Error for Jacobian of in-place function that internally creates complex numbers

Hi there,

MWE

newton_f(x, p) = x^p[1] - 1.0
newton_df(x, p)= p[1]*x^(p[1] - 1.0)
function newton_map(dz, z, p, n)
    z1 = z[1] + im*z[2]
    dz1 = newton_f(z1, p)/newton_df(z1, p)
    z1 = z1 - dz1
    dz[1] = real(z1)
    dz[2] = imag(z1)
    return
end

import ForwardDiff
s = [0.1, 0.2]
p = [3.0]
dum = deepcopy(s)
inplace_f_2args = (y, x) -> newton_map(y, x, p, 0)
cfg = ForwardDiff.JacobianConfig(inplace_f_2args, dum, s)
jac! = (J, u, p, t) -> ForwardDiff.jacobian!(
    J, inplace_f_2args, dum, u, cfg, Val{false}()
)

jac!(zeros(2, 2), s, p, 0)

        
ERROR: LoadError: MethodError: no method matching Int64(::ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2})
Closest candidates are:
  (::Type{T})(::T) where T<:Number at boot.jl:760
  (::Type{T})(::AbstractChar) where T<:Union{AbstractChar, Number} at char.jl:50
  (::Type{T})(::BigInt) where T<:Union{Int128, Int16, Int32, Int64, Int8} at gmp.jl:356
  ...
Stacktrace:
  [1] convert(#unused#::Type{Int64}, x::ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2})
    @ Base .\number.jl:7
  [2] _cpow(z::Complex{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}, p::ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2})
    @ Base .\complex.jl:740
  [3] ^
    @ .\complex.jl:809 [inlined]
  [4] ^
    @ .\complex.jl:824 [inlined]
  [5] newton_f
    @ c:\Users\datse\.julia\dev\DynamicalSystemsBase\test\dynsys_types.jl:111 [inlined]
  [6] newton_map(dz::Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}, z::Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}, p::Vector{Float64}, 
n::Int64)
    @ Main c:\Users\datse\.julia\dev\DynamicalSystemsBase\test\dynsys_types.jl:115
  [7] (::var"#7#8")(y::Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}, x::Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}})
    @ Main c:\Users\datse\.julia\dev\DynamicalSystemsBase\test\dynsys_types.jl:135
  [8] vector_mode_dual_eval
    @ C:\Users\datse\.julia\packages\ForwardDiff\QOqCN\src\apiutils.jl:44 [inlined]
  [9] vector_mode_jacobian!(result::Matrix{Float64}, f!::var"#7#8", y::Vector{Float64}, x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}}})
    @ ForwardDiff C:\Users\datse\.julia\packages\ForwardDiff\QOqCN\src\jacobian.jl:172
 [10] jacobian!(result::Matrix{Float64}, f!::Function, y::Vector{Float64}, x::Vector{Float64}, cfg::ForwardDiff.JacobianConfig{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2, Tuple{Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#7#8", Float64}, Float64, 2}}}}, ::Val{false})
    @ ForwardDiff C:\Users\datse\.julia\packages\ForwardDiff\QOqCN\src\jacobian.jl:78

I need to create the Jacobian in this format of (J, x, p, t), which is why all these anonymous functions got there. Making a function that does not internally make complex numbers works with the exact same syntax as above.

Reproduce the MWE from test/dynsys_types.jl, then inspect ForwardDiff/src/apiutils.jl and src/jacobian.jl around vector-mode Jacobian evaluation. Compare this with the complex-power conversion failure shown in the Julia stack trace, and determine the expected behavior for in-place functions that create complex values. Done means the behavior is covered by a regression test or the limitation is clearly documented.