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
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@@ -1254,7 +1254,7 @@ mod tests {
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);
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assert_ulps_eq!(
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p[1][0],
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Gaussian::from_ms(19.287197, 7.243465),
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Gaussian::from_ms(19.287198285, 7.243465848),
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epsilon = 1e-6
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);
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assert_ulps_eq!(
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@@ -1314,7 +1314,7 @@ mod tests {
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assert_ulps_eq!(
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p[0][0],
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Gaussian::from_ms(31.674697, 7.501180),
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Gaussian::from_ms(31.674698083, 7.501180037),
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epsilon = 1e-6
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);
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assert_ulps_eq!(
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