feat: support forward-mode Mooncake [experimental]

[gdalle]

Warning

Do not merge, this is just for testing.

Adds support for Mooncake in forward mode, through the AutoMooncakeForward backend object.

Relies on:

• Start forward mode AD chalk-lab/Mooncake.jl#389
• feat: Add forward mode Mooncake SciML/ADTypes.jl#110

To try it out, install the following branches first:

using Pkg
Pkg.add(; url="https://github.com/gdalle/Mooncake.jl", rev="gd/forward")
Pkg.add(; url="https://github.com/SciML/ADTypes.jl", rev="gd/forward_mooncake")
Pkg.add(;
    url="https://github.com/JuliaDiff/DifferentiationInterface.jl",
    rev="gd/forward_mooncake",
)

This currently passes all the tests that I subject reverse-mode Mooncake to.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 9.20%. Comparing base (8c234e6) to head (6df2ac6).

❗ There is a different number of reports uploaded between BASE (8c234e6) and HEAD (6df2ac6). Click for more details.

HEAD has 28 uploads less than BASE

Flag	BASE (8c234e6)	HEAD (6df2ac6)
DI	22	0
DIT	7	1

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #813       +/-   ##
==========================================
- Coverage   88.42%   9.20%   -79.23%     
==========================================
  Files         128      20      -108     
  Lines        7726    2096     -5630     
==========================================
- Hits         6832     193     -6639     
- Misses        894    1903     +1009     

Flag	Coverage Δ
DI	?
DIT	9.20% <ø> (-86.41%)	⬇️

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