prime/src/Runtime/e_rem_pio2.c

/* @(#)e_rem_pio2.c 1.3 95/01/04 */
/*
 * ====================================================
 * 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.
 * ====================================================
 *
 */

/* __ieee754_rem_pio2(x,y)
 *
 * return the remainder of x rem pi/2 in y[0]+y[1]
 * use __kernel_rem_pio2()
 */

#include "fdlibm.h"

/*
 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
 */

#ifdef __STDC__
static const _INT32 two_over_pi[] = {
#else
static _INT32 two_over_pi[] = {
#endif
    0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C, 0x439041, 0xFE5163,
    0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
    0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C,
    0x845F8B, 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
    0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, 0xF17B3D, 0x0739F7, 0x8A5292,
    0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
    0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA,
    0x73A8C9, 0x60E27B, 0xC08C6B,
};

#ifdef __STDC__
static const _INT32 npio2_hw[] = {
#else
static _INT32 npio2_hw[] = {
#endif
    0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, 0x4025FDBB, 0x402921FB,
    0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB,
    0x403AB41B, 0x403C463A, 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
    0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, 0x404858EB, 0x404921FB,
};

/*
 * invpio2:  53 bits of 2/pi
 * pio2_1:   first  33 bit of pi/2
 * pio2_1t:  pi/2 - pio2_1
 * pio2_2:   second 33 bit of pi/2
 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
 * pio2_3:   third  33 bit of pi/2
 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
 */

#ifdef __STDC__
static const double
#else
static double
#endif
    zero = 0.00000000000000000000e+00,    /* 0x00000000, 0x00000000 */
    half = 5.00000000000000000000e-01,    /* 0x3FE00000, 0x00000000 */
    two24 = 1.67772160000000000000e+07,   /* 0x41700000, 0x00000000 */
    invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
    pio2_1 = 1.57079632673412561417e+00,  /* 0x3FF921FB, 0x54400000 */
    pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
    pio2_2 = 6.07710050630396597660e-11,  /* 0x3DD0B461, 0x1A600000 */
    pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
    pio2_3 = 2.02226624871116645580e-21,  /* 0x3BA3198A, 0x2E000000 */
    pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */

#ifdef __STDC__
_INT32 __ieee754_rem_pio2(double x, double* y)
#else
_INT32 __ieee754_rem_pio2(x, y)
double x, y[];
#endif
{
  double z, w, t, r, fn;
  double tx[3];
  _INT32 e0, i, j, nx, n, ix, hx;

  hx = __HI(x); /* high word of x */
  ix = hx & 0x7fffffff;
  if (ix <= 0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
  {
    y[0] = x;
    y[1] = 0;
    return 0;
  }
  if (ix < 0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
    if (hx > 0) {
      z = x - pio2_1;
      if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
        y[0] = z - pio2_1t;
        y[1] = (z - y[0]) - pio2_1t;
      } else { /* near pi/2, use 33+33+53 bit pi */
        z -= pio2_2;
        y[0] = z - pio2_2t;
        y[1] = (z - y[0]) - pio2_2t;
      }
      return 1;
    } else { /* negative x */
      z = x + pio2_1;
      if (ix != 0x3ff921fb) { /* 33+53 bit pi is good enough */
        y[0] = z + pio2_1t;
        y[1] = (z - y[0]) + pio2_1t;
      } else { /* near pi/2, use 33+33+53 bit pi */
        z += pio2_2;
        y[0] = z + pio2_2t;
        y[1] = (z - y[0]) + pio2_2t;
      }
      return -1;
    }
  }
  if (ix <= 0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
    t = fabs(x);
    n = (_INT32)(t * invpio2 + half);
    fn = (double)n;
    r = t - fn * pio2_1;
    w = fn * pio2_1t; /* 1st round good to 85 bit */
    if (n < 32 && ix != npio2_hw[n - 1]) {
      y[0] = r - w; /* quick check no cancellation */
    } else {
      j = ix >> 20;
      y[0] = r - w;
      i = j - (((__HI(y[0])) >> 20) & 0x7ff);
      if (i > 16) { /* 2nd iteration needed, good to 118 */
        t = r;
        w = fn * pio2_2;
        r = t - w;
        w = fn * pio2_2t - ((t - r) - w);
        y[0] = r - w;
        i = j - (((__HI(y[0])) >> 20) & 0x7ff);
        if (i > 49) { /* 3rd iteration need, 151 bits acc */
          t = r;      /* will cover all possible cases */
          w = fn * pio2_3;
          r = t - w;
          w = fn * pio2_3t - ((t - r) - w);
          y[0] = r - w;
        }
      }
    }
    y[1] = (r - y[0]) - w;
    if (hx < 0) {
      y[0] = -y[0];
      y[1] = -y[1];
      return -n;
    } else
      return n;
  }
  /*
   * all other (large) arguments
   */
  if (ix >= 0x7ff00000) { /* x is inf or NaN */
    y[0] = y[1] = x - x;
    return 0;
  }
  /* set z = scalbn(|x|,ilogb(x)-23) */
  __LO(z) = __LO(x);
  e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */
  __HI(z) = ix - (e0 << 20);
  for (i = 0; i < 2; i++) {
    tx[i] = (double)((_INT32)(z));
    z = (z - tx[i]) * two24;
  }
  tx[2] = z;
  nx = 3;
  while (tx[nx - 1] == zero)
    nx--; /* skip zero term */
  n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
  if (hx < 0) {
    y[0] = -y[0];
    y[1] = -y[1];
    return -n;
  }
  return n;
}