[devmotion]

Alternative to #862, based on JuliaMath/SpecialFunctions.jl#568.

#862 avoids the overflow of exp(-x) * x^(a - 1) / gamma(a) by computing the x-derivative of gamma_inc in log space, but the cancellation in the exponent makes it inaccurate for large a (relative error ~7e-10 at a = 1e6 and ~1e-5 at a = 1e10 in Float64; ~16% at a = 1e6 in Float32).

SpecialFunctions#568 adds a public gamma_inc_dx(a, x) = x^(a - 1) e^(-x) / Γ(a) (the x-derivative of gamma_inc(a, x)[1]), computed without cancellation, together with internal helpers for its derivatives in a and x. This PR:

• uses SpecialFunctions.gamma_inc_dx for the derivative of gamma_inc(a, x, ind) with respect to x, if it is available, and falls back to the current formula otherwise (to be removed once SpecialFunctions >= 2.10 is required);
• defines Dual rules for gamma_inc_dx in both arguments (via @define_binary_dual_op, so mixed tags work), which makes nested derivatives of gamma_inc in x accurate, including at x = 0 and at the poles of digamma(a);
• throws an ArgumentError for derivatives of gamma_inc with respect to a, instead of a MethodError from SpecialFunctions internals; Duals with zero partials for a are supported.

Tests compare first derivatives (Float32 and Float64), gradients, Hessians and nested mixed derivatives against BigFloat references, and cover NaN-safe mode. Tests that require gamma_inc_dx are guarded on isdefined(SpecialFunctions, :gamma_inc_dx).

Dependencies:

• Add gamma_inc_dx and fix ChainRules for gamma_inc and loggamma(a, x) JuliaMath/SpecialFunctions.jl#568 (until it is released, CI exercises only the fallback path);
• Define floatmin, floatmax, maxintfloat and issubnormal for Dual instances #855: with SpecialFunctions >= 2.10, nested derivatives in x call floatmin on a Dual instance.

Locally (with a stand-in for #855), test/DualTest.jl and test/DerivativeTest.jl pass against the SpecialFunctions#568 branch and against SpecialFunctions 2.9.1, with NaN-safe mode on and off, and there are no new method ambiguities.

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[codecov]

Codecov Report

❌ Patch coverage is 91.66667% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 91.20%. Comparing base (8c66edc) to head (36121d3).

Files with missing lines	Patch %	Lines
src/dual.jl	91.66%	1 Missing ⚠️

Additional details and impacted files

@@                     Coverage Diff                     @@
##           dw/dual-float-constants     #871      +/-   ##
===========================================================
- Coverage                    91.22%   91.20%   -0.02%     
===========================================================
  Files                           11       11              
  Lines                         1083     1092       +9     
===========================================================
+ Hits                           988      996       +8     
- Misses                          95       96       +1     

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