diff --git a/src/dual.jl b/src/dual.jl
index ad0af94..d71fa34 100644
--- a/src/dual.jl
+++ b/src/dual.jl
@@ -170,17 +170,17 @@ end
 Base.convert(::Type{Dual}, z::Dual) = z
 Base.convert(::Type{Dual}, x::Number) = Dual(x)
 
-Base.:(==)(z::Dual, w::Dual) = value(z) == value(w)
-Base.:(==)(z::Dual, x::Number) = value(z) == x
-Base.:(==)(x::Number, z::Dual) = value(z) == x
+Base.:(==)(z::Dual, w::Dual) = value(z) == value(w) && epsilon(z) == epsilon(w)
+Base.:(==)(z::Dual, x::Number) = value(z) == x && iszero(epsilon(z))
+Base.:(==)(x::Number, z::Dual) = z == x
 
 Base.isequal(z::Dual, w::Dual) = isequal(value(z),value(w)) && isequal(epsilon(z), epsilon(w))
 Base.isequal(z::Dual, x::Number) = isequal(value(z), x) && isequal(epsilon(z), zero(x))
 Base.isequal(x::Number, z::Dual) = isequal(z, x)
 
-Base.isless(z::Dual{<:Real},w::Dual{<:Real}) = value(z) < value(w)
-Base.isless(z::Real,w::Dual{<:Real}) = z < value(w)
-Base.isless(z::Dual{<:Real},w::Real) = value(z) < w
+Base.isless(z::Dual{<:Real},w::Dual{<:Real}) = isless(value(z), value(w)) || (isequal(value(z), value(w)) && isless(epsilon(z), epsilon(w)))
+Base.isless(z::Real,w::Dual{<:Real}) = isless(z, value(w)) || (isequal(z, value(w)) && isless(zero(epsilon(w)), epsilon(w)))
+Base.isless(z::Dual{<:Real},w::Real) = isless(value(z), w) || (isequal(value(z), w) && isless(epsilon(z), zero(epsilon(z))))
 
 Base.hash(z::Dual) = (x = hash(value(z)); epsilon(z)==0 ? x : bitmix(x,hash(epsilon(z))))
 
diff --git a/test/automatic_differentiation_test.jl b/test/automatic_differentiation_test.jl
index 2af2b2a..2b43cb2 100644
--- a/test/automatic_differentiation_test.jl
+++ b/test/automatic_differentiation_test.jl
@@ -43,20 +43,26 @@ powwrap(z, n, epspart=0) = Dual(z, epspart)^n
 @test powwrap(1, -1) == powwrap(1.0, -1) # special case is handled
 @test powwrap(1, -2) == powwrap(1.0, -2) # special case is handled
 @test powwrap(1, -123) == powwrap(1.0, -123) # special case is handled
-@test powwrap(1, 0) == Dual(1, 1)
-@test powwrap(123, 0) == Dual(1, 1)
+@test powwrap(1, 0) == Dual(1, 0)
+@test powwrap(1, 0) != Dual(1, 1)
+@test powwrap(123, 0) == Dual(1, 0)
+@test powwrap(123, 0) != Dual(1, 1)
 for i ∈ -3:3
-  @test powwrap(1, i) == Dual(1, i)
+  @test powwrap(1, i) == Dual(1, 0)
+  @test i == 0 || (powwrap(1, i) != Dual(1, i))
 end
 
 # this no longer throws 1/0 DomainError
-@test powwrap(0, Dual(0, 1)) == Dual(1, 0)
+@test powwrap(0, Dual(0, 1)) == Dual(1, -Inf)
+@test powwrap(0, Dual(0, 1)) != Dual(1, 0)
 # this never did DomainError because it starts off with a float
-@test 0.0^Dual(0, 1) == Dual(1.0, NaN)
+@test 0.0^Dual(0, 1) == Dual(1.0, -Inf)
+@test 0.0^Dual(0, 1) != Dual(1.0, NaN)
 # and Dual^Dual uses a log and is now type stable
 # because the log promotes ints to floats for all values
 @test typeof(value(powwrap(0, Dual(0, 1)))) == Float64
-@test Dual(0, 1)^Dual(0, 1) == Dual(1, 0)
+@test Dual(0, 1)^Dual(0, 1) == Dual(1, -Inf)
+@test Dual(0, 1)^Dual(0, 1) != Dual(1, 0)
 
 y = Dual(2.0, 1)^UInt64(0)
 @test !isnan(epsilon(y))