Module numerics_host 

Host-side scalar special functions shared by the CPU parity references of the GPU backends.

The CUDA kernels emit their own NVRTC-visible numerics (see crate::numerics_device); this module is the matching host side used by the CPU parity oracles (bms_flex_row’s test oracle) and the CPU reference path (pirls_row’s probit CDF). Keeping a single definition here means the host erfc cannot drift between backends.

Constants

INV_SQRT_2PI

1/√(2π), matching INV_SQRT_2PI in the kernel source bit-for-bit.

INV_SQRT_PI

1/√π, matching inv_sqrt_pi in the kernel source bit-for-bit.

LN_2

ln(2), matching LN_2 in the kernel source bit-for-bit.

SQRT_2

√2, matching SQRT_2 in the kernel source bit-for-bit.

SQRT_2_OVER_PI

√(2/π), matching sqrt_2_over_pi in the kernel source bit-for-bit.

Functions

erfc

Complementary error function erfc(x) = 1 − erf(x) evaluated on the host.

erfcx_nonnegative

Scaled complementary error function erfcx(x) = exp(x²)·erfc(x) for x ≥ 0, the host oracle for the device erfcx_nonnegative. Returns 0.0 at +∞; negative inputs and NaN return NaN because they violate the restricted domain. For 0 ≤ x < 26 evaluates exp(x²)·erfc(x) directly; beyond that it switches to the same six-correction asymptotic expansion as the kernel.

log_ndtr

log Φ(x) for the standard normal CDF, the host oracle for the device log_ndtr. For x < 0 uses the erfcx representation log Φ(x) = −u² + log(½·erfcx(u)), u = −x/√2, keeping digits into the deep left tail; for x ≥ 0 uses log1p(−½·erfc(x/√2)), retaining the negative tail after the CDF rounds to one. Propagates ±∞/NaN exactly as the device path does.

log_ndtr_and_mills

Joint (log Φ(x), Mills ratio φ(x)/Φ(x)), the host oracle for the device log_ndtr_and_mills. The x < 0 branch computes the Mills ratio as √(2/π)/erfcx(u), which stays finite even when Φ(x) underflows; the x ≥ 0 branch forms pdf/cdf directly. Boundary values mirror the kernel: (+0, +0) at +∞, (−∞, +∞) at −∞, (NaN, NaN) at NaN.

log_ndtr_mills_curvature

Joint (log Φ(x), φ(x)/Φ(x), −d²log Φ(x)/dx²) host oracle.