A trait-based system to handle Wirtinger derivatives

[YingboMa]

While writing ForwardDiff2.jl I have encountered a problem with nested differentiation with the Wirtinger derivative.

Say if I have

x = Dual(Dual(1+1im, 1+0im), 1+0im)

how can I make hypot(value(x)) dispatch to the right place if there is

@scalar_rule(hypot(x::Real), sign(x))
@scalar_rule(hypot(x::Complex), Wirtinger(x' / 2Ω, x / 2Ω))

Neither Dual <: Number or Dual <: Real would work.

https://github.com/JuliaDiff/ChainRules.jl/blob/146d031235b28265772b8c0b98ed54b812c39316/src/rulesets/Base/base.jl#L59-L60

Maybe we need to have a trait-based system to handle Wirtinger derivatives?

[nickrobinson251]

Do you have some initial thoughts on what the traits would be?

[oxinabox]

cf #40, cf #54 Wirtinger stuff is hard in general.

In general you basically want Dual{T} <: T I think
Which is kinda sad that julia doesn't let us have.

But as such, it might actually be to have a trait on all inputs for hmmm semantic_type
So that we could then have semantic_type(x::T) where T = T
and semantic_type(x::Dual{T}) where T = T
Though also isn't this to be solved/already solve in ForwardDiff2?
Arn't you using Cassette to allow calls to f on Dual{Float64} to hit f(::Float64)'s code?
So would the same trick not apply here also?
Call to frule on Dual{Complex} to hit frule(f, Complex} ?

[YingboMa]

Maybe something like

frule(f, args...) = frule(semantic_type(args), f, args...)
frule(::Type{<:Complex}, f, args...)
frule(::Type{<:Real}, f, args...)

[simeonschaub]

The type can't be the first argument, since we want to allow frules for constructors too, but it could be a keyword argument. Although that's definitely one of the smaller issues with Wirtinger right now 😉