text: /* @(#)e_rem_pio2.c 5.1 93/09/24 */
  /*
   * ====================================================
   * 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.
   * ====================================================
   */
  
  #if defined(LIBM_SCCS) && !defined(lint)
  static const char rcsid[] =
      "$NetBSD: e_rem_pio2.c,v 1.8 1995/05/10 20:46:02 jtc Exp $";
  #endif
  
  /* __ieee754_rem_pio2(x,y)
   *
   * return the remainder of x rem pi/2 in y[0]+y[1]
   * use __kernel_rem_pio2()
   */
  
  #include "math.h"
  #include "math_private.h"
  
  libm_hidden_proto(fabs)
  
  /*
   * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
   */
  #ifdef __STDC__
       static const int32_t two_over_pi[] = {
  #else
       static int32_t 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_t npio2_hw[] = {
  #else
  static int32_t 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_t attribute_hidden
  __ieee754_rem_pio2(double x, double *y)
  #else
  int32_t attribute_hidden
  __ieee754_rem_pio2(x, y)
       double x, y[];
  #endif
  {
      double z = 0.0, w, t, r, fn;
      double tx[3];
      int32_t e0, i, j, nx, n, ix, hx;
      u_int32_t low;
  
      GET_HIGH_WORD(hx, 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) (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 {
              u_int32_t high;
              j = ix >> 20;
              y[0] = r - w;
              GET_HIGH_WORD(high, y[0]);
              i = j - ((high >> 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;
                  GET_HIGH_WORD(high, y[0]);
                  i = j - ((high >> 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) */
      GET_LOW_WORD(low, x);
      SET_LOW_WORD(z, low);
      e0 = (ix >> 20) - 1046;     /* e0 = ilogb(z)-23; */
      SET_HIGH_WORD(z, ix - ((int32_t) (e0 << 20)));
      for (i = 0; i < 2; i++) {
          tx[i] = (double) ((int32_t) (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;
  }
  