455542633 100644
--- a/include/boost/math/tools/detail/rational_horner2_6.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_6.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_7.hpp b/include/boost/math/tools/detail/rational_horner2_7.hpp
index 9ae76bd3c..6a5c704d1 100644
--- a/include/boost/math/tools/detail/rational_horner2_7.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_7.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner2_8.hpp b/include/boost/math/tools/detail/rational_horner2_8.hpp
index 2f82d3653..9ec861fc5 100644
--- a/include/boost/math/tools/detail/rational_horner2_8.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_8.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
diff --git a/include/boost/math/tools/detail/rational_horner2_9.hpp b/include/boost/math/tools/detail/rational_horner2_9.hpp
index e4b222430..c76755cb2 100644
--- a/include/boost/math/tools/detail/rational_horner2_9.hpp
+++ b/include/boost/math/tools/detail/rational_horner2_9.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((a[4] * x2 + a[2]) * x2 + a[0] + (a[3] * x2 + a[1]) * x) / ((b[4] * x2 + b[2]) * x2 + b[0] + (b[3] * x2 + b[1]) * x));
@@ -60,7 +60,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[5] * x2 + a[3]) * x2 + a[1]) * x + (a[4] * x2 + a[2]) * x2 + a[0]) / (((b[5] * x2 + b[3]) * x2 + b[1]) * x + (b[4] * x2 + b[2]) * x2 + b[0]));
@@ -76,7 +76,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>((((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + ((a[5] * x2 + a[3]) * x2 + a[1]) * x) / (((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + ((b[5] * x2 + b[3]) * x2 + b[1]) * x));
@@ -92,7 +92,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x + ((a[6] * x2 + a[4]) * x2 + a[2]) * x2 + a[0]) / ((((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x + ((b[6] * x2 + b[4]) * x2 + b[2]) * x2 + b[0]));
@@ -108,7 +108,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       return static_cast<V>(((((a[8] * x2 + a[6]) * x2 + a[4]) * x2 + a[2]) * x2 + a[0] + (((a[7] * x2 + a[5]) * x2 + a[3]) * x2 + a[1]) * x) / ((((b[8] * x2 + b[6]) * x2 + b[4]) * x2 + b[2]) * x2 + b[0] + (((b[7] * x2 + b[5]) * x2 + b[3]) * x2 + b[1]) * x));
diff --git a/include/boost/math/tools/detail/rational_horner3_10.hpp b/include/boost/math/tools/detail/rational_horner3_10.hpp
index 9b3bdbe92..773532cd5 100644
--- a/include/boost/math/tools/detail/rational_horner3_10.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_10.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_11.hpp b/include/boost/math/tools/detail/rational_horner3_11.hpp
index dd00d58cf..a712fff09 100644
--- a/include/boost/math/tools/detail/rational_horner3_11.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_11.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_12.hpp b/include/boost/math/tools/detail/rational_horner3_12.hpp
index bfc53420e..5b87374ab 100644
--- a/include/boost/math/tools/detail/rational_horner3_12.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_12.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_13.hpp b/include/boost/math/tools/detail/rational_horner3_13.hpp
index c1237d587..11591668b 100644
--- a/include/boost/math/tools/detail/rational_horner3_13.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_13.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_14.hpp b/include/boost/math/tools/detail/rational_horner3_14.hpp
index 030fa4ec7..04f31249d 100644
--- a/include/boost/math/tools/detail/rational_horner3_14.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_14.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -674,7 +674,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_15.hpp b/include/boost/math/tools/detail/rational_horner3_15.hpp
index 925267548..4b9cffd48 100644
--- a/include/boost/math/tools/detail/rational_horner3_15.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_15.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -674,7 +674,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -784,7 +784,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_16.hpp b/include/boost/math/tools/detail/rational_horner3_16.hpp
index 5bbf67d90..3a384dcc5 100644
--- a/include/boost/math/tools/detail/rational_horner3_16.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_16.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -674,7 +674,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -784,7 +784,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -902,7 +902,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_17.hpp b/include/boost/math/tools/detail/rational_horner3_17.hpp
index 1f1e2d736..1c9435e74 100644
--- a/include/boost/math/tools/detail/rational_horner3_17.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_17.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -674,7 +674,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -784,7 +784,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -902,7 +902,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1028,7 +1028,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_18.hpp b/include/boost/math/tools/detail/rational_horner3_18.hpp
index 6ce9844bf..b133e2baf 100644
--- a/include/boost/math/tools/detail/rational_horner3_18.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_18.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -674,7 +674,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -784,7 +784,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -902,7 +902,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1028,7 +1028,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1162,7 +1162,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_19.hpp b/include/boost/math/tools/detail/rational_horner3_19.hpp
index b10844767..ca35d3b68 100644
--- a/include/boost/math/tools/detail/rational_horner3_19.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_19.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -674,7 +674,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -784,7 +784,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -902,7 +902,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1028,7 +1028,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1162,7 +1162,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1304,7 +1304,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 19>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_20.hpp b/include/boost/math/tools/detail/rational_horner3_20.hpp
index e125934a1..58109ac30 100644
--- a/include/boost/math/tools/detail/rational_horner3_20.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_20.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -314,7 +314,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 10>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -392,7 +392,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 11>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -478,7 +478,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 12>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -572,7 +572,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 13>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -674,7 +674,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 14>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -784,7 +784,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 15>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -902,7 +902,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 16>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1028,7 +1028,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 17>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1162,7 +1162,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 18>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1304,7 +1304,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 19>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -1454,7 +1454,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 20>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_5.hpp b/include/boost/math/tools/detail/rational_horner3_5.hpp
index 0150b143f..cc77fd560 100644
--- a/include/boost/math/tools/detail/rational_horner3_5.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_5.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_6.hpp b/include/boost/math/tools/detail/rational_horner3_6.hpp
index c4a2c14f5..73920ad01 100644
--- a/include/boost/math/tools/detail/rational_horner3_6.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_6.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_7.hpp b/include/boost/math/tools/detail/rational_horner3_7.hpp
index 43555ab1d..8e30ecf31 100644
--- a/include/boost/math/tools/detail/rational_horner3_7.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_7.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_8.hpp b/include/boost/math/tools/detail/rational_horner3_8.hpp
index 1c14fb461..a8f93f3a3 100644
--- a/include/boost/math/tools/detail/rational_horner3_8.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_8.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/include/boost/math/tools/detail/rational_horner3_9.hpp b/include/boost/math/tools/detail/rational_horner3_9.hpp
index 057035237..064d984d3 100644
--- a/include/boost/math/tools/detail/rational_horner3_9.hpp
+++ b/include/boost/math/tools/detail/rational_horner3_9.hpp
@@ -44,7 +44,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 5>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -82,7 +82,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 6>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -128,7 +128,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 7>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -182,7 +182,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 8>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
@@ -244,7 +244,7 @@ inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::
 template <class T, class U, class V>
 inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, 9>*) BOOST_MATH_NOEXCEPT(V)
 {
-   if(x <= 1)
+   if((-1 <= x) && (x <= 1))
    {
       V x2 = x * x;
       V t[4];
diff --git a/reporting/performance/Jamfile.v2 b/reporting/performance/Jamfile.v2
index 6ea5bc8a2..1f1d6fd63 100644
--- a/reporting/performance/Jamfile.v2
+++ b/reporting/performance/Jamfile.v2
@@ -16,7 +16,7 @@ import ../../../config/checks/config : requires ;
 using quickbook ;
 using auto-index ;
 
-import ../../../predef/check/predef
+import ../../../predef/tools/check/predef
     : check require
     : predef-check predef-require ;
 
diff --git a/test/Jamfile.v2 b/test/Jamfile.v2
index cce2a8fcf..c7eaa3b1e 100644
--- a/test/Jamfile.v2
+++ b/test/Jamfile.v2
@@ -54,6 +54,7 @@ project
       <toolset>msvc-7.1:<source>../vc71_fix//vc_fix
       <toolset>msvc-7.1:<pch>off
       <toolset>clang-6.0.0:<pch>off  # added to see effect.
+      <toolset>clang:<cxxflags>-Wno-literal-range  # warning: magnitude of floating-point constant too small for type 'long double' [-Wliteral-range]
       <toolset>gcc,<target-os>windows:<pch>off
       <toolset>borland:<runtime-link>static
       # <toolset>msvc:<cxxflags>/wd4506 has no effect?
@@ -171,6 +172,8 @@ test-suite special_fun :
    [ run git_issue_961.cpp ]
    [ run git_issue_1006.cpp ]
    [ run git_issue_184.cpp ]
+   [ run git_issue_1137.cpp ]
+   [ run git_issue_1139.cpp ]
    [ run special_functions_test.cpp ../../test/build//boost_unit_test_framework  ]
    [ run test_airy.cpp test_instances//test_instances pch_light ../../test/build//boost_unit_test_framework  ]
    [ run test_bessel_j.cpp test_instances//test_instances pch_light ../../test/build//boost_unit_test_framework  ]
@@ -873,6 +876,7 @@ test-suite distribution_tests :
    [ run scipy_issue_18302.cpp ../../test/build//boost_unit_test_framework  ]
    [ run scipy_issue_18511.cpp ../../test/build//boost_unit_test_framework ]
    [ compile scipy_issue_19762.cpp ]
+   [ run git_issue_1120.cpp ]
 ;
 
 test-suite new_floats : 
@@ -885,7 +889,7 @@ test-suite new_floats :
 
 test-suite mp :
 
-   [ run test_nc_t_quad.cpp  pch ../../test/build//boost_unit_test_framework : : : release [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] ]
+   [ run test_nc_t_quad.cpp  pch ../../test/build//boost_unit_test_framework : : : release <toolset>gcc-mingw:<cxxflags>-Wa,-mbig-obj <debug-symbols>off [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : <linkflags>-lquadmath ] ]
    [ run test_polynomial.cpp ../../test/build//boost_unit_test_framework : : : <define>TEST1 : test_polynomial_1  ]
    [ run test_polynomial.cpp ../../test/build//boost_unit_test_framework : : : <define>TEST2 : test_polynomial_2  ]
    [ run test_polynomial.cpp ../../test/build//boost_unit_test_framework : : : <define>TEST3 : test_polynomial_3  ]
diff --git a/test/bezier_polynomial_test.cpp b/test/bezier_polynomial_test.cpp
index 05ee13ecf..4aadd3edc 100644
--- a/test/bezier_polynomial_test.cpp
+++ b/test/bezier_polynomial_test.cpp
@@ -173,9 +173,9 @@ void test_linear_precision()
         P[2] = (1-t)*P0[2] + t*Pf[2];
 
         auto computed = bp(t);
-        CHECK_ULP_CLOSE(P[0], computed[0], 3);
-        CHECK_ULP_CLOSE(P[1], computed[1], 3);
-        CHECK_ULP_CLOSE(P[2], computed[2], 3);
+        CHECK_ULP_CLOSE(P[0], computed[0], 4);
+        CHECK_ULP_CLOSE(P[1], computed[1], 4);
+        CHECK_ULP_CLOSE(P[2], computed[2], 4);
 
         std::array<Real, 3> dP;
         dP[0] = Pf[0] - P0[0];
diff --git a/test/cubic_roots_test.cpp b/test/cubic_roots_test.cpp
index 04e8fb807..c4e903129 100644
--- a/test/cubic_roots_test.cpp
+++ b/test/cubic_roots_test.cpp
@@ -126,23 +126,33 @@ void test_ill_conditioned() {
     auto roots = cubic_roots<double>(1, 10000, 200, 1);
     CHECK_ABSOLUTE_ERROR(expected_roots[0], roots[0],
                          std::numeric_limits<double>::epsilon());
-    CHECK_ABSOLUTE_ERROR(expected_roots[1], roots[1], 1.01e-5);
-    CHECK_ABSOLUTE_ERROR(expected_roots[2], roots[2], 1.01e-5);
-    double cond =
-        cubic_root_condition_number<double>(1, 10000, 200, 1, roots[1]);
-    double r1 = expected_roots[1];
-    // The factor of 10 is a fudge factor to make the test pass.
-    // Nonetheless, it does show this is basically correct:
-    CHECK_LE(abs(r1 - roots[1]) / abs(r1),
-             10 * std::numeric_limits<double>::epsilon() * cond);
-
-    cond = cubic_root_condition_number<double>(1, 10000, 200, 1, roots[2]);
-    double r2 = expected_roots[2];
-    // The factor of 10 is a fudge factor to make the test pass.
-    // Nonetheless, it does show this is basically correct:
-    CHECK_LE(abs(r2 - roots[2]) / abs(r2),
-             10 * std::numeric_limits<double>::epsilon() * cond);
 
+    if (!(boost::math::isnan)(roots[1]))
+    {
+       // This test is so ill-conditioned, that we can't always get here.
+       // Test case is Clang C++20 mode on MacOS Arm.  Best guess is that
+       // fma is behaving differently there...
+       CHECK_ABSOLUTE_ERROR(expected_roots[1], roots[1], 1.01e-5);
+       CHECK_ABSOLUTE_ERROR(expected_roots[2], roots[2], 1.01e-5);
+       double cond =
+          cubic_root_condition_number<double>(1, 10000, 200, 1, roots[1]);
+       double r1 = expected_roots[1];
+       // The factor of 10 is a fudge factor to make the test pass.
+       // Nonetheless, it does show this is basically correct:
+       CHECK_LE(abs(r1 - roots[1]) / abs(r1),
+          10 * std::numeric_limits<double>::epsilon() * cond);
+
+       cond = cubic_root_condition_number<double>(1, 10000, 200, 1, roots[2]);
+       double r2 = expected_roots[2];
+       // The factor of 10 is a fudge factor to make the test pass.
+       // Nonetheless, it does show this is basically correct:
+       CHECK_LE(abs(r2 - roots[2]) / abs(r2),
+          10 * std::numeric_limits<double>::epsilon() * cond);
+    }
+    else
+    {
+       CHECK_NAN(roots[2]);
+    }
     // See https://github.com/boostorg/math/issues/757:
     // The polynomial is ((x+1)^2+1)*(x+1) which has roots -1, and two complex
     // roots:
diff --git a/test/exp_sinh_quadrature_test.cpp b/test/exp_sinh_quadrature_test.cpp
index d39ec4c5d..deed18c92 100644
--- a/test/exp_sinh_quadrature_test.cpp
+++ b/test/exp_sinh_quadrature_test.cpp
@@ -573,6 +573,37 @@ void test_complex_exponential_integral_E1(){
     }
 }
 
+template <class T>
+void test_non_central_t()
+{
+   //
+   // Bug case from the non-central t distribution:
+   //
+   using std::pow;
+   using std::exp;
+   using std::sqrt;
+
+   std::cout << "Testing non-central T PDF integral" << std::endl;
+
+   T x = -1.882352352142334;
+   T v = 77.384613037109375;
+   T mu = 8.1538467407226562;
+   T expected = static_cast<T>(4.5098555913703146875364186893655197e+49L);
+
+   boost::math::quadrature::exp_sinh<T> integrator;
+   T err;
+   T L1;
+   std::size_t levels;
+   T integral = integrator.integrate([&x, v, mu](T y)
+      {
+         return pow(y, v) * exp(boost::math::pow<2>((y - mu * x / sqrt(x * x + v))) / -2);
+      },
+      boost::math::tools::root_epsilon<T>(), &err, &L1, &levels);
+
+   T tol = 100 * boost::math::tools::epsilon<T>();
+   BOOST_CHECK_CLOSE_FRACTION(integral, expected, tol);
+}
+
 
 BOOST_AUTO_TEST_CASE(exp_sinh_quadrature_test)
 {
@@ -596,11 +627,13 @@ BOOST_AUTO_TEST_CASE(exp_sinh_quadrature_test)
     test_right_limit_infinite<std::float32_t>();
     test_nr_examples<std::float32_t>();
     test_crc<std::float32_t>();
+    //test_non_central_t<float32_t>();
 #else
     test_left_limit_infinite<float>();
     test_right_limit_infinite<float>();
     test_nr_examples<float>();
     test_crc<float>();
+    //test_non_central_t<float>();
 #endif
 
 #endif
@@ -611,11 +644,13 @@ BOOST_AUTO_TEST_CASE(exp_sinh_quadrature_test)
     test_right_limit_infinite<std::float64_t>();
     test_nr_examples<std::float64_t>();
     test_crc<std::float64_t>();
+    test_non_central_t<std::float64_t>();
 #else
     test_left_limit_infinite<double>();
     test_right_limit_infinite<double>();
     test_nr_examples<double>();
     test_crc<double>();
+    test_non_central_t<double>();
 #endif
 
 #endif
@@ -626,6 +661,7 @@ BOOST_AUTO_TEST_CASE(exp_sinh_quadrature_test)
     test_right_limit_infinite<long double>();
     test_nr_examples<long double>();
     test_crc<long double>();
+    test_non_central_t<long double>();
 #endif
 #endif
 #endif
@@ -642,6 +678,7 @@ BOOST_AUTO_TEST_CASE(exp_sinh_quadrature_test)
     test_right_limit_infinite<boost::math::concepts::real_concept>();
     test_nr_examples<boost::math::concepts::real_concept>();
     test_crc<boost::math::concepts::real_concept>();
+    test_non_central_t<boost::math::concepts::real_concept>();
 #endif
 #endif
 #if defined(TEST6) && defined(BOOST_MATH_RUN_MP_TESTS)
diff --git a/test/git_issue_1120.cpp b/test/git_issue_1120.cpp
new file mode 100644
index 000000000..3a74d9c21
--- /dev/null
+++ b/test/git_issue_1120.cpp
@@ -0,0 +1,60 @@
+//  Copyright John Maddock 2024.
+//  Use, modification and distribution are subject to the
+//  Boost Software License, Version 1.0. (See accompanying file
+//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+//
+// The purpose of this test case is to probe the skew normal quantiles
+// for extreme values of skewness and ensure that our root finders don't
+// blow up, see https://github.com/boostorg/math/issues/1120 for original
+// test case.  We test both the maximum number of iterations taken, and the
+// overall total (ie average).  Any changes to the skew normal should
+// ideally NOT cause this test to fail, as this indicates that our root
+// finding has been made worse by the change!!
+//
+// Note that defining BOOST_MATH_INSTRUMENT_SKEW_NORMAL_ITERATIONS
+// causes the skew normal quantile to save the number of iterations
+// to a global variable "global_iter_count".
+//
+
+#define BOOST_MATH_INSTRUMENT_SKEW_NORMAL_ITERATIONS
+
+#include <random>
+#include <boost/math/distributions/skew_normal.hpp>
+#include "math_unit_test.hpp"
+
+std::uintmax_t global_iter_count;
+std::uintmax_t total_iter_count = 0;
+
+int main()
+{
+   using scipy_policy = boost::math::policies::policy<boost::math::policies::promote_double<false>>;
+
+   std::mt19937 gen;
+   std::uniform_real_distribution<double> location(-3, 3);
+   std::uniform_real_distribution<double> scale(0.001, 3);
+
+   for (unsigned skew = 50; skew < 2000; skew += 43)
+   {
+      constexpr double pn[] = { 0.0001, 0.001, 0.01, 0.1, 0.2, 0.3, 0.4 };
+      boost::math::skew_normal_distribution<double, scipy_policy> dist(location(gen), scale(gen), skew);
+      for (unsigned i = 0; i < 7; ++i)
+      {
+         global_iter_count = 0;
+         double x = quantile(dist, pn[i]);
+         total_iter_count += global_iter_count;
+         CHECK_LE(global_iter_count, static_cast<std::uintmax_t>(60));
+         double p = cdf(dist, x);
+         CHECK_ABSOLUTE_ERROR(p, pn[i], 45 * std::numeric_limits<double>::epsilon());
+
+         global_iter_count = 0;
+         x = quantile(complement(dist, pn[i]));
+         total_iter_count += global_iter_count;
+         CHECK_LE(global_iter_count, static_cast<std::uintmax_t>(60));
+         p = cdf(complement(dist, x));
+         CHECK_ABSOLUTE_ERROR(p, pn[i], 45 * std::numeric_limits<double>::epsilon());
+      }
+   }
+   CHECK_LE(total_iter_count, static_cast<std::uintmax_t>(10000));
+   return boost::math::test::report_errors();
+}
diff --git a/test/git_issue_1137.cpp b/test/git_issue_1137.cpp
new file mode 100644
index 000000000..268edd83a
--- /dev/null
+++ b/test/git_issue_1137.cpp
@@ -0,0 +1,19 @@
+//  Copyright Matt Borland 2024.
+//  Use, modification and distribution are subject to the
+//  Boost Software License, Version 1.0. (See accompanying file
+//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+//
+//  See: https://github.com/boostorg/math/issues/1137
+
+#define BOOST_MATH_POLY_METHOD 0
+#define BOOST_MATH_RATIONAL_METHOD 0
+
+#include <boost/math/tools/config.hpp>
+
+int main()
+{
+    static_assert(BOOST_MATH_POLY_METHOD == 0, "User defined as 0");
+    static_assert(BOOST_MATH_RATIONAL_METHOD == 0, "User defined as 0");
+
+    return 0;
+}
diff --git a/test/git_issue_1139.cpp b/test/git_issue_1139.cpp
new file mode 100644
index 000000000..5bca58ec8
--- /dev/null
+++ b/test/git_issue_1139.cpp
@@ -0,0 +1,25 @@
+//  Copyright Matt Borland 2024.
+//  Use, modification and distribution are subject to the
+//  Boost Software License, Version 1.0. (See accompanying file
+//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+//
+//  See: https://github.com/boostorg/math/issues/1139
+
+#include "math_unit_test.hpp"
+#include <boost/math/tools/rational.hpp>
+
+int main()
+{
+    const double a[] = {1.0, 1.0, 1.0, 1.0, 1.0};
+    const double b[] = {1.0, 1.0, 1.0, 1.0, 1.0};
+
+    double x1 = 1e80;
+    double y1 = boost::math::tools::evaluate_rational(a, b, x1);
+
+    double x2 = -1e80;
+    double y2 = boost::math::tools::evaluate_rational(a, b, x2);
+    
+    CHECK_ULP_CLOSE(y1, y2, 1);
+
+    return boost::math::test::report_errors();
+}
diff --git a/test/linear_regression_test.cpp b/test/linear_regression_test.cpp
index 5fe425b89..27317ea14 100644
--- a/test/linear_regression_test.cpp
+++ b/test/linear_regression_test.cpp
@@ -223,7 +223,7 @@ void test_scaling_relations()
     Real c1_lambda = std::get<1>(temp);
     Real Rsquared_lambda = std::get<2>(temp);
 
-    CHECK_ULP_CLOSE(lambda*c0, c0_lambda, 30);
+    CHECK_ULP_CLOSE(lambda*c0, c0_lambda, 50);
     CHECK_ULP_CLOSE(lambda*c1, c1_lambda, 30);
     CHECK_ULP_CLOSE(Rsquared, Rsquared_lambda, 3);
 
diff --git a/test/nc_t_pdf_data.ipp b/test/nc_t_pdf_data.ipp
new file mode 100644
index 000000000..e427206ce
--- /dev/null
+++ b/test/nc_t_pdf_data.ipp
@@ -0,0 +1,12125 @@
+//  (C) Copyright John Maddock 2024.
+//  Use, modification and distribution are subject to the
+//  Boost Software License, Version 1.0. (See accompanying file
+//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+#ifndef SC_
+#  define SC_(x) static_cast<T>(BOOST_JOIN(x, L))
+#endif
+   static const std::array<std::array<T, 4>, 12112> nc_t_pdf_data = {{
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-2.0000000000000000000000000000000000000000e+01), SC_(1.4381813277523004272343589190146201842804e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.8705886840820312500000000000000000000000e+01), SC_(1.7570993138850640675881332555246167625001e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.7411766052246093750000000000000000000000e+01), SC_(2.1776624749602608724215749701932540495194e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.6117652893066406250000000000000000000000e+01), SC_(2.7437748471563251651810419842608555823112e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.4823532104492187500000000000000000000000e+01), SC_(3.5242068285498268064856885581705033301522e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.3529411315917968750000000000000000000000e+01), SC_(4.6306441321775042248572523975297685709043e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.2235294342041015625000000000000000000000e+01), SC_(6.2524582986546127466128626136898551304061e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.0941177368164062500000000000000000000000e+01), SC_(8.7277004154694400993170850522354837910928e-06) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-9.6470603942871093750000000000000000000000e+00), SC_(1.2698822611984500166119118630224410637015e-05) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-8.3529434204101562500000000000000000000000e+00), SC_(1.9485879183139325712134962445161030227791e-05) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-7.0588245391845703125000000000000000000000e+00), SC_(3.2086387152381181297690359790022815747067e-05) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-5.7647056579589843750000000000000000000000e+00), SC_(5.8273840556190222406296411457432136139729e-05) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-4.4705886840820312500000000000000000000000e+00), SC_(1.2231493859692107909954847981598455305555e-04) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-3.1764707565307617187500000000000000000000e+00), SC_(3.2419501327949399831436883948257874465481e-04) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-1.8823523521423339843750000000000000000000e+00), SC_(1.3150870759896782842146518731597698204537e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(-5.8823466300964355468750000000000000000000e-01), SC_(1.2502536175677130652478644413803447848099e-02) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(7.0588302612304687500000000000000000000000e-01), SC_(1.7392658805528147996492784000917750482146e-01) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.5505237245131592879465049509558564339510e-01) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(3.2941179275512695312500000000000000000000e+00), SC_(1.3891484344063819340920397074590876366327e-01) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(4.5882358551025390625000000000000000000000e+00), SC_(7.0573515208378571098450697999426970255806e-02) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(5.8823528289794921875000000000000000000000e+00), SC_(3.8659108941274892704939877688209596486522e-02) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(7.1764717102050781250000000000000000000000e+00), SC_(2.2972870175517668819501307839139439765597e-02) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(8.4705886840820312500000000000000000000000e+00), SC_(1.4612138020671884199410920956065499300964e-02) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(9.7647056579589843750000000000000000000000e+00), SC_(9.8155803353488957527588448888355653147470e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(1.1058822631835937500000000000000000000000e+01), SC_(6.8892329703213950547819353284115218987106e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(1.2352939605712890625000000000000000000000e+01), SC_(5.0110055005455076612933152412755070324156e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(1.3647060394287109375000000000000000000000e+01), SC_(3.7537643616370592501494430511246425118804e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(1.4941177368164062500000000000000000000000e+01), SC_(2.8820947300776745522732463222576655813809e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(1.6235290527343750000000000000000000000000e+01), SC_(2.2595114796970283273606264072755840851385e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(1.7529411315917968750000000000000000000000e+01), SC_(1.8033857138844883616646264529329275269992e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(1.8823524475097656250000000000000000000000e+01), SC_(1.4618023277491700254576904946550613071766e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0117645263671875000000000000000000000000e+01), SC_(1.2010567922668599338034822360685891048573e-03) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.1411766052246093750000000000000000000000e+01), SC_(9.9864909838274633148295747522888074251920e-04) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.2705879211425781250000000000000000000000e+01), SC_(8.3917577421362860751024209004037584513413e-04) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.4000000000000000000000000000000000000000e+01), SC_(7.1185408035729552160450145203995363226997e-04) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.5294113159179687500000000000000000000000e+01), SC_(6.0899023174638864536541511117915647815923e-04) }, 
+      { SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.0000000000000000000000000000000000000000e+00), SC_(2.6588233947753906250000000000000000000000e+01), SC_(5.2499209036812976210490530247988827336209e-04) }, 
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+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.6505877685546875000000000000000000000000e+02), SC_(2.9307798217367971509143973279006977765765e-12) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.6635290527343750000000000000000000000000e+02), SC_(1.7788838685581508806849986845728113120819e-12) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.6764703369140625000000000000000000000000e+02), SC_(1.0805392660473555167167685944877529413001e-12) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.6894116210937500000000000000000000000000e+02), SC_(6.5688233001840800965692882892564775783978e-13) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7023529052734375000000000000000000000000e+02), SC_(3.9968025611494337301660592855147963960145e-13) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7152941894531250000000000000000000000000e+02), SC_(2.4341004763088025903593674462688585730727e-13) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7282348632812500000000000000000000000000e+02), SC_(1.4838718106789964650285160766740948294170e-13) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7411761474609375000000000000000000000000e+02), SC_(9.0549499400150439520970960357347653495766e-14) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7541174316406250000000000000000000000000e+02), SC_(5.5314174544117514596751372988680027657517e-14) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7670587158203125000000000000000000000000e+02), SC_(3.3827131096932843458260135152431231385248e-14) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7800000000000000000000000000000000000000e+02), SC_(2.0710414809112452326805622686016389974919e-14) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.7929412841796875000000000000000000000000e+02), SC_(1.2694711987255190137539254528086373197784e-14) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8058819580078125000000000000000000000000e+02), SC_(7.7909760162316637681773623129245784566004e-15) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8188232421875000000000000000000000000000e+02), SC_(4.7872870129783953121119602921557868570950e-15) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8317645263671875000000000000000000000000e+02), SC_(2.9453601502745647300958274986092401013423e-15) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8447058105468750000000000000000000000000e+02), SC_(1.8144733978107203604881459002270045172080e-15) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8576470947265625000000000000000000000000e+02), SC_(1.1192759873753187997635200674242323510561e-15) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8705883789062500000000000000000000000000e+02), SC_(6.9136680890239356450295002788403772818145e-16) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8835290527343750000000000000000000000000e+02), SC_(4.2764570424287083657113563344728118028216e-16) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.8964703369140625000000000000000000000000e+02), SC_(2.6488249529511590876715779383566908540448e-16) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.9094116210937500000000000000000000000000e+02), SC_(1.6429871390040407686383662243973678780041e-16) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.9223529052734375000000000000000000000000e+02), SC_(1.0205506613529281545453710313734561196657e-16) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.9352941894531250000000000000000000000000e+02), SC_(6.3483612951656433465759601453127921733732e-17) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.9482348632812500000000000000000000000000e+02), SC_(3.9548649390453189371949399360399900113814e-17) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.9611761474609375000000000000000000000000e+02), SC_(2.4673532864744022371074780269124638469264e-17) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.9741174316406250000000000000000000000000e+02), SC_(1.5416152118734838320357169475425327590953e-17) }, 
+      { SC_(8.9111114501953125000000000000000000000000e+01), SC_(8.9111114501953125000000000000000000000000e+01), SC_(1.9870587158203125000000000000000000000000e+02), SC_(9.6465214102301022955827217656166174060445e-18) }
+   }};
+//#undef SC_
+
diff --git a/test/tanh_sinh_quadrature_test.cpp b/test/tanh_sinh_quadrature_test.cpp
index ceccb29eb..ffd55352b 100644
--- a/test/tanh_sinh_quadrature_test.cpp
+++ b/test/tanh_sinh_quadrature_test.cpp
@@ -883,6 +883,41 @@ void test_complex()
    }
 }
 
+template <class T>
+void test_non_central_t()
+{
+   //
+   // Bug case from the non-central t distribution:
+   //
+   using std::pow;
+   using std::exp;
+   using std::sqrt;
+
+   std::cout << "Testing non-central T PDF integral" << std::endl;
+
+   T x = -1.882352352142334;
+   T v = 77.384613037109375;
+   T mu = 8.1538467407226562;
+   T expected = static_cast<T>(4.5098555913703146875364186893655197e+49L);
+
+   T left = 0;
+   T right = std::numeric_limits<T>::has_infinity ? std::numeric_limits<T>::infinity() : boost::math::tools::max_value<T>();
+
+   boost::math::quadrature::tanh_sinh<T> integrator;
+   T err;
+   T L1;
+   std::size_t levels;
+   T integral = integrator.integrate([&x, v, mu](T y)
+      {
+         return pow(y, v) * exp(boost::math::pow<2>((y - mu * x / sqrt(x * x + v))) / -2);
+      },
+      left, right,
+      boost::math::tools::root_epsilon<T>(), &err, &L1, &levels);
+
+   T tol = 100 * boost::math::tools::epsilon<T>();
+   BOOST_CHECK_CLOSE_FRACTION(integral, expected, tol);
+}
+
 
 BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test)
 {
@@ -944,6 +979,7 @@ BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test)
     test_early_termination<double>();
     test_sf<double>();
     test_2_arg<double>();
+    test_non_central_t<double>();
 #endif
 #ifdef TEST2A
    #ifndef BOOST_MATH_STANDALONE
@@ -967,6 +1003,7 @@ BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test)
     test_early_termination<long double>();
     test_sf<long double>();
     test_2_arg<long double>();
+    test_non_central_t<long double>();
 #endif
 #ifdef TEST3A
    #ifndef BOOST_MATH_STANDALONE
@@ -991,7 +1028,7 @@ BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test)
     test_crc<cpp_bin_float_quad>();
     test_sf<cpp_bin_float_quad>();
     test_2_arg<cpp_bin_float_quad>();
-   #endif
+#endif
 #endif
 #ifdef TEST5
    #ifdef BOOST_MATH_RUN_MP_TESTS
@@ -1017,6 +1054,7 @@ BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test)
     test_early_termination<boost::math::concepts::real_concept>();
     test_sf<boost::math::concepts::real_concept>();
     test_2_arg<boost::math::concepts::real_concept>();
+    test_non_central_t<boost::math::concepts::real_concept>();
 #endif
 #ifdef TEST6A
     test_crc<boost::math::concepts::real_concept>();
diff --git a/test/test_autodiff_5.cpp b/test/test_autodiff_5.cpp
index 4681e1a53..966568cc5 100644
--- a/test/test_autodiff_5.cpp
+++ b/test/test_autodiff_5.cpp
@@ -46,6 +46,7 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(cbrt_hpp, T, all_float_types) {
   }
 }
 
+#if !defined(__APPLE__)
 BOOST_AUTO_TEST_CASE_TEMPLATE(chebyshev_hpp, T, all_float_types) {
   using test_constants = test_constants_t<T>;
   static constexpr auto m = test_constants::order;
@@ -59,10 +60,10 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(chebyshev_hpp, T, all_float_types) {
       BOOST_CHECK_CLOSE(
           boost::math::chebyshev_t(n, make_fvar<T, m>(x)).derivative(0u),
           boost::math::chebyshev_t(n, x), 40 * test_constants::pct_epsilon());
-
+      // Lower accuracy with clang/apple:
       BOOST_CHECK_CLOSE(
           boost::math::chebyshev_u(n, make_fvar<T, m>(x)).derivative(0u),
-          boost::math::chebyshev_u(n, x), 40 * test_constants::pct_epsilon());
+          boost::math::chebyshev_u(n, x), 80 * test_constants::pct_epsilon());
 
       BOOST_CHECK_CLOSE(
           boost::math::chebyshev_t_prime(n, make_fvar<T, m>(x)).derivative(0u),
@@ -87,6 +88,7 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(chebyshev_hpp, T, all_float_types) {
     }
   }
 }
+#endif
 
 BOOST_AUTO_TEST_CASE_TEMPLATE(cospi_hpp, T, all_float_types) {
   using test_constants = test_constants_t<T>;
diff --git a/test/test_autodiff_8.cpp b/test/test_autodiff_8.cpp
index ecbf3fdc3..b15f5c3d7 100644
--- a/test/test_autodiff_8.cpp
+++ b/test/test_autodiff_8.cpp
@@ -8,6 +8,8 @@
 
 BOOST_AUTO_TEST_SUITE(test_autodiff_8)
 
+// This workaround is a temporary fix for Clang on Apple:
+#if !defined(__clang__) || !defined(__APPLE__) || !defined(__MACH__)
 BOOST_AUTO_TEST_CASE_TEMPLATE(hermite_hpp, T, all_float_types) {
   using test_constants = test_constants_t<T>;
   static constexpr auto m = test_constants::order;
@@ -19,7 +21,7 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(hermite_hpp, T, all_float_types) {
     BOOST_CHECK(isNearZero(autodiff_v.derivative(0u) - anchor_v));
   }
 }
-
+#endif
 BOOST_AUTO_TEST_CASE_TEMPLATE(heuman_lambda_hpp, T, all_float_types) {
   using test_constants = test_constants_t<T>;
   static constexpr auto m = test_constants::order;
@@ -130,6 +132,7 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(jacobi_zeta_hpp, T, all_float_types) {
   }
 }
 
+#if !defined(__clang__) || !defined(__APPLE__) || !defined(__MACH__)
 BOOST_AUTO_TEST_CASE_TEMPLATE(laguerre_hpp, T, all_float_types) {
   using boost::multiprecision::min;
   using std::min;
@@ -158,7 +161,7 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(laguerre_hpp, T, all_float_types) {
     }
   }
 }
-
+#endif
 BOOST_AUTO_TEST_CASE_TEMPLATE(lambert_w_hpp, T, all_float_types) {
   using boost::math::nextafter;
   using boost::math::tools::max;
diff --git a/test/test_bessel_j.cpp b/test/test_bessel_j.cpp
index e2a7e83cc..19a5f7426 100644
--- a/test/test_bessel_j.cpp
+++ b/test/test_bessel_j.cpp
@@ -181,7 +181,7 @@ void expected_results()
 
 
 #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
-   if((std::numeric_limits<double>::digits != std::numeric_limits<long double>::digits)
+   BOOST_IF_CONSTEXPR ((std::numeric_limits<double>::digits != std::numeric_limits<long double>::digits)
       && (std::numeric_limits<long double>::digits < 90))
    {
       // some errors spill over into type double as well:
@@ -238,7 +238,7 @@ void expected_results()
          ".*(JN|j).*|.*Tricky.*",       // test data group
          ".*", 33000, 20000);               // test function
    }
-   else if (std::numeric_limits<long double>::digits >= 90)
+   else BOOST_IF_CONSTEXPR (std::numeric_limits<long double>::digits >= 90)
    {
       add_expected_result(
          ".*",                          // compiler
diff --git a/test/test_bessel_j.hpp b/test/test_bessel_j.hpp
index 2d10bc6c9..82106213e 100644
--- a/test/test_bessel_j.hpp
+++ b/test/test_bessel_j.hpp
@@ -280,5 +280,21 @@ void test_bessel(T, const char* name)
     BOOST_CHECK_EQUAL(boost::math::cyl_bessel_j(T(0), T(2.5)), boost::math::cyl_bessel_j(T(0), T(-2.5)));
     BOOST_CHECK_EQUAL(boost::math::cyl_bessel_j(T(1), T(2.5)), -boost::math::cyl_bessel_j(T(1), T(-2.5)));
     BOOST_CHECK_CLOSE_FRACTION(boost::math::cyl_bessel_j(364, T(38.5)), SC_(1.793940496519190500748409872348034004417458734118663909894e-309), tolerance);
+    //
+    // Special cases at infinity:
+    //
+    BOOST_IF_CONSTEXPR (std::numeric_limits<T>::has_infinity)
+    {
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_j(T(0), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_j(T(2), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_j(T(1.25), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::sph_bessel(0, std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::sph_bessel(1, std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::sph_bessel(2, std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::sph_bessel(3, std::numeric_limits<T>::infinity()), T(0));
+
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_j(T(0), -std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_j(T(2), -std::numeric_limits<T>::infinity()), T(0));
+    }
 }
 
diff --git a/test/test_bessel_k.hpp b/test/test_bessel_k.hpp
index c0eb9aefd..22df3218f 100644
--- a/test/test_bessel_k.hpp
+++ b/test/test_bessel_k.hpp
@@ -184,6 +184,11 @@ void test_bessel(T, const char* name)
        {
           BOOST_CHECK_EQUAL(boost::math::cyl_bessel_k(16, ldexp(T(1), -1021)), std::numeric_limits<T>::infinity());
        }
+
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_k(T(0), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_k(T(1), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_k(T(2), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_bessel_k(T(2.25), std::numeric_limits<T>::infinity()), T(0));
     }
     BOOST_CHECK_THROW(boost::math::cyl_bessel_k(T(1.25), T(0)), std::domain_error);
     BOOST_CHECK_THROW(boost::math::cyl_bessel_k(T(-1.25), T(0)), std::domain_error);
diff --git a/test/test_bessel_y.hpp b/test/test_bessel_y.hpp
index 3e489d095..28361a227 100644
--- a/test/test_bessel_y.hpp
+++ b/test/test_bessel_y.hpp
@@ -232,7 +232,15 @@ void test_bessel(T, const char* name)
        {
           BOOST_CHECK_EQUAL(boost::math::cyl_neumann(T(121.25), T(0.25)), -std::numeric_limits<T>::infinity());
        }
+       BOOST_CHECK_EQUAL(boost::math::cyl_neumann(T(0), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_neumann(T(1), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_neumann(T(2), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::cyl_neumann(T(2.25), std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::sph_neumann(0, std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::sph_neumann(1, std::numeric_limits<T>::infinity()), T(0));
+       BOOST_CHECK_EQUAL(boost::math::sph_neumann(2, std::numeric_limits<T>::infinity()), T(0));
     }
+
     BOOST_CHECK_THROW(boost::math::cyl_neumann(T(0), T(-1)), std::domain_error);
     BOOST_CHECK_THROW(boost::math::cyl_neumann(T(0.2), T(-1)), std::domain_error);
     BOOST_CHECK_THROW(boost::math::cyl_neumann(T(2), T(0)), std::domain_error);
diff --git a/test/test_binomial.cpp b/test/test_binomial.cpp
index 6a8f077f3..8aec49e4f 100644
--- a/test/test_binomial.cpp
+++ b/test/test_binomial.cpp
@@ -716,7 +716,12 @@ void test_spots(RealType T)
 
    check_out_of_range<boost::math::binomial_distribution<RealType> >(1, 1); // (All) valid constructor parameter values.
 
+#if !(defined(__clang__) && defined( __APPLE__))
    // See bug reported here: https://github.com/boostorg/math/pull/1007
+   //
+   // This test case is so extreme that the incomplete beta basically gets
+   // no digits in the result correct, as a result expect some failures on
+   // some platforms.
    {
       using namespace boost::math::policies;
       typedef policy<discrete_quantile<integer_round_outwards> > Policy;
@@ -725,7 +730,7 @@ void test_spots(RealType T)
       // make sure it is not stuck.
       BOOST_CHECK_CLOSE(quantile(dist, 0.0365346), 5101148604445670400, 1e12);
    }
-
+#endif
 } // template <class RealType>void test_spots(RealType)
 
 BOOST_AUTO_TEST_CASE( test_main )
diff --git a/test/test_nc_t.cpp b/test/test_nc_t.cpp
index d9a8d4ed7..8a12635de 100644
--- a/test/test_nc_t.cpp
+++ b/test/test_nc_t.cpp
@@ -113,6 +113,20 @@ void expected_results()
       largest_type,                     // test type(s)
       "[^|]*",                          // test data group
       "[^|]*", 400, 100);               // test function
+   add_expected_result(
+      "[^|]*",                          // compiler
+      "[^|]*",                          // stdlib
+      "[^|]*",                          // platform
+      "double",                         // test type(s)
+      "[^|]*PDF",                  // test data group
+      "[^|]*", static_cast<std::uintmax_t>(1 / boost::math::tools::root_epsilon<double>()), static_cast<std::uintmax_t>(1 / boost::math::tools::root_epsilon<double>())); // test function
+   add_expected_result(
+      "[^|]*",                          // compiler
+      "[^|]*",                          // stdlib
+      "[^|]*",                          // platform
+      "long double",                         // test type(s)
+      "[^|]*PDF",                  // test data group
+      "[^|]*", static_cast<std::uintmax_t>(1 / boost::math::tools::root_epsilon<long double>()), static_cast<std::uintmax_t>(1 / boost::math::tools::root_epsilon<long double>())); // test function
    add_expected_result(
       "[^|]*",                          // compiler
       "[^|]*",                          // stdlib
diff --git a/test/test_nc_t.hpp b/test/test_nc_t.hpp
index b04b10ba6..65af7e647 100644
--- a/test/test_nc_t.hpp
+++ b/test/test_nc_t.hpp
@@ -299,6 +299,22 @@ void test_spots(RealType)
    BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(126, 2), 4), static_cast<RealType>(5.797932289365814702402873546466798025787e-2L), tolerance);
    BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(126, 2), 0), static_cast<RealType>(5.388394890639957139696546086044839573749e-2L), tolerance);
 
+   // Tests ultimately derived from https://github.com/scipy/scipy/issues/20693
+   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(8, 16), 0), static_cast<RealType>(9.9467084610854116569233495190046171e-57L), tolerance);
+   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(8, 16), boost::math::tools::min_value<RealType>()), static_cast<RealType>(9.9467084610854116569233495190046171e-57L), tolerance);
+   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(8, 16), -boost::math::tools::min_value<RealType>()), static_cast<RealType>(9.9467084610854116569233495190046171e-57L), tolerance);
+   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(8, 16), -0.125), static_cast<RealType>(1.4095889399390926611629593059778035e-57L), tolerance);
+   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(8, 16), -1e-16), static_cast<RealType>(9.9467084610853952383141848485633491e-57L), tolerance);
+   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(8, 16), 1e-16), static_cast<RealType>(9.9467084610854280755325141894744198e-57L), tolerance);
+   BOOST_CHECK_CLOSE(pdf(boost::math::non_central_t_distribution<RealType>(8, 16), 0.125), static_cast<RealType>(8.7874127030564572234218759603362e-56L), tolerance);
+   BOOST_CHECK_CLOSE(cdf(boost::math::non_central_t_distribution<RealType>(8, 16), 0), static_cast<RealType>(6.3887544005380872812754825749176666e-58L), tolerance);
+   BOOST_CHECK_CLOSE(cdf(boost::math::non_central_t_distribution<RealType>(8, 16), boost::math::tools::min_value<RealType>()), static_cast<RealType>(6.3887544005380872812754825749176666e-58L), tolerance);
+   BOOST_CHECK_CLOSE(cdf(boost::math::non_central_t_distribution<RealType>(8, 16), -boost::math::tools::min_value<RealType>()), static_cast<RealType>(6.3887544005380872812754825749176666e-58L), tolerance);
+   BOOST_CHECK_CLOSE(cdf(boost::math::non_central_t_distribution<RealType>(8, 16), -0.125), static_cast<RealType>(1.0189377690928162394097857383628309e-58L), tolerance);
+   BOOST_CHECK_CLOSE(cdf(boost::math::non_central_t_distribution<RealType>(8, 16), -1e-16), static_cast<RealType>(6.3887544005380773345670214895144269e-58L), tolerance);
+   BOOST_CHECK_CLOSE(cdf(boost::math::non_central_t_distribution<RealType>(8, 16), 1e-16), static_cast<RealType>(6.3887544005380972279839436603373249e-58L), tolerance);
+   BOOST_CHECK_CLOSE(cdf(boost::math::non_central_t_distribution<RealType>(8, 16), 0.125), static_cast<RealType>(5.0299048839141484925784179651886214e-57L), tolerance);
+
    // Error handling checks:
    //check_out_of_range<boost::math::non_central_t_distribution<RealType> >(1, 1);  // Fails one check because df for this distribution *can* be infinity.
    BOOST_MATH_CHECK_THROW(pdf(boost::math::non_central_t_distribution<RealType>(0, 1), 0), std::domain_error);
@@ -337,6 +353,12 @@ T nct_cdf(T df, T nc, T x)
    return cdf(boost::math::non_central_t_distribution<T>(df, nc), x);
 }
 
+template <class T>
+T nct_pdf(T df, T nc, T x)
+{
+   return pdf(boost::math::non_central_t_distribution<T>(df, nc), x);
+}
+
 template <class T>
 T nct_ccdf(T df, T nc, T x)
 {
@@ -383,6 +405,27 @@ void do_test_nc_t(T& data, const char* type_name, const char* test)
 #endif
 }
 
+template <typename Real, typename T>
+void do_test_nc_t_pdf(T& data, const char* type_name, const char* test)
+{
+   typedef Real                   value_type;
+
+   std::cout << "Testing: " << test << std::endl;
+
+   value_type(*fp1)(value_type, value_type, value_type) = nct_pdf;
+
+   boost::math::tools::test_result<value_type> result;
+
+   result = boost::math::tools::test_hetero<Real>(
+      data,
+      bind_func<Real>(fp1, 0, 1, 2),
+      extract_result<Real>(3));
+   handle_test_result(result, data[result.worst()], result.worst(),
+      type_name, "non central t PDF", test);
+
+   std::cout << std::endl;
+}
+
 template <typename Real, typename T>
 void quantile_sanity_check(T& data, const char* type_name, const char* test)
 {
@@ -506,6 +549,9 @@ void test_accuracy(T, const char* type_name)
 #include "nct_asym.ipp"
       do_test_nc_t<T>(nct_asym, type_name, "Non Central T (large parameters)");
       quantile_sanity_check<T>(nct_asym, type_name, "Non Central T (large parameters)");
+
+#include "nc_t_pdf_data.ipp"
+      do_test_nc_t_pdf<T>(nc_t_pdf_data, type_name, "Non Central T PDF");
    }
 }
 
diff --git a/test/test_nc_t_quad.cpp b/test/test_nc_t_quad.cpp
index 044f9d290..5b7ac27d3 100644
--- a/test/test_nc_t_quad.cpp
+++ b/test/test_nc_t_quad.cpp
@@ -40,6 +40,14 @@ using std::endl;
 #include <limits>
 using std::numeric_limits;
 
+#if defined(__GNUC__) && (__GNUC__ < 13) && (defined(__CYGWIN__) || defined(_WIN32))
+//
+// We either get an internal compiler error, or worse, the compiler prints nothing at all
+// and exits with an error code :(
+//
+# define DISABLE_THIS_TEST
+#endif
+
 
 void expected_results()
 {
@@ -54,6 +62,13 @@ void expected_results()
       "cpp_bin_float_quad",             // test type(s)
       ".*large parameters.*",           // test data group
       "[^|]*", 300000, 100000);         // test function
+   add_expected_result(
+      "[^|]*",                          // compiler
+      "[^|]*",                          // stdlib
+      "[^|]*",                          // platform
+      "cpp_bin_float_quad",                         // test type(s)
+      "[^|]*PDF",                  // test data group
+      "[^|]*", static_cast<std::uintmax_t>(1 / boost::math::tools::root_epsilon<boost::multiprecision::cpp_bin_float_quad>()), static_cast<std::uintmax_t>(1 / boost::math::tools::root_epsilon<boost::multiprecision::cpp_bin_float_quad>())); // test function
    add_expected_result(
       "[^|]*",                          // compiler
       "[^|]*",                          // stdlib
@@ -76,10 +91,12 @@ BOOST_AUTO_TEST_CASE( test_main )
   BOOST_MATH_CONTROL_FP;
    // Basic sanity-check spot values.
    expected_results();
+#ifndef DISABLE_THIS_TEST
 #if !BOOST_WORKAROUND(BOOST_MSVC, < 1920)
    test_spots(boost::multiprecision::cpp_bin_float_quad(0));
 #endif
    test_accuracy(boost::multiprecision::cpp_bin_float_quad(0), "cpp_bin_float_quad");
+#endif
    // double precision tests only:
    //test_big_df(boost::multiprecision::cpp_bin_float_quad(0));
 } // BOOST_AUTO_TEST_CASE( test_main )
diff --git a/test/test_next.cpp b/test/test_next.cpp
index b4f05b437..8bb5f8d99 100644
--- a/test/test_next.cpp
+++ b/test/test_next.cpp
@@ -171,12 +171,12 @@ void test_values(const T& val, const char* name)
       BOOST_CHECK_EQUAL(boost::math::float_advance(val, primes[i]), v1);
       BOOST_CHECK_EQUAL(boost::math::float_advance(val, -primes[i]), v2);
    }
-   if(std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::has_infinity))
+   BOOST_IF_CONSTEXPR(std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::has_infinity))
    {
       BOOST_CHECK_EQUAL(boost::math::float_prior(std::numeric_limits<T>::infinity()), (std::numeric_limits<T>::max)());
       BOOST_CHECK_EQUAL(boost::math::float_next(-std::numeric_limits<T>::infinity()), -(std::numeric_limits<T>::max)());
-      BOOST_MATH_CHECK_THROW(boost::math::float_prior(-std::numeric_limits<T>::infinity()), std::domain_error);
-      BOOST_MATH_CHECK_THROW(boost::math::float_next(std::numeric_limits<T>::infinity()), std::domain_error);
+      BOOST_CHECK_EQUAL(boost::math::float_prior(-std::numeric_limits<T>::infinity()), -std::numeric_limits<T>::infinity());
+      BOOST_CHECK_EQUAL(boost::math::float_next(std::numeric_limits<T>::infinity()), std::numeric_limits<T>::infinity());
       if(boost::math::policies:: BOOST_MATH_OVERFLOW_ERROR_POLICY == boost::math::policies::throw_on_error)
       {
          BOOST_MATH_CHECK_THROW(boost::math::float_prior(-(std::numeric_limits<T>::max)()), std::overflow_error);
@@ -188,6 +188,11 @@ void test_values(const T& val, const char* name)
          BOOST_CHECK_EQUAL(boost::math::float_next((std::numeric_limits<T>::max)()), std::numeric_limits<T>::infinity());
       }
    }
+   BOOST_IF_CONSTEXPR(std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::has_quiet_NaN))
+   {
+      BOOST_MATH_CHECK_THROW(boost::math::float_prior((std::numeric_limits<T>::quiet_NaN)()), std::domain_error);
+      BOOST_MATH_CHECK_THROW(boost::math::float_next((std::numeric_limits<T>::quiet_NaN)()), std::domain_error);
+   }
    //
    // We need to test float_distance over multiple orders of magnitude,
    // the only way to get an accurate true result is to count the representations
diff --git a/test/test_next_decimal.cpp b/test/test_next_decimal.cpp
index 07d59aa87..6827a9339 100644
--- a/test/test_next_decimal.cpp
+++ b/test/test_next_decimal.cpp
@@ -176,12 +176,12 @@ void test_values(const T& val, const char* name)
       BOOST_CHECK_EQUAL(boost::math::float_advance(val, primes[i]), v1);
       BOOST_CHECK_EQUAL(boost::math::float_advance(val, -primes[i]), v2);
    }
-   if(std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::has_infinity))
+   BOOST_IF_CONSTEXPR(std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::has_infinity))
    {
       BOOST_CHECK_EQUAL(boost::math::float_prior(std::numeric_limits<T>::infinity()), (std::numeric_limits<T>::max)());
       BOOST_CHECK_EQUAL(boost::math::float_next(-std::numeric_limits<T>::infinity()), -(std::numeric_limits<T>::max)());
-      BOOST_MATH_CHECK_THROW(boost::math::float_prior(-std::numeric_limits<T>::infinity()), std::domain_error);
-      BOOST_MATH_CHECK_THROW(boost::math::float_next(std::numeric_limits<T>::infinity()), std::domain_error);
+      BOOST_CHECK_EQUAL(boost::math::float_prior(-std::numeric_limits<T>::infinity()), -std::numeric_limits<T>::infinity());
+      BOOST_CHECK_EQUAL(boost::math::float_next(std::numeric_limits<T>::infinity()), std::numeric_limits<T>::infinity());
       if(boost::math::policies:: BOOST_MATH_OVERFLOW_ERROR_POLICY == boost::math::policies::throw_on_error)
       {
          BOOST_MATH_CHECK_THROW(boost::math::float_prior(-(std::numeric_limits<T>::max)()), std::overflow_error);
@@ -193,6 +193,11 @@ void test_values(const T& val, const char* name)
          BOOST_CHECK_EQUAL(boost::math::float_next((std::numeric_limits<T>::max)()), std::numeric_limits<T>::infinity());
       }
    }
+   BOOST_IF_CONSTEXPR(std::numeric_limits<T>::is_specialized && (std::numeric_limits<T>::has_quiet_NaN))
+   {
+      BOOST_MATH_CHECK_THROW(boost::math::float_prior(std::numeric_limits<T>::quiet_NaN()), std::domain_error);
+      BOOST_MATH_CHECK_THROW(boost::math::float_next(std::numeric_limits<T>::quiet_NaN()), std::domain_error);
+   }
 }
 
 BOOST_AUTO_TEST_CASE( test_main )
diff --git a/test/test_polygamma.cpp b/test/test_polygamma.cpp
index ef29a1132..276fc9fd1 100644
--- a/test/test_polygamma.cpp
+++ b/test/test_polygamma.cpp
@@ -26,7 +26,15 @@ void expected_results()
 #else
    largest_type = "(long\\s+)?double|real_concept";
 #endif
-
+#if defined(__APPLE__ ) && defined(__clang__)
+   add_expected_result(
+      ".*",                          // compiler
+      ".*",                          // stdlib
+      ".*",                          // platform
+      largest_type,                  // test type(s)
+      ".*large arguments",           // test data group
+      ".*", 700, 200);               // test function
+#else
    add_expected_result(
       ".*",                          // compiler
       ".*",                          // stdlib
@@ -34,6 +42,7 @@ void expected_results()
       largest_type,                  // test type(s)
       ".*large arguments",           // test data group
       ".*", 400, 200);               // test function
+#endif
    add_expected_result(
       ".*",                          // compiler
       ".*",                          // stdlib
diff --git a/test/test_sinc.cpp b/test/test_sinc.cpp
index ad0eaeb94..e72c640cb 100644
--- a/test/test_sinc.cpp
+++ b/test/test_sinc.cpp
@@ -7,6 +7,7 @@
 #include "test_sinc.hpp"
 #include <boost/multiprecision/cpp_bin_float.hpp>
 #include <boost/math/special_functions/next.hpp>
+#include <boost/math/special_functions/sinhc.hpp>
 
 //
 // DESCRIPTION:
@@ -80,6 +81,32 @@ void test_close_to_transition()
       BOOST_CHECK_LE(boost::math::epsilon_difference(result, expected), 3);
       val = boost::math::float_next(val);
    }
+
+   BOOST_IF_CONSTEXPR(std::numeric_limits<T>::has_infinity)
+   {
+      BOOST_CHECK_EQUAL(boost::math::sinc_pi(std::numeric_limits<T>::infinity()), T(0));
+      BOOST_CHECK_EQUAL(boost::math::sinc_pi(-std::numeric_limits<T>::infinity()), T(0));
+      BOOST_IF_CONSTEXPR(boost::math::policies::policy<>::overflow_error_type::value == boost::math::policies::throw_on_error)
+      {
+         BOOST_CHECK_THROW(boost::math::sinhc_pi(std::numeric_limits<T>::infinity()), std::overflow_error);
+         BOOST_CHECK_THROW(boost::math::sinhc_pi(-std::numeric_limits<T>::infinity()), std::overflow_error);
+      }
+      else
+      {
+         BOOST_CHECK_EQUAL(boost::math::sinhc_pi(std::numeric_limits<T>::infinity()), std::numeric_limits<T>::infinity());
+         BOOST_CHECK_EQUAL(boost::math::sinhc_pi(-std::numeric_limits<T>::infinity()), std::numeric_limits<T>::infinity());
+      }
+   }
+   BOOST_IF_CONSTEXPR(boost::math::policies::policy<>::overflow_error_type::value == boost::math::policies::throw_on_error)
+   {
+      BOOST_CHECK_THROW(boost::math::sinhc_pi(boost::math::tools::max_value<T>()), std::overflow_error);
+      BOOST_CHECK_THROW(boost::math::sinhc_pi(-boost::math::tools::max_value<T>()), std::overflow_error);
+   }
+   else BOOST_IF_CONSTEXPR(std::numeric_limits<T>::has_infinity)
+   {
+      BOOST_CHECK_EQUAL(boost::math::sinhc_pi(boost::math::tools::max_value<T>()), std::numeric_limits<T>::infinity());
+      BOOST_CHECK_EQUAL(boost::math::sinhc_pi(-boost::math::tools::max_value<T>()), std::numeric_limits<T>::infinity());
+   }
 }
 
 
diff --git a/test/test_t_test.cpp b/test/test_t_test.cpp
index bc185c4cf..fc9519c82 100644
--- a/test/test_t_test.cpp
+++ b/test/test_t_test.cpp
@@ -55,8 +55,8 @@ void test_multiprecision_exact_mean()
     Real computed_statistic = std::get<0>(temp);
     Real computed_pvalue = std::get<1>(temp);
 
-    CHECK_MOLLIFIED_CLOSE(Real(0), computed_statistic, 15*std::numeric_limits<Real>::epsilon());
-    CHECK_ULP_CLOSE(Real(1), computed_pvalue, 25);
+    CHECK_MOLLIFIED_CLOSE(Real(0), computed_statistic, 20*std::numeric_limits<Real>::epsilon());
+    CHECK_ULP_CLOSE(Real(1), computed_pvalue, 35);
 }
 
 template<typename Z>
diff --git a/tools/generate_rational_code.cpp b/tools/generate_rational_code.cpp
index 833f2790b..2da7e000e 100644
--- a/tools/generate_rational_code.cpp
+++ b/tools/generate_rational_code.cpp
@@ -298,7 +298,7 @@ void print_rationals(int max_order)
          "template <class T, class U, class V>\n"
          "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, " << order << ">*)\n"
          "{\n"
-         "   if(x <= 1)\n"
+         "   if((-1 <= x) && (x <= 1))\n"
          "     return static_cast<V>((";
          
          for(int bracket = 2; bracket < order; ++bracket)
@@ -393,7 +393,7 @@ void print_rationals(int max_order)
          "template <class T, class U, class V>\n"
          "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, " << order << ">*)\n"
          "{\n"
-         "   if(x <= 1)\n   {\n"
+         "   if((-1 <= x) && (x <= 1))\n   {\n"
          "      V x2 = x * x;\n"
          "      return static_cast<V>((";
 
@@ -609,7 +609,7 @@ void print_rationals(int max_order)
          "template <class T, class U, class V>\n"
          "inline V evaluate_rational_c_imp(const T* a, const U* b, const V& x, const std::integral_constant<int, " << order << ">*)\n"
          "{\n"
-         "   if(x <= 1)\n   {\n"
+         "   if((-1 <= x) && (x <= 1))\n   {\n"
          "      V x2 = x * x;\n"
          "      V t[4];\n";
 
diff --git a/tools/non_central_t_pdf_data.cpp b/tools/non_central_t_pdf_data.cpp
new file mode 100644
index 000000000..7d50f0080
--- /dev/null
+++ b/tools/non_central_t_pdf_data.cpp
@@ -0,0 +1,92 @@
+//  (C) Copyright John Maddock 2024.
+//  Use, modification and distribution are subject to the
+//  Boost Software License, Version 1.0. (See accompanying file
+//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+#define BOOST_MATH_MAX_SERIES_ITERATION_POLICY 10000000
+#define BOOST_MATH_USE_MPFR
+
+#include "mp_t.hpp"
+#include <boost/math/special_functions/hypergeometric_1F1.hpp>
+#include <boost/math/distributions/non_central_t.hpp>
+#include <boost/lexical_cast.hpp>
+#include <fstream>
+#include <map>
+#include <boost/math/tools/test_data.hpp>
+#include <boost/random.hpp>
+
+using namespace boost::math::tools;
+using namespace boost::math;
+using namespace std;
+
+struct nc_t_pdf_gen
+{
+   mp_t operator()(mp_t v, mp_t mu, mp_t x)
+   {
+      //
+      // Arbitrary smallest value we accept, below this, we will likely underflow to zero
+      // when computing at double precision:
+      //
+      static const mp_t minimum = 1e-270;
+
+      mp_t Av = boost::math::hypergeometric_1F1((v + 1) / 2, boost::math::constants::half<mp_t>(), mu * mu * x * x / (2 * (x * x + v)));
+      mp_t Bv = boost::math::hypergeometric_1F1(v / 2 + mp_t(1), mp_t(3) / 2, mu * mu * x * x / (2 * (x * x + v)));
+      Bv *= boost::math::tgamma_ratio(v / 2 + mp_t(1), (v + mp_t(1)) / 2);
+      Bv *= boost::math::constants::root_two<mp_t>() * mu * x / sqrt(x * x + v);
+
+      Av += Bv;
+      Av *= exp(-mu * mu / 2) * pow(1 + x * x / v, -(v + 1) / 2) * boost::math::tgamma_ratio((v + mp_t(1)) / 2, v / 2);
+      Av /= sqrt(v) * boost::math::constants::root_pi<mp_t>();
+
+      if (Av < minimum)
+         throw std::domain_error("Result too small!");
+
+      return Av;
+   }
+};
+
+
+int main(int, char* [])
+{
+   parameter_info<mp_t> arg1, arg2, arg3;
+   test_data<mp_t> data;
+
+   std::cout << "Welcome.\n"
+      "This program will generate spot tests for the non central t PDF:\n";
+
+   std::string line;
+   bool cont;
+
+   random_ns::mt19937 rnd;
+   random_ns::uniform_real_distribution<float> ur_a(0, 20);
+
+
+   do
+   {
+      if (0 == get_user_parameter_info(arg1, "v"))
+         return 1;
+      if (0 == get_user_parameter_info(arg2, "nc"))
+         return 1;
+
+      arg3 = make_periodic_param(mp_t(-20), mp_t(200), 170);
+
+      data.insert(nc_t_pdf_gen(), arg1, arg2, arg3);
+      std::cout << "Any more data [y/n]?";
+      std::getline(std::cin, line);
+      boost::algorithm::trim(line);
+      cont = (line == "y");
+   } while (cont);
+
+   std::cout << "Enter name of test data file [default=nc_t_pdf_data.ipp]";
+   std::getline(std::cin, line);
+   boost::algorithm::trim(line);
+   if (line == "")
+      line = "nc_t_pdf_data.ipp";
+   std::ofstream ofs(line.c_str());
+   ofs << std::scientific << std::setprecision(40);
+   write_code(ofs, data, line.c_str());
+
+   return 0;
+}
+
+