Fix DiffRules-based definitions for complex-valued functions

[devmotion]

Currently, complex-valued functions (with real inputs!) for which derivatives are defined in DiffRules don't work with ForwardDiff since the return values for DiffRules-based definitions is enforced to be a Dual but should be a Complex{<:Dual} in these cases. E.g., the following example fails currently (and is therefore excluded from the tests):

julia> using ForwardDiff, SpecialFunctions

julia> hankelh1(1, 2.0)
0.5767248077568733 - 0.10703243154093756im

julia> hankelh1(1, ForwardDiff.Dual(2.0))
ERROR: ArgumentError: Cannot create a dual over scalar type Any. If the type behaves as a scalar, define ForwardDiff.can_dual(::Type{Any}) = true.
Stacktrace:
 [1] throw_cannot_dual(V::Type)
   @ ForwardDiff ~/.julia/packages/ForwardDiff/PBzup/src/dual.jl:41
 [2] ForwardDiff.Dual{Nothing, Any, 1}(value::ComplexF64, partials::ForwardDiff.Partials{1, Any})
   @ ForwardDiff ~/.julia/packages/ForwardDiff/PBzup/src/dual.jl:18
 [3] Dual
   @ ~/.julia/packages/ForwardDiff/PBzup/src/dual.jl:60 [inlined]
 [4] Dual
   @ ~/.julia/packages/ForwardDiff/PBzup/src/dual.jl:64 [inlined]
 [5] Dual
   @ ~/.julia/packages/ForwardDiff/PBzup/src/dual.jl:67 [inlined]
 [6] Dual
   @ ~/.julia/packages/ForwardDiff/PBzup/src/dual.jl:71 [inlined]
 [7] hankelh1(x::Int64, y::ForwardDiff.Dual{Nothing, Float64, 0})
   @ ForwardDiff ~/.julia/packages/ForwardDiff/PBzup/src/dual.jl:249
 [8] top-level scope
   @ REPL[9]:1

This PR fixes this issue and returns values of type Complex{<:Dual} if the primal function value is a Complex number. This fixes the example above:

julia> using ForwardDiff, SpecialFunctions

julia> hankelh1(1, 2.0)
0.5767248077568733 - 0.10703243154093756im

julia> hankelh1(1, ForwardDiff.Dual(2.0))
Dual{Nothing}(0.5767248077568733) - Dual{Nothing}(0.10703243154093756)*im

[devmotion]

Bump 🙂

[devmotion]

Bump 🙂 It would be very helpful, in DiffRules and downstream packages, to fix this bug.

[codecov-commenter]

Codecov Report

Merging #577 (1043a59) into master (e11936c) will decrease coverage by 0.62%.
The diff coverage is 65.21%.

@@            Coverage Diff             @@
##           master     #577      +/-   ##
==========================================
- Coverage   85.43%   84.80%   -0.63%     
==========================================
  Files           9        9              
  Lines         865      882      +17     
==========================================
+ Hits          739      748       +9     
- Misses        126      134       +8     

Impacted Files	Coverage Δ
src/dual.jl	73.76% <65.21%> (-1.16%)	⬇️

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[andreasnoack]

There have been a couple of discussions of support for complex numbers. Might be good to address the issues here.

[devmotion]

Might be good to address the issues here.

Which issues do you have in mind?

Complex numbers (in intermediate calculations!) are already supported, e.g.,

julia> ForwardDiff.gradient([1.0, 2.0]) do x
           abs(log(x[1] + im * x[2]))
       end
2-element Vector{Float64}:
 -0.20597271407900614
  0.39695748015993854

julia> (x -> log(x[1] + im*x[2]))([ForwardDiff.Dual(1.0), ForwardDiff.Dual(2.0)])
Dual{Nothing}(0.8047189562170501) + Dual{Nothing}(1.1071487177940904)*im

This PR just fixes incorrect ForwardDiff-definitions of rules present in DiffRules but doesn't change the support for complex numbers in general. In particular the PR does not add support for complex numbers as inputs, it just makes it possible to use e.g. hankelh1 in a chain of calculations with real-valued inputs and outputs.

[andreasnoack]

Okay. Then let's get this one in.