[jmh530]

Fixes #11108.

For small positive shape parameters and x < 1, gammaIncompleteCompl computes Q by subtracting P from one. Since P is close to one, cancellation can overwhelm the upper tail. For example, gammaIncompleteCompl(1e-20L, 0.5L) returns approximately 2.92735e-18 on Windows DMD instead of 5.597735947761608e-21.

Evaluate Q directly for a <= 1/8 and x < 1 using the series in DLMF 8.7.3. Use expm1 and evaluate logGamma(1+a)/a without forming 1+a. For tiny arguments, factor out a to preserve representable subnormal results. Other parameter ranges retain their existing algorithms.

Validation

• DMD 2.113.0: std.internal.math.gammafunction and std.mathspecial unit tests pass, with and without optimization.
• LDC 1.43.0: the new regression tests pass with -O2 --noasm. The complete module suite still encounters an existing igammaTemmeLarge assertion (original line 295), also observed on the baseline.
• An independent 72-case comparison against references calculated with 100 decimal digits passes on both compilers, including branch boundaries. Additional regression tests check representable subnormal tails.
• Tested Windows real with 64 mantissa bits under DMD and 53 under LDC; 113-bit real was not tested.

Performance

Compared with original commit 1877367354fadf001365292d74e39ffd15c21bbc, in ns/call (original → patched):

Inputs	DMD	LDC
a = 0.01, x = 0.1..0.9	837.6 → 655.9 (-21.7%)	77.6 → 60.1 (-22.6%)
a = 1e-20, x = 0.1..0.9	806.3 → 400.6 (-50.3%)	76.0 → 38.7 (-49.1%)
a = 1e-20, x around 1e-100	783.4 → 288.1 (-63.2%)	56.2 → 15.2 (-73.0%)
a = 0.125, x = 0.1..0.9	996.8 → 710.0 (-28.8%)	69.8 → 60.5 (-13.4%)
Subnormal a, x = 0.5	10804.2 → 5501.3 (-49.1%)	589.0 → 150.0 (-74.5%)
Unchanged: a = 2.5, x = 0.1..0.9	840.4 → 837.4 (-0.4%)	70.1 → 69.4 (-1.0%)
Unchanged: a = 1e-20, x = 1..5	3282.4 → 3133.7 (-4.5%)	189.1 → 191.9 (+1.5%)

Measured on Windows x64, Intel Core Ultra 7 258V, using DMD -O -inline and LDC -O2 --noasm. Both implementations run in one process pinned to one logical processor, with 512 input pairs repeated 100 times per sample. Values are medians of ten samples after one warm-up, alternating execution order.

The direct series avoids the general logGamma evaluation. The tiny-shape shortcut also avoids rational-approximation arithmetic on subnormal operands. All affected benchmark cases improved.

These are local microbenchmarks, and the original results in the affected range are inaccurate. The small differences in unchanged paths, including the 1.5% LDC slowdown for x >= 1, do not establish a systematic performance change.

This PR was written entirely by an LLM/AI.

AI-Assisted-By: Codex (GPT-6 Astra Medium)