fusl/src/math/j1f.c

/* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.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.
 * ====================================================
 */

#define _GNU_SOURCE
#include "libm.h"

static float ponef(float), qonef(float);

static const float invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
    tpi = 6.3661974669e-01;                      /* 0x3f22f983 */

static float common(uint32_t ix, float x, int y1, int sign) {
  double z, s, c, ss, cc;

  s = sinf(x);
  if (y1)
    s = -s;
  c = cosf(x);
  cc = s - c;
  if (ix < 0x7f000000) {
    ss = -s - c;
    z = cosf(2 * x);
    if (s * c > 0)
      cc = z / ss;
    else
      ss = z / cc;
    if (ix < 0x58800000) {
      if (y1)
        ss = -ss;
      cc = ponef(x) * cc - qonef(x) * ss;
    }
  }
  if (sign)
    cc = -cc;
  return invsqrtpi * cc / sqrtf(x);
}

/* R0/S0 on [0,2] */
static const float r00 = -6.2500000000e-02, /* 0xbd800000 */
    r01 = 1.4070566976e-03,                 /* 0x3ab86cfd */
    r02 = -1.5995563444e-05,                /* 0xb7862e36 */
    r03 = 4.9672799207e-08,                 /* 0x335557d2 */
    s01 = 1.9153760746e-02,                 /* 0x3c9ce859 */
    s02 = 1.8594678841e-04,                 /* 0x3942fab6 */
    s03 = 1.1771846857e-06,                 /* 0x359dffc2 */
    s04 = 5.0463624390e-09,                 /* 0x31ad6446 */
    s05 = 1.2354227016e-11;                 /* 0x2d59567e */

float j1f(float x) {
  float z, r, s;
  uint32_t ix;
  int sign;

  GET_FLOAT_WORD(ix, x);
  sign = ix >> 31;
  ix &= 0x7fffffff;
  if (ix >= 0x7f800000)
    return 1 / (x * x);
  if (ix >= 0x40000000) /* |x| >= 2 */
    return common(ix, fabsf(x), 0, sign);
  if (ix >= 0x32000000) { /* |x| >= 2**-27 */
    z = x * x;
    r = z * (r00 + z * (r01 + z * (r02 + z * r03)));
    s = 1 + z * (s01 + z * (s02 + z * (s03 + z * (s04 + z * s05))));
    z = 0.5f + r / s;
  } else
    /* raise inexact if x!=0 */
    z = 0.5f + x;
  return z * x;
}

static const float U0[5] = {
    -1.9605709612e-01, /* 0xbe48c331 */
    5.0443872809e-02,  /* 0x3d4e9e3c */
    -1.9125689287e-03, /* 0xbafaaf2a */
    2.3525259166e-05,  /* 0x37c5581c */
    -9.1909917899e-08, /* 0xb3c56003 */
};
static const float V0[5] = {
    1.9916731864e-02, /* 0x3ca3286a */
    2.0255257550e-04, /* 0x3954644b */
    1.3560879779e-06, /* 0x35b602d4 */
    6.2274145840e-09, /* 0x31d5f8eb */
    1.6655924903e-11, /* 0x2d9281cf */
};

float y1f(float x) {
  float z, u, v;
  uint32_t ix;

  GET_FLOAT_WORD(ix, x);
  if ((ix & 0x7fffffff) == 0)
    return -1 / 0.0f;
  if (ix >> 31)
    return 0 / 0.0f;
  if (ix >= 0x7f800000)
    return 1 / x;
  if (ix >= 0x40000000) /* |x| >= 2.0 */
    return common(ix, x, 1, 0);
  if (ix < 0x32000000) /* x < 2**-27 */
    return -tpi / x;
  z = x * x;
  u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * U0[4])));
  v = 1.0f + z * (V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * V0[4]))));
  return x * (u / v) + tpi * (j1f(x) * logf(x) - 1.0f / x);
}

/* For x >= 8, the asymptotic expansions of pone is
 *      1 + 15/128 s^2 - 4725/2^15 s^4 - ...,   where s = 1/x.
 * We approximate pone by
 *      pone(x) = 1 + (R/S)
 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 *        S = 1 + ps0*s^2 + ... + ps4*s^10
 * and
 *      | pone(x)-1-R/S | <= 2  ** ( -60.06)
 */

static const float pr8[6] = {
    /* for x in [inf, 8]=1/[0,0.125] */
    0.0000000000e+00, /* 0x00000000 */
    1.1718750000e-01, /* 0x3df00000 */
    1.3239480972e+01, /* 0x4153d4ea */
    4.1205184937e+02, /* 0x43ce06a3 */
    3.8747453613e+03, /* 0x45722bed */
    7.9144794922e+03, /* 0x45f753d6 */
};
static const float ps8[5] = {
    1.1420736694e+02, /* 0x42e46a2c */
    3.6509309082e+03, /* 0x45642ee5 */
    3.6956207031e+04, /* 0x47105c35 */
    9.7602796875e+04, /* 0x47bea166 */
    3.0804271484e+04, /* 0x46f0a88b */
};

static const float pr5[6] = {
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    1.3199052094e-11, /* 0x2d68333f */
    1.1718749255e-01, /* 0x3defffff */
    6.8027510643e+00, /* 0x40d9b023 */
    1.0830818176e+02, /* 0x42d89dca */
    5.1763616943e+02, /* 0x440168b7 */
    5.2871520996e+02, /* 0x44042dc6 */
};
static const float ps5[5] = {
    5.9280597687e+01, /* 0x426d1f55 */
    9.9140142822e+02, /* 0x4477d9b1 */
    5.3532670898e+03, /* 0x45a74a23 */
    7.8446904297e+03, /* 0x45f52586 */
    1.5040468750e+03, /* 0x44bc0180 */
};

static const float pr3[6] = {
    3.0250391081e-09, /* 0x314fe10d */
    1.1718686670e-01, /* 0x3defffab */
    3.9329774380e+00, /* 0x407bb5e7 */
    3.5119403839e+01, /* 0x420c7a45 */
    9.1055007935e+01, /* 0x42b61c2a */
    4.8559066772e+01, /* 0x42423c7c */
};
static const float ps3[5] = {
    3.4791309357e+01, /* 0x420b2a4d */
    3.3676245117e+02, /* 0x43a86198 */
    1.0468714600e+03, /* 0x4482dbe3 */
    8.9081134033e+02, /* 0x445eb3ed */
    1.0378793335e+02, /* 0x42cf936c */
};

static const float pr2[6] = {
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    1.0771083225e-07, /* 0x33e74ea8 */
    1.1717621982e-01, /* 0x3deffa16 */
    2.3685150146e+00, /* 0x401795c0 */
    1.2242610931e+01, /* 0x4143e1bc */
    1.7693971634e+01, /* 0x418d8d41 */
    5.0735230446e+00, /* 0x40a25a4d */
};
static const float ps2[5] = {
    2.1436485291e+01, /* 0x41ab7dec */
    1.2529022980e+02, /* 0x42fa9499 */
    2.3227647400e+02, /* 0x436846c7 */
    1.1767937469e+02, /* 0x42eb5bd7 */
    8.3646392822e+00, /* 0x4105d590 */
};

static float ponef(float x) {
  const float *p, *q;
  float_t z, r, s;
  uint32_t ix;

  GET_FLOAT_WORD(ix, x);
  ix &= 0x7fffffff;
  if (ix >= 0x41000000) {
    p = pr8;
    q = ps8;
  } else if (ix >= 0x40f71c58) {
    p = pr5;
    q = ps5;
  } else if (ix >= 0x4036db68) {
    p = pr3;
    q = ps3;
  } else /*ix >= 0x40000000*/ {
    p = pr2;
    q = ps2;
  }
  z = 1.0f / (x * x);
  r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
  s = 1.0f + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))));
  return 1.0f + r / s;
}

/* For x >= 8, the asymptotic expansions of qone is
 *      3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 * We approximate pone by
 *      qone(x) = s*(0.375 + (R/S))
 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 *        S = 1 + qs1*s^2 + ... + qs6*s^12
 * and
 *      | qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 */

static const float qr8[6] = {
    /* for x in [inf, 8]=1/[0,0.125] */
    0.0000000000e+00,  /* 0x00000000 */
    -1.0253906250e-01, /* 0xbdd20000 */
    -1.6271753311e+01, /* 0xc1822c8d */
    -7.5960174561e+02, /* 0xc43de683 */
    -1.1849806641e+04, /* 0xc639273a */
    -4.8438511719e+04, /* 0xc73d3683 */
};
static const float qs8[6] = {
    1.6139537048e+02,  /* 0x43216537 */
    7.8253862305e+03,  /* 0x45f48b17 */
    1.3387534375e+05,  /* 0x4802bcd6 */
    7.1965775000e+05,  /* 0x492fb29c */
    6.6660125000e+05,  /* 0x4922be94 */
    -2.9449025000e+05, /* 0xc88fcb48 */
};

static const float qr5[6] = {
    /* for x in [8,4.5454]=1/[0.125,0.22001] */
    -2.0897993405e-11, /* 0xadb7d219 */
    -1.0253904760e-01, /* 0xbdd1fffe */
    -8.0564479828e+00, /* 0xc100e736 */
    -1.8366960144e+02, /* 0xc337ab6b */
    -1.3731937256e+03, /* 0xc4aba633 */
    -2.6124443359e+03, /* 0xc523471c */
};
static const float qs5[6] = {
    8.1276550293e+01,  /* 0x42a28d98 */
    1.9917987061e+03,  /* 0x44f8f98f */
    1.7468484375e+04,  /* 0x468878f8 */
    4.9851425781e+04,  /* 0x4742bb6d */
    2.7948074219e+04,  /* 0x46da5826 */
    -4.7191835938e+03, /* 0xc5937978 */
};

static const float qr3[6] = {
    -5.0783124372e-09, /* 0xb1ae7d4f */
    -1.0253783315e-01, /* 0xbdd1ff5b */
    -4.6101160049e+00, /* 0xc0938612 */
    -5.7847221375e+01, /* 0xc267638e */
    -2.2824453735e+02, /* 0xc3643e9a */
    -2.1921012878e+02, /* 0xc35b35cb */
};
static const float qs3[6] = {
    4.7665153503e+01,  /* 0x423ea91e */
    6.7386511230e+02,  /* 0x4428775e */
    3.3801528320e+03,  /* 0x45534272 */
    5.5477290039e+03,  /* 0x45ad5dd5 */
    1.9031191406e+03,  /* 0x44ede3d0 */
    -1.3520118713e+02, /* 0xc3073381 */
};

static const float qr2[6] = {
    /* for x in [2.8570,2]=1/[0.3499,0.5] */
    -1.7838172539e-07, /* 0xb43f8932 */
    -1.0251704603e-01, /* 0xbdd1f475 */
    -2.7522056103e+00, /* 0xc0302423 */
    -1.9663616180e+01, /* 0xc19d4f16 */
    -4.2325313568e+01, /* 0xc2294d1f */
    -2.1371921539e+01, /* 0xc1aaf9b2 */
};
static const float qs2[6] = {
    2.9533363342e+01,  /* 0x41ec4454 */
    2.5298155212e+02,  /* 0x437cfb47 */
    7.5750280762e+02,  /* 0x443d602e */
    7.3939318848e+02,  /* 0x4438d92a */
    1.5594900513e+02,  /* 0x431bf2f2 */
    -4.9594988823e+00, /* 0xc09eb437 */
};

static float qonef(float x) {
  const float *p, *q;
  float_t s, r, z;
  uint32_t ix;

  GET_FLOAT_WORD(ix, x);
  ix &= 0x7fffffff;
  if (ix >= 0x40200000) {
    p = qr8;
    q = qs8;
  } else if (ix >= 0x40f71c58) {
    p = qr5;
    q = qs5;
  } else if (ix >= 0x4036db68) {
    p = qr3;
    q = qs3;
  } else /*ix >= 0x40000000*/ {
    p = qr2;
    q = qs2;
  }
  z = 1.0f / (x * x);
  r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))));
  s = 1.0f +
      z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))));
  return (.375f + r / s) / x;
}