Module probability 

crate::probability → distributional primitives now in gam-math.

Functions

erfcx_nonnegative

Scaled complementary error function erfcx(x) = exp(x²) · erfc(x), specialized to x ≥ 0. Returns 1.0 for x ≤ 0 and 0.0 for x = +∞. For 0 < x < 26 uses the direct exp(x²)·erfc(x) form; beyond that the (otherwise overflowing) exp(x²) is replaced by a 4-term asymptotic expansion (1/(x√π))·(1 − 1/(2x²) + 3/(4x⁴) − …), keeping relative accuracy near machine epsilon. The non-negative restriction lets the caller skip the reflection identity.

log1mexp_positive

Computes log(1 - exp(-a)) for a >= 0 without cancellation.

normal_cdf

Standard normal CDF Phi(x) evaluated via the exact special-function identity

normal_logcdf

Numerically stable ln Φ(x) for the standard normal CDF. For x ≥ 0 computes ln(Φ(x)) directly with a small floor against underflow; for x < 0 rewrites ln Φ(x) = −u² + ln(½·erfcx(u)), u = −x/√2, which preserves digits all the way into the deep left tail (no ln(0)). Returns ±∞ and NaN at the corresponding inputs.

normal_logsf

Numerically stable ln(1 − Φ(x)) = ln Φ(−x) for the standard normal survival function. Delegates to normal_logcdf(-x) so the deep-right tail benefits from the same erfcx-based representation.

normal_pdf

Standard normal PDF phi(x).

signed_log_sum_exp

Numerically stable signed log-sum-exp. Given pairs (log|aⱼ|, sign(aⱼ)) (with signs[j] ∈ {−1, 0, +1}), returns (log|S|, sign(S)) for S = Σⱼ signs[j]·exp(log_mags[j]). Positive and negative magnitudes are reduced separately with the standard log-sum-exp trick (subtract the max, sum, log, add back); the two partial sums are then combined via log(|p − n|) = max(log p, log n) + log1mexp(|log p − log n|), preserving accuracy even when p ≈ n (catastrophic cancellation regime). When all signs are zero or all magnitudes are −∞, returns (NEG_INFINITY, 0.0).

signed_probit_logcdf_and_mills_ratio

Joint evaluation of ln Φ(x) and the Mills-ratio analogue φ(x) / Φ(x), signed for the symmetric branch. Used by the latent probit families where the inverse-link gradient needs the ratio and the likelihood needs the log-CDF on the same x; computing both in one call shares the erfcx evaluation that dominates the cost in the deep tail.

standard_normal_quantile

Standard normal quantile Φ⁻¹(p) using Acklam’s rational approximation.