[jmh530]

Rationale

betaIncomplete can lose substantial relative accuracy near x = 1 when the second shape is tiny. For a = 1, b = 1e-20, and the largest double below one, the expected result is approximately 3.673680056967710e-19, but the existing implementation returns approximately 1.314235314789e-19: about 64% relative error, although the absolute error is small.

Reuse the stable complementary series and coordinate handling already used by betaIncompleteComplSmallX. Both entry points now use a shared implementation with a runtime nearOne parameter. The ordinary path remains in use for x <= 0.95.

The regression tests cover the exact shape-one identity through the last representable real below one, together with independent high-precision references for fractional and larger first shapes.

Feature request or issue tracking

Closes #11133.

Performance

Compared three implementations: upstream master, an initial fix retaining separate template instantiations, and the final shared implementation. Measurements used Windows DMD 2.113.0 (-O -inline -release) and LDC 1.43.0 (-O3 -enable-inlining -release), without fast-math.

Each dataset contains 128 varied inputs. The benchmark runs all three implementations in the same process through equivalent function pointers, rotates their measurement order, and pins execution to one logical CPU. Results below are median nanoseconds per call across three process runs, with nine retained batches per run after warmup.

Compiler	Dataset	Original	Initial fix	Shared implementation
LDC	Reported tiny-shape tail	2506.8	16.7	16.8
LDC	Fractional first shape, tiny tail	2509.8	192.1	191.6
LDC	Large first shape, tiny tail	2599.8	112.3	112.8
LDC	Ordinary near-one inputs	33.4	34.5	33.2
LDC	Inputs around the dispatch boundary	1058.8	1057.0	1064.0
LDC	Ordinary power series	37.3	37.9	37.0
LDC	Ordinary continued fraction	289.9	280.3	268.8
LDC	Complementary CDF, tiny input	57.6	57.9	57.7
LDC	Complementary CDF, ordinary input	35.4	35.4	35.4
LDC	Inverse CDF, central probabilities	580.7	582.3	589.8
LDC	Inverse CDF, upper probabilities	538.9	544.3	542.5
DMD	Reported tiny-shape tail	26882.3	142.8	144.2
DMD	Fractional first shape, tiny tail	26821.6	1372.3	1389.8
DMD	Large first shape, tiny tail	26688.1	1027.1	1029.5
DMD	Ordinary near-one inputs	421.9	413.2	426.6
DMD	Inputs around the dispatch boundary	8241.0	7879.1	7599.0
DMD	Ordinary power series	447.9	431.0	433.3
DMD	Ordinary continued fraction	2047.7	2003.4	1980.3
DMD	Complementary CDF, tiny input	492.0	501.2	498.0
DMD	Complementary CDF, ordinary input	442.6	452.1	444.6
DMD	Inverse CDF, central probabilities	3747.1	3797.1	3787.8
DMD	Inverse CDF, upper probabilities	3580.3	3534.8	3505.2

Both fixes avoid the expensive, inaccurate continued-fraction calculation in the affected cases. The shared implementation performs similarly to the initial fix: median paired timing changes across datasets range from −2.7% to +0.6% with LDC and −0.6% to +2.2% with DMD. These are local microbenchmarks, and small differences vary between runs.

Sharing the implementation also reduces generated code size. Summed code-section sizes for the optimized module object, excluding data and metadata, are:

Compiler	Original	Initial fix	Shared implementation
LDC	38,967 bytes	40,262 bytes	38,348 bytes
DMD	50,573 bytes	51,234 bytes	48,888 bytes

Validation

• Full std.internal.math.gammafunction module tests pass with DMD.
• Focused regression tests pass with LDC.
• A separate 294-case sweep per compiler passes against 100-digit mpmath references, covering both sides of the dispatch boundary, tiny second shapes, and fractional through large first shapes. The comparison allows relative error of 1e-11 or one result ULP for rounded subnormal results.
• The original implementation fails 24 DMD cases and 28 LDC cases in that sweep; the updated implementation passes all cases.
• Benchmark result checksums agree between the initial fix and shared implementation.

The full Phobos suite was not run. Full-module LDC testing has a previously reproduced baseline assertion failure in igammaTemmeLarge; LDC validation here used focused tests. A platform with 113-bit real was not available.

Pre-review checklist

• I have performed a self-review of my code.
• If my PR fixes a bug or introduces a new feature, I have added thorough tests.
• If my changes are non-trivial and do not concern a reported issue, I have added a changelog entry.

The changelog condition does not apply because this fixes a reported issue.

LLM/AI disclosure

This PR was written with the assistance of an LLM/AI.

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