[jmh530]

Rationale

With 80-bit real, erfc(106.625L) returns zero instead of approximately 1.892705840797887e-4940, a representable subnormal value. The existing cutoff uses the normal-range limit and discards part of the subnormal output range.

Replace that cutoff with a conservative zero bound derived from the floating-point format. In the extreme tail, evaluate z = exp(-x*x/2) and form (z * erfce(x)) * z, keeping intermediate factors normal and rounding into the subnormal range only at the final multiplication. Ordinary inputs retain the original calculation.

Feature request or issue tracking

Closes #11123.

Validation

• Error-function module tests passed in debug and optimized builds with Windows DMD 2.113.0 (80-bit real), Windows LDC 1.43.0 (64-bit real), and Linux LDC 1.36.0 (80-bit real).
• Focused public-API checks passed on all three configurations.
• Added regression tests for subnormal results, the transition to zero, negative arguments, and large finite inputs.
• Checked 2,001 boundary inputs per configuration against independent high-precision references, plus 2,001 additional fractional inputs and three adjacent inputs around the new path's threshold on each Windows compiler. All reference checks passed, with no sampled lost tails or incorrect nonzero results below half the smallest subnormal.

These are targeted checks, not a full Phobos test-suite run or a correctly-rounded accuracy guarantee. Reference tolerances allow the existing exponential's argument-dependent error. IEEE quadruple precision was not tested locally.

Performance

Optimized local microbenchmarks compared this commit with 6f9665c6c. Each sample made 100,000 calls through function pointers. Times below are medians of six alternating baseline/PR pairs after discarding the first pair, in nanoseconds per call.

Configuration	Input	Original	PR
Windows LDC, 64-bit real	2	17.24	18.27
Windows LDC, 64-bit real	26.625	130.18	65.28
Windows LDC, 64-bit real	27	130.62	66.10
Windows DMD, 80-bit real	106.5	361.64	352.84
Windows DMD, 80-bit real	106.625	107.50	513.03
Linux LDC, 80-bit real	106.5	175.57	180.00
Linux LDC, 80-bit real	106.625	8.00	302.56

Sampled ordinary-case timings varied approximately -7% to +6%. Windows LDC subnormal cases improved by about 2x, consistent with avoiding subnormal intermediate arithmetic. The substantial slowdown at 106.625 with 80-bit real replaces an incorrect immediate zero with a computed subnormal result. These short local measurements are not performance guarantees.

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. (Not applicable: this fixes erfc returns zero for representable subnormal results with 80-bit real #11123.)

LLM/AI disclosure

This PR was written entirely by an LLM/AI.

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