[jmh530]

Rationale

expm1(double) can lose one smallest-subnormal unit—a tiny absolute error that can represent a substantial relative error for subnormal inputs. For example, an input of −5 minimum-subnormal units produces −4 units instead of −5. The smallest positive and negative inputs produce signed zero.

The portable double-precision approximation effectively halves these inputs before doubling the rounded intermediate. That intermediate rounding discards information that the final multiplication cannot recover.

Extend the existing zero shortcut to return x when |x| < epsilon / 4, where the correction in expm1(x) = x + x²/2 + … rounds away under round-to-nearest. This preserves signed zero and avoids the problematic intermediate arithmetic. The wider-format algorithms remain unchanged.

Feature request or issue tracking

Closes #11140.

Validation

• Full exponential-module unittests pass with DMD 2.113.0 and LDC 1.43.0 on Windows, using -preview=dip1000. LDC testing disables assembly to exercise the portable implementation.
• New regressions cover both signs, signed zero, subnormal inputs, the normal/subnormal boundary, and values immediately around the shortcut cutoff.
• The new regression fails against the original implementation.
• A separate LDC sweep of 20,000 signed subnormal inputs passes with the fix; the original implementation fails 10,000 cases.

Module testing used the source under a temporary module name with installed runtime libraries. The full Phobos suite was not run. The shortcut is justified for round-to-nearest; this change does not establish correctly rounded behavior under directed rounding modes.

Performance

A local paired benchmark uses 128 varied inputs per dataset, alternates measurement order, and reports median nanoseconds per call over six batches after warmup. DMD uses -O -inline -release; LDC uses -O3 -release --noasm.

Compiler	Inputs	Before	After
DMD	Subnormal	123.0 ns	3.9 ns
DMD	Tiny normal	27.7 ns	3.9 ns
DMD	Ordinary approximation range	28.1 ns	27.3 ns
DMD	Wider range	56.6 ns	55.1 ns
LDC	Subnormal	144.5 ns	1.7 ns
LDC	Tiny normal	1.9 ns	1.7 ns
LDC	Ordinary approximation range	1.9 ns	1.9 ns
LDC	Wider range	7.7 ns	8.1 ns

Tiny-input calls benefit from bypassing the approximation, while ordinary-input timings remain broadly similar. These are local microbenchmark results; small differences should not be interpreted as general 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.

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)