Error functions

Chelis provides correctly rounded erf and erfc builtins. Call them directly without an import from Nautilus. They accept scalar or tensor inputs at the active float dtypes. Nautilus provides erfinv for the inverse error function at f32 and f64.

erf

Chelis builtin: erf(x) preserves the input shape and dtype.

Computes the error function erf(x) = (2/sqrt(pi)) * integral(0, x, exp(-t^2) dt). The function is odd: erf(-x) = -erf(x).

• Domain: all reals
• Range: [-1, 1] at finite precision; values round to the endpoints for sufficiently large magnitudes
• Precision: correctly rounded at f32 and f64

val = erf(cast(0.5, f32))        -- approximately 0.5205
neg_val = erf(cast(-1.0, f32))   -- approximately -0.8427

erfc

Chelis builtin: erfc(x) preserves the input shape and dtype.

Computes the complementary error function erfc(x) = 1 - erf(x) directly. It keeps relative precision in the positive tail, where the subtraction loses it.

• Domain: all reals
• Range: [0, 2] at finite precision; the result can round to either endpoint
• Precision: correctly rounded at f32 and f64

tail = erfc(cast(2.0, f32))      -- approximately 0.0047

erfinv

Signature: [prec: {f32, f64}](x: prec) -> prec

Computes the inverse error function: if y = erf(x), then x = erfinv(y). Internally delegates to the Acklam rational approximation of the normal inverse CDF, then rescales by 1/sqrt(2).

• Domain: (-1, 1) strictly; values at or beyond the boundary produce inf/NaN
• Precision: ~1e-8 relative

import Nautilus.Special (erfinv)

x = erfinv(cast(0.5, f32))      -- approximately 0.4769

Edge cases

Input	erf result	erfc result	erfinv result
0.0	0.0	1.0	0.0
+large	approaches 1.0	approaches 0.0	n/a (outside domain)
-large	approaches -1.0	approaches 2.0	n/a (outside domain)
NaN	NaN	NaN	NaN
1.0	n/a	n/a	+inf
-1.0	n/a	n/a	-inf