fix: replace the erfc approximation with libm, for free

#41 asked whether the Numerical Recipes `erfcc` approximation — 1.2e-7
relative, and the binding accuracy constraint on the whole crate — was
worth replacing, given it sits in the inference hot loop. It is, and it
costs nothing.

Measured, against an independent incomplete-gamma reference:

    range        previous (NR)        libm
    [-3, 0]            7.95e-8     2.15e-14
    [0, 0.5]           8.69e-8     4.70e-14
    [0.5, 2]           9.38e-8     1.24e-12
    [2, 6]             1.04e-7     5.20e-14
    [6, 26]            1.07e-7     1.75e-13

    erfc(0)          1.00000003          1.0  (exactly)
    |erfc(z)+erfc(-z)-2|  6.00e-8     2.22e-16

Performance, on `benches/batch.rs`: change [-3.34% +2.26%], p = 0.89 —
no change detected.

That result is counterintuitive, because libm's erfc is 1.65x slower
when swept uniformly over [-2.5, 2.5]. The sweep was the wrong input
distribution. Capturing the arguments inference actually passes:

    |x|<0.5    96.16%
    0.5-0.84    2.05%
    0.84-1.25   1.24%
    1.25-2      0.54%
    2-6         0.00%

98% fall below 0.84375, which is exactly where FDLIBM skips the
exponential entirely — while the NR form always pays for one. On the
real trace libm is the faster of the two (3.23 vs 3.78 ns/call).

An ad-hoc `Instant` harness reported a 16% end-to-end speedup; that was
an artifact of its own setup allocating and leaking per run, and
criterion's verdict of "no change" is the one to believe.

What it bought:

- `compute_margin` against exact quantiles: 8.4e-8 -> 1.7e-16.
- `cdf(mu, mu, sigma)` is now exactly 0.5; it was 1.5e-8 out.
- `sf + cdf` sums to one within a ULP, from 3e-8.
- `erfcx`'s two branches now agree to round-off across the crossover
  rather than to 1e-7, so the log-space evidence path and the linear one
  are consistent.
- Ten test tolerances tightened from 1e-6 to 1e-13..1e-15, and the
  prediction floor is now the integrator's rather than `cdf`'s.

Five goldens moved, by 2.4e-9 to 6e-7 — the magnitude of the removed
error, and `test_env_ttt`'s mu still rounds to the same six decimals.
Re-recorded with more digits so future drift stays visible. Verified as
movement toward truth per the goldens policy: every value now derives
from a primitive checked against an independent reference and satisfying
the exact identities, which the previous one did not.

Adds `libm` — zero transitive dependencies, rust-lang maintained.

Closes #41

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ
This commit is contained in:
2026-09-07 22:25:00 +02:00
co-authored by Claude Opus 5
parent 7e289ee834
commit 3dd659307a
5 changed files with 50 additions and 52 deletions
+1
View File
@@ -51,6 +51,7 @@ harness = false
[dependencies]
approx = { version = "0.5.1", optional = true }
libm = "0.2.16"
rayon = { version = "1", optional = true }
smallvec = "1"