erfinvl.c

/*
 * Extended precision inverse error function.
 *
 * Copyright (c) 2023-2024, Arm Limited.
 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
 */
#define _GNU_SOURCE
#include <math.h>
#include <stdbool.h>
#include <float.h>

#include "math_config.h"
#include "poly_scalar_f64.h"

#define SQRT_PIl 0x1.c5bf891b4ef6aa79c3b0520d5db9p0l
#define HF_SQRT_PIl 0x1.c5bf891b4ef6aa79c3b0520d5db9p-1l

const static struct
{
  /*  We use P_N and Q_N to refer to arrays of coefficients, where P_N is the
      coeffs of the numerator in table N of Blair et al, and Q_N is the coeffs
      of the denominator.  */
  double P_17[7], Q_17[7], P_37[8], Q_37[8], P_57[9], Q_57[10];
} data = {
  .P_17 = { 0x1.007ce8f01b2e8p+4, -0x1.6b23cc5c6c6d7p+6, 0x1.74e5f6ceb3548p+7,
	    -0x1.5200bb15cc6bbp+7, 0x1.05d193233a849p+6, -0x1.148c5474ee5e1p+3,
	    0x1.689181bbafd0cp-3 },
  .Q_17 = { 0x1.d8fb0f913bd7bp+3, -0x1.6d7f25a3f1c24p+6, 0x1.a450d8e7f4cbbp+7,
	    -0x1.bc3480485857p+7, 0x1.ae6b0c504ee02p+6, -0x1.499dfec1a7f5fp+4,
	    0x1p+0 },
  .P_37 = { -0x1.f3596123109edp-7, 0x1.60b8fe375999ep-2, -0x1.779bb9bef7c0fp+1,
	    0x1.786ea384470a2p+3, -0x1.6a7c1453c85d3p+4, 0x1.31f0fc5613142p+4,
	    -0x1.5ea6c007d4dbbp+2, 0x1.e66f265ce9e5p-3 },
  .Q_37 = { -0x1.636b2dcf4edbep-7, 0x1.0b5411e2acf29p-2, -0x1.3413109467a0bp+1,
	    0x1.563e8136c554ap+3, -0x1.7b77aab1dcafbp+4, 0x1.8a3e174e05ddcp+4,
	    -0x1.4075c56404eecp+3, 0x1p+0 },
  .P_57 = { 0x1.b874f9516f7f1p-14, 0x1.5921f2916c1c4p-7, 0x1.145ae7d5b8fa4p-2,
	    0x1.29d6dcc3b2fb7p+1, 0x1.cabe2209a7985p+2, 0x1.11859f0745c4p+3,
	    0x1.b7ec7bc6a2ce5p+2, 0x1.d0419e0bb42aep+1, 0x1.c5aa03eef7258p-1 },
  .Q_57 = { 0x1.b8747e12691f1p-14, 0x1.59240d8ed1e0ap-7, 0x1.14aef2b181e2p-2,
	    0x1.2cd181bcea52p+1, 0x1.e6e63e0b7aa4cp+2, 0x1.65cf8da94aa3ap+3,
	    0x1.7e5c787b10a36p+3, 0x1.0626d68b6cea3p+3, 0x1.065c5f193abf6p+2,
	    0x1p+0 }
};

/* Inverse error function approximation, based on rational approximation as
   described in
   J. M. Blair, C. A. Edwards, and J. H. Johnson,
   "Rational Chebyshev approximations for the inverse of the error function",
   Math. Comp. 30, pp. 827--830 (1976).
   https://doi.org/10.1090/S0025-5718-1976-0421040-7.  */
static inline double
__erfinv (double x)
{
  if (x == 1.0)
    return __math_oflow (0);
  if (x == -1.0)
    return __math_oflow (1);

  double a = fabs (x);
  if (a > 1)
    return __math_invalid (x);

  if (a <= 0.75)
    {
      double t = x * x - 0.5625;
      return x * horner_6_f64 (t, data.P_17) / horner_6_f64 (t, data.Q_17);
    }

  if (a <= 0.9375)
    {
      double t = x * x - 0.87890625;
      return x * horner_7_f64 (t, data.P_37) / horner_7_f64 (t, data.Q_37);
    }

  double t = 1.0 / (sqrtl (-log1pl (-a)));
  return horner_8_f64 (t, data.P_57)
	 / (copysign (t, x) * horner_9_f64 (t, data.Q_57));
}

/* Extended-precision variant, which uses the above (or asymptotic estimate) as
   starting point for Newton refinement. This implementation is a port to C of
   the version in the SpecialFunctions.jl Julia package, with relaxed stopping
   criteria for the Newton refinement.  */
long double
erfinvl (long double x)
{
  if (x == 0)
    return 0;

  double yf = __erfinv (x);
  long double y;
  if (isfinite (yf))
    y = yf;
  else
    {
      /* Double overflowed, use asymptotic estimate instead.  */
      y = copysignl (sqrtl (-logl (1.0l - fabsl (x)) * SQRT_PIl), x);
      if (!isfinite (y))
	return y;
    }

  double eps = fabs (yf - nextafter (yf, 0));
  while (true)
    {
      long double dy = HF_SQRT_PIl * (erfl (y) - x) * exp (y * y);
      y -= dy;
      /* Stopping criterion is different to Julia implementation, but is enough
	 to ensure result is accurate when rounded to double-precision.  */
      if (fabsl (dy) < eps)
	break;
    }
  return y;
}