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Preserve small shape precision in gamma tail recovery - #182
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When x/scale is subnormal, the gamma CDF and CCDF recover the lower probability in logarithmic form. Forming 1 + shape for logGamma loses small shape bits even if logGamma itself is accurate. At shape 1e-8 and the smallest positive double argument, this rounding alone causes about 4.7e-12 relative error in the complementary probability. Extend the existing small-shape approximation to include the quadratic term when shape squared is at most the type's epsilon. The omitted cubic term is negligible relative to shape * log(x/scale) in this region. Keep the change within the subnormal-ratio recovery path. Add independent high-precision CDF and CCDF references for the affected case. AI-Assisted-By: Codex (GPT-6 Astra Medium)
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Improve gamma CDF/CCDF accuracy when x/scale is subnormal and the shape is small. The existing logarithmic recovery forms
1 + shapebefore callinglogGamma, losing shape bits even iflogGammaitself is accurate. At shape1e-8and the smallest positive double argument, this addition alone causes approximately4.7e-12relative error in the complementary probability.Extend the existing small-shape approximation with the quadratic term of
logGamma(1 + shape)whenshape² <= epsilon. The omitted cubic term is negligible relative toshape * log(x/scale)in this region. The change stays within the existing subnormal-ratio recovery path; ordinary CDF/CCDF evaluation is unchanged.Add independent high-precision CDF and CCDF regression references. This addresses argument rounding in mir-stat separately from the Phobos
logGammaevaluation issue reported in dlang/phobos#11128.Validation:
AI-Assisted-By: Codex (GPT-6 Astra Medium)