fusl/src/math/erff.c

/* origin: FreeBSD /usr/src/lib/msun/src/s_erff.c */
/*
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#include "libm.h"

static const float erx = 8.4506291151e-01, /* 0x3f58560b */
                                           /*
                                            * Coefficients for approximation to  erf on [0,0.84375]
                                            */
    efx8 = 1.0270333290e+00,               /* 0x3f8375d4 */
    pp0 = 1.2837916613e-01,                /* 0x3e0375d4 */
    pp1 = -3.2504209876e-01,               /* 0xbea66beb */
    pp2 = -2.8481749818e-02,               /* 0xbce9528f */
    pp3 = -5.7702702470e-03,               /* 0xbbbd1489 */
    pp4 = -2.3763017452e-05,               /* 0xb7c756b1 */
    qq1 = 3.9791721106e-01,                /* 0x3ecbbbce */
    qq2 = 6.5022252500e-02,                /* 0x3d852a63 */
    qq3 = 5.0813062117e-03,                /* 0x3ba68116 */
    qq4 = 1.3249473704e-04,                /* 0x390aee49 */
    qq5 = -3.9602282413e-06,               /* 0xb684e21a */
                                           /*
                                            * Coefficients for approximation to  erf  in [0.84375,1.25]
                                            */
    pa0 = -2.3621185683e-03,               /* 0xbb1acdc6 */
    pa1 = 4.1485610604e-01,                /* 0x3ed46805 */
    pa2 = -3.7220788002e-01,               /* 0xbebe9208 */
    pa3 = 3.1834661961e-01,                /* 0x3ea2fe54 */
    pa4 = -1.1089469492e-01,               /* 0xbde31cc2 */
    pa5 = 3.5478305072e-02,                /* 0x3d1151b3 */
    pa6 = -2.1663755178e-03,               /* 0xbb0df9c0 */
    qa1 = 1.0642088205e-01,                /* 0x3dd9f331 */
    qa2 = 5.4039794207e-01,                /* 0x3f0a5785 */
    qa3 = 7.1828655899e-02,                /* 0x3d931ae7 */
    qa4 = 1.2617121637e-01,                /* 0x3e013307 */
    qa5 = 1.3637083583e-02,                /* 0x3c5f6e13 */
    qa6 = 1.1984500103e-02,                /* 0x3c445aa3 */
                                           /*
                                            * Coefficients for approximation to  erfc in [1.25,1/0.35]
                                            */
    ra0 = -9.8649440333e-03,               /* 0xbc21a093 */
    ra1 = -6.9385856390e-01,               /* 0xbf31a0b7 */
    ra2 = -1.0558626175e+01,               /* 0xc128f022 */
    ra3 = -6.2375331879e+01,               /* 0xc2798057 */
    ra4 = -1.6239666748e+02,               /* 0xc322658c */
    ra5 = -1.8460508728e+02,               /* 0xc3389ae7 */
    ra6 = -8.1287437439e+01,               /* 0xc2a2932b */
    ra7 = -9.8143291473e+00,               /* 0xc11d077e */
    sa1 = 1.9651271820e+01,                /* 0x419d35ce */
    sa2 = 1.3765776062e+02,                /* 0x4309a863 */
    sa3 = 4.3456588745e+02,                /* 0x43d9486f */
    sa4 = 6.4538726807e+02,                /* 0x442158c9 */
    sa5 = 4.2900814819e+02,                /* 0x43d6810b */
    sa6 = 1.0863500214e+02,                /* 0x42d9451f */
    sa7 = 6.5702495575e+00,                /* 0x40d23f7c */
    sa8 = -6.0424413532e-02,               /* 0xbd777f97 */
                                           /*
                                            * Coefficients for approximation to  erfc in [1/.35,28]
                                            */
    rb0 = -9.8649431020e-03,               /* 0xbc21a092 */
    rb1 = -7.9928326607e-01,               /* 0xbf4c9dd4 */
    rb2 = -1.7757955551e+01,               /* 0xc18e104b */
    rb3 = -1.6063638306e+02,               /* 0xc320a2ea */
    rb4 = -6.3756646729e+02,               /* 0xc41f6441 */
    rb5 = -1.0250950928e+03,               /* 0xc480230b */
    rb6 = -4.8351919556e+02,               /* 0xc3f1c275 */
    sb1 = 3.0338060379e+01,                /* 0x41f2b459 */
    sb2 = 3.2579251099e+02,                /* 0x43a2e571 */
    sb3 = 1.5367296143e+03,                /* 0x44c01759 */
    sb4 = 3.1998581543e+03,                /* 0x4547fdbb */
    sb5 = 2.5530502930e+03,                /* 0x451f90ce */
    sb6 = 4.7452853394e+02,                /* 0x43ed43a7 */
    sb7 = -2.2440952301e+01;               /* 0xc1b38712 */

static float erfc1(float x) {
  float_t s, P, Q;

  s = fabsf(x) - 1;
  P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 + s * pa6)))));
  Q = 1 + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 + s * qa6)))));
  return 1 - erx - P / Q;
}

static float erfc2(uint32_t ix, float x) {
  float_t s, R, S;
  float z;

  if (ix < 0x3fa00000) /* |x| < 1.25 */
    return erfc1(x);

  x = fabsf(x);
  s = 1 / (x * x);
  if (ix < 0x4036db6d) { /* |x| < 1/0.35 */
    R = ra0 +
        s * (ra1 +
             s * (ra2 +
                  s * (ra3 + s * (ra4 + s * (ra5 + s * (ra6 + s * ra7))))));
    S = 1.0f +
        s * (sa1 +
             s * (sa2 +
                  s * (sa3 +
                       s * (sa4 +
                            s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
  } else { /* |x| >= 1/0.35 */
    R = rb0 +
        s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 + s * rb6)))));
    S = 1.0f +
        s * (sb1 +
             s * (sb2 +
                  s * (sb3 + s * (sb4 + s * (sb5 + s * (sb6 + s * sb7))))));
  }
  GET_FLOAT_WORD(ix, x);
  SET_FLOAT_WORD(z, ix & 0xffffe000);
  return expf(-z * z - 0.5625f) * expf((z - x) * (z + x) + R / S) / x;
}

float erff(float x) {
  float r, s, z, y;
  uint32_t ix;
  int sign;

  GET_FLOAT_WORD(ix, x);
  sign = ix >> 31;
  ix &= 0x7fffffff;
  if (ix >= 0x7f800000) {
    /* erf(nan)=nan, erf(+-inf)=+-1 */
    return 1 - 2 * sign + 1 / x;
  }
  if (ix < 0x3f580000) {   /* |x| < 0.84375 */
    if (ix < 0x31800000) { /* |x| < 2**-28 */
      /*avoid underflow */
      return 0.125f * (8 * x + efx8 * x);
    }
    z = x * x;
    r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
    s = 1 + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
    y = r / s;
    return x + x * y;
  }
  if (ix < 0x40c00000) /* |x| < 6 */
    y = 1 - erfc2(ix, x);
  else
    y = 1 - 0x1p-120f;
  return sign ? -y : y;
}

float erfcf(float x) {
  float r, s, z, y;
  uint32_t ix;
  int sign;

  GET_FLOAT_WORD(ix, x);
  sign = ix >> 31;
  ix &= 0x7fffffff;
  if (ix >= 0x7f800000) {
    /* erfc(nan)=nan, erfc(+-inf)=0,2 */
    return 2 * sign + 1 / x;
  }

  if (ix < 0x3f580000) { /* |x| < 0.84375 */
    if (ix < 0x23800000) /* |x| < 2**-56 */
      return 1.0f - x;
    z = x * x;
    r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
    s = 1.0f + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
    y = r / s;
    if (sign || ix < 0x3e800000) /* x < 1/4 */
      return 1.0f - (x + x * y);
    return 0.5f - (x - 0.5f + x * y);
  }
  if (ix < 0x41e00000) { /* |x| < 28 */
    return sign ? 2 - erfc2(ix, x) : erfc2(ix, x);
  }
  return sign ? 2 - 0x1p-120f : 0x1p-120f * 0x1p-120f;
}