Bug(Mooncake): jacobian for real→complex functions returns Float64 instead of ComplexF64

Description

When using AutoMooncake() with functions that have real input and complex output, jacobian returns a Float64 matrix containing only partial information instead of the expected ComplexF64 matrix. The imaginary components of the derivatives are lost.

Note: This is likely a Mooncake limitation that should be upstreamed, as the native Mooncake API exhibits the same behavior.

MWE

using DifferentiationInterface
using Mooncake: Mooncake
using ADTypes: AutoMooncake

backend = AutoMooncake(; config=nothing)

# Real input → Complex output
f(x) = [complex(x[1], x[2]), complex(x[2], x[1])]
x = [1.0, 2.0]

J = jacobian(f, backend, x)

Output:

2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

Expected:

2×2 Matrix{ComplexF64}:
 1.0+0.0im  0.0+1.0im
 0.0+1.0im  1.0+0.0im

The jacobian should have imaginary components (the derivative of x[2] appearing in the imaginary part of y[1] is im, not 0).

Native Mooncake API Comparison

The native Mooncake API returns the same incomplete result:

using Mooncake

f(x) = [complex(x[1], x[2]), complex(x[2], x[1])]
x = [1.0, 2.0]

cache = Mooncake.prepare_pullback_cache(f, x)
dy = ComplexF64[1.0, 0.0]

y, (_, dx) = Mooncake.value_and_pullback!!(cache, dy, f, x)
# dx = [1.0, 0.0]  (Float64, not ComplexF64)
# Expected: [1.0, im] or similar complex representation

Mooncake's tangent type appears to match the input type (Float64) rather than accounting for the complex output, losing the imaginary gradient components.

Enzyme Comparison (Correct Output)

Enzyme handles this case correctly:

using DifferentiationInterface
using Enzyme: Enzyme
using ADTypes: AutoEnzyme

f(x) = [complex(x[1], x[2]), complex(x[2], x[1])]
x = [1.0, 2.0]

J = jacobian(f, AutoEnzyme(), x)
# 2×2 Matrix{ComplexF64}:
#  1.0+0.0im  0.0+1.0im
#  0.0+1.0im  1.0+0.0im

Interesting Note

Mooncake's derivative for scalar real→complex works correctly:

g(t) = complex(t, 2*t)
d = derivative(g, AutoMooncake(; config=nothing), 1.0)
# Returns: 1.0 + 2.0im ✓

Only jacobian (which uses pullbacks internally) exhibits this issue.

Recommendation

This should likely be reported upstream to Mooncake, as DI correctly calls the Mooncake API which returns limited results. Possible solutions:

1. Mooncake could support complex tangents for real inputs when output is complex
2. DI could document this as a known Mooncake limitation
3. DI could detect this case and warn users

Environment

• Julia 1.12.5
• DifferentiationInterface v0.7.18
• Mooncake v0.5.28

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🤖 I am a robot. This is an experiment in agentic bug-catching under the supervision of @adrhill and @gdalle (#1008). Contents may be hallucinated.

Reproduce the real-input/complex-output MWE with jacobian and AutoMooncake, then compare it with derivative and Mooncake's prepare_pullback_cache/value_and_pullback!! entry points. Trace whether DifferentiationInterface transforms or merely returns Mooncake's tangent, and determine whether a DI change is possible or the limitation belongs upstream. Done means a confirmed fix, warning, or documented limitation with regression coverage if this repository owns the behavior.