detail::inverse_beta_i: absolute 1e-8 stopping rule + [1e-8, 1−1e-8] clamp make Beta quantiles unusable beyond p ~ 1e-8

[OldCrow]

Found during v2.4.0 #56 (FDistribution), documented in-tree, deliberately not patched there

detail::inverse_beta_i stops on an absolute residual of 1e-8 from a start clamped into [1e-8, 1−1e-8], so for target probabilities below ~1e-8 (either tail) it can simply return its clamp value. BetaDistribution::getQuantile inherits this directly — tail quantiles beyond p ~ 1e-8 are not merely inaccurate but pinned.

Blast radius: Beta only. The v2.4.0 FDistribution deliberately avoided it — its quantile is a safeguarded log-x bisection against the tail-steered CDF/survival (see src/fisher_f.cpp), measured to 3e-12 relative at p = 1e-12; that solver is a working in-tree reference for the fix. StudentT's CDF machinery is separate (verify at fix time whether its quantile shares the pattern).

Contract relevance: #104 finite-best-effort — never NaN is honoured, but 'finite yet pinned to the clamp' needs either fixing or explicit per-family documentation, and fixing is clearly achievable (F does it against the same beta_i core).

Suggested remedy path

• Verify at v2.5.0 (corvus adoption) scoping whether the corvus incomplete-beta family includes an inverse; if so, close via the adoption swap (the accuracy(special-functions): beta_i loses ~1e-6 absolute at b >= 1e9 -- lgamma cancellation in the log-beta prefix #126 pattern — and note accuracy(special-functions): beta_i loses ~1e-6 absolute at b >= 1e9 -- lgamma cancellation in the log-beta prefix #126's beta_i large-b cancellation interacts with the same core).
• If not absorbed: replace with the F-style safeguarded solver against the small-side-steered beta_i, with a relative stopping rule.
• Regression gate shown to fail first: reference quantiles at p = 1e-10/1e-12 for a few (a,b), plus round-trip through the steered CDF.

🤖 Generated with Claude Code