Fix flaky gamma_inc partials test

[ChrisRackauckas-Claude]

Summary

The gamma_inc partials assertions in test/DualTest.jl fail sporadically on CI —
roughly one job per full-matrix run, on arbitrary OS/Julia-version combinations (e.g.
"Julia lts - windows-latest" in the 2026-06-22 master run, "Julia 1 - ubuntu-latest" in
the 2026-06-07 run), always of the form

partials(pq[i]) ≈ PARTIALS * Calculus.derivative(x -> gamma_inc(a, x, ind...)[i], 1 + PRIMAL) rtol=tol

with V === Float32 and random a/PRIMAL narrowly missing rtol = 5f-4.

Diagnosis

The reference, not ForwardDiff, is at fault, for two reasons:

1. For V === Float32, Calculus.derivative central-differences the Float32
   evaluation of gamma_inc, whose finite-difference error is of the same order as the
   5f-4 tolerance — so random draws sporadically cross it (~4% of draws, measured).
2. For ind = 1 / ind = 2, gamma_inc deliberately evaluates a reduced-accuracy
   approximation (~14 / ~6 significant digits). Finite-differencing a function with
   ~1e-6 intrinsic noise amplifies that noise by 1/h; the measured reference error
   reaches 2000–3000% of the true derivative in the worst draws. The existing
   tol^(1/2^ind) loosening (up to rtol ≈ 0.15) was compensating for this.

In every sampled failing draw, ForwardDiff's Float32 derivative was within 4e-8
relative of the true derivative (Float64 dual evaluation), while the finite-difference
reference was off by 2%–3000%.

Fix

ForwardDiff's rule (src/dual.jl) is the analytic exp(-x)·x^(a-1)/Γ(a), independent
of ind. So compare the partials against a finite difference of the full-accuracy
(ind = 0) Float64 evaluation, at the base tolerance — the ind-dependent
loosening remains only on the value comparison, where it belongs (values genuinely
differ by the requested approximation accuracy). Float64(1 + PRIMAL) converts after
the V-precision addition, so the reference is evaluated at exactly the primal of the
dual input.

Measured over 300000 random (a, PRIMAL, PARTIALS) draws × all ind variants
(2.4M assertions): old reference fails ~4% of draws; new reference fails 0, even at
the base (unloosened) tolerance.

test/DualTest.jl passes locally (Julia 1.12, --check-bounds=yes).

---

Note

Opened as a draft by an agent on behalf of @ChrisRackauckas. Please ignore
until reviewed by @ChrisRackauckas.

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 90.74%. Comparing base (090ddbb) to head (cb0e8ac).

Additional details and impacted files

@@           Coverage Diff           @@
##           master     #822   +/-   ##
=======================================
  Coverage   90.74%   90.74%           
=======================================
  Files          11       11           
  Lines        1070     1070           
=======================================
  Hits          971      971           
  Misses         99       99           

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[ChrisRackauckas-Claude]

A fresh occurrence of this flake just hit CI on #823 ("Julia min-patch - macOS - NaN-safe disabled", 2026-07-03): Partials(0.0029103528, 0.0038809243, 0.002506475) ≈ Partials(0.0029121826, 0.0038833646, 0.0025080508) at rtol=0.0005 — a ~6e-4 relative miss of the Float32 finite-difference reference, exactly the failure mode this PR removes.