Hi all!

I am trying to `ForwardDiff`

a function that is defined by solving a non-linear equation that depends on parameters. The code below reproduces the error. It first redefines `exp`

by solving `log(x) - a = 0`

```
using Roots, ForwardDiff
f( x, a ) = log( x ) - a
∂f_x( x, a ) = ForwardDiff.derivative( t -> f( t, a ), x )
mexp( a ) = find_zero(
(
x -> f( x, a ),
x -> ∂f_x( x, a )
),
1, Roots.Newton()
)
# Test
mexp( log( π ) ) ≈ π # true
# Differentiate
dmexp(a) = ForwardDiff.derivative( mexp, a )
dmexp( 1 ) # error!
```

Full stacktrace:

```
ERROR: MethodError: no method matching Float64(::ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Float64, 1})
Closest candidates are:
(::Type{T})(::Real, ::RoundingMode) where T<:AbstractFloat at rounding.jl:200
(::Type{T})(::T) where T<:Number at boot.jl:772
(::Type{T})(::AbstractChar) where T<:Union{AbstractChar, Number} at char.jl:50
...
Stacktrace:
[1] convert(#unused#::Type{Float64}, x::ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Float64, 1})
@ Base ./number.jl:7
[2] init_state(M::Roots.Newton, F::Roots.Callable_Function{Val{2}, Val{true}, Tuple{var"#11#13"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}, var"#12#14"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}}, Nothing}, x::Int64)
@ Roots ~/.julia/packages/Roots/LYDH4/src/Derivative/newton.jl:63
[3] init(𝑭𝑿::ZeroProblem{Tuple{var"#11#13"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}, var"#12#14"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}}, Int64}, M::Roots.Newton, p′::Nothing; p::Nothing, verbose::Bool, tracks::Roots.NullTracks, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
@ Roots ~/.julia/packages/Roots/LYDH4/src/find_zero.jl:283
[4] solve(𝑭𝑿::ZeroProblem{Tuple{var"#11#13"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}, var"#12#14"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}}, Int64}, M::Roots.Newton, p::Nothing; verbose::Bool, kwargs::Base.Pairs{Symbol, Union{Nothing, Roots.NullTracks}, Tuple{Symbol, Symbol}, NamedTuple{(:p, :tracks), Tuple{Nothing, Roots.NullTracks}}})
@ Roots ~/.julia/packages/Roots/LYDH4/src/find_zero.jl:469
[5] find_zero(f::Tuple{var"#11#13"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}, var"#12#14"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}}, x0::Int64, M::Roots.Newton; p::Nothing, verbose::Bool, tracks::Roots.NullTracks, kwargs::Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}})
@ Roots ~/.julia/packages/Roots/LYDH4/src/find_zero.jl:215
[6] find_zero(f::Tuple{var"#11#13"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}, var"#12#14"{ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1}}}, x0::Int64, M::Roots.Newton)
@ Roots ~/.julia/packages/Roots/LYDH4/src/find_zero.jl:206
[7] mexp(a::ForwardDiff.Dual{ForwardDiff.Tag{typeof(mexp), Int64}, Int64, 1})
@ Main ~/Dropbox/Research/dynamic_games/hvb/tests.jl:21
[8] derivative
@ ~/.julia/packages/ForwardDiff/pDtsf/src/derivative.jl:14 [inlined]
[9] dmexp(a::Int64)
@ Main ~/Dropbox/Research/dynamic_games/hvb/tests.jl:31
[10] top-level scope
@ ~/Dropbox/Research/dynamic_games/hvb/tests.jl:32
```

Output of `versioninfo`

:

```
Julia Version 1.8.0
Commit 5544a0fab76 (2022-08-17 13:38 UTC)
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 20 × 12th Gen Intel(R) Core(TM) i9-12900HK
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-13.0.1 (ORCJIT, goldmont)
Threads: 10 on 20 virtual cores
Environment:
JULIA_EDITOR = code
JULIA_NUM_THREADS = 10
```

`Roots v2.0.4`

and `ForwardDiff v0.10.32`

.

Many thanks for your help!