Function erfInv

src/compute-engine/numerics/special-functions.ts:332–359

(x: number)

/**
 * Winitzki's approximation for the inverse error function, accurate to
 * ~2e-3 relative over (-1, 1). Used as the Newton seed for `erfInv()` and
 * `bigErfInv()`.
 */
function erfInvApprox(x: number): number {
  const a = 0.147;
  const ln1mx2 = Math.log(1 - x * x);
  const b = 2 / (Math.PI * a) + ln1mx2 / 2;
  return Math.sign(x) * Math.sqrt(Math.sqrt(b * b - ln1mx2 / a) - b);
}

/**
 * Inverse Error Function, accurate to full machine (double) precision.
 *
 * Winitzki's approximation (~3 correct digits) refined with Newton's
 * method on the full-precision `erf()`:
 *    y ← y − (erf(y) − x)·(√π/2)·e^{y²}
 * Each iteration doubles the number of correct digits, so 4 iterations
 * reach machine precision.
 *
 * (Previously used a 6-term truncated Maclaurin series, which was only
 * ~4-digit accurate at x = 0.5 and diverged badly for |x| → 1.)
 */
export function erfInv(x: number): number {
  if (Number.isNaN(x) || x < -1 || x > 1) return NaN;
  if (x === 0) return 0;
  if (x === 1) return Infinity;
  if (x === -1) return -Infinity;

  const sign = x < 0 ? -1 : 1;
  const ax = Math.abs(x);