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Preserve small shape precision in gamma tail recovery #182

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jmh530 merged 1 commit into from
Oct 19, 2026
Merged

Preserve small shape precision in gamma tail recovery #182

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23 changes: 19 additions & 4 deletions source/mir/stat/distribution/gamma.d
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Original file line number Diff line number Diff line change
Expand Up @@ -906,10 +906,12 @@ T gammaSmallRatioLogCDF(T)(const T x, const T shape, const T scale)
pragma(inline, false);
import mir.math.common: log;
import std.mathspecial: logGamma;
// logGamma(1+a)=-EulerGamma*a+O(a*a). Avoid rounding 1+a to 1
// when this first-order term matters to a small complementary tail.
const T logNext = shape < T.epsilon
? -T(0.577215664901532860606512090082402431L) * shape
// Forming 1+a loses small shape bits even with an accurate logGamma.
// For a*a <= epsilon, the quadratic expansion omits O(a^3), at most
// O(epsilon*a), which is negligible relative to a*log(x/scale) here.
const T logNext = shape * shape <= T.epsilon
? shape * (-T(0.577215664901532860606512090082402431L)
+ shape * T(0.822467033424113218236207583323012595L))
: T(logGamma(1 + shape));
return shape * (log(x) - log(scale)) - logNext;
}
Expand Down Expand Up @@ -1039,3 +1041,16 @@ version(mir_stat_test)
static foreach (T; AliasSeq!(float, double, real))
assert(gammaPDF(T(1), size_t(1_000_000_000)) == 0);
}

// Preserve small shape bits before evaluating logGamma(1+shape). References
// use the exact binary double inputs and high-precision gamma arithmetic.
version(mir_stat_test)
@safe pure nothrow @nogc unittest
{
import std.math: nextUp;
import mir.math.common: approxEqual;
assert(approxEqual(gammaCCDF(nextUp(0.0), 1e-8),
7.438600896118198025322571949534157411e-6, 4e-15, 0.0));
assert(approxEqual(gammaCDF(nextUp(0.0), 1e-8),
0.999992561399103881801974677428050466, 4e-15, 0.0));
}
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