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e_rem_pio2l.c

/* Quad-precision floating point argument reduction.
   Copyright (C) 1999 Free Software Foundation, Inc.
   This file is part of the GNU C Library.
   Contributed by Jakub Jelinek <jj@ultra.linux.cz>

   The GNU C Library is free software; you can redistribute it and/or
   modify it under the terms of the GNU Lesser General Public
   License as published by the Free Software Foundation; either
   version 2.1 of the License, or (at your option) any later version.

   The GNU C Library is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
   Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public
   License along with the GNU C Library; if not, write to the Free
   Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
   02111-1307 USA.  */

#include "math.h"
#include "math_private.h"

/*
 * Table of constants for 2/pi, 5628 hexadecimal digits of 2/pi 
 */
static const int32_t two_over_pi[] = {
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, 
0x47c419, 0xc367cd, 0xdce809, 0x2a8359, 0xc4768b, 0x961ca6, 
0xddaf44, 0xd15719, 0x053ea5, 0xff0705, 0x3f7e33, 0xe832c2, 
0xde4f98, 0x327dbb, 0xc33d26, 0xef6b1e, 0x5ef89f, 0x3a1f35, 
0xcaf27f, 0x1d87f1, 0x21907c, 0x7c246a, 0xfa6ed5, 0x772d30, 
0x433b15, 0xc614b5, 0x9d19c3, 0xc2c4ad, 0x414d2c, 0x5d000c, 
0x467d86, 0x2d71e3, 0x9ac69b, 0x006233, 0x7cd2b4, 0x97a7b4, 
0xd55537, 0xf63ed7, 0x1810a3, 0xfc764d, 0x2a9d64, 0xabd770, 
0xf87c63, 0x57b07a, 0xe71517, 0x5649c0, 0xd9d63b, 0x3884a7, 
0xcb2324, 0x778ad6, 0x23545a, 0xb91f00, 0x1b0af1, 0xdfce19, 
0xff319f, 0x6a1e66, 0x615799, 0x47fbac, 0xd87f7e, 0xb76522, 
0x89e832, 0x60bfe6, 0xcdc4ef, 0x09366c, 0xd43f5d, 0xd7de16, 
0xde3b58, 0x929bde, 0x2822d2, 0xe88628, 0x4d58e2, 0x32cac6, 
0x16e308, 0xcb7de0, 0x50c017, 0xa71df3, 0x5be018, 0x34132e, 
0x621283, 0x014883, 0x5b8ef5, 0x7fb0ad, 0xf2e91e, 0x434a48, 
0xd36710, 0xd8ddaa, 0x425fae, 0xce616a, 0xa4280a, 0xb499d3, 
0xf2a606, 0x7f775c, 0x83c2a3, 0x883c61, 0x78738a, 0x5a8caf, 
0xbdd76f, 0x63a62d, 0xcbbff4, 0xef818d, 0x67c126, 0x45ca55, 
0x36d9ca, 0xd2a828, 0x8d61c2, 0x77c912, 0x142604, 0x9b4612, 
0xc459c4, 0x44c5c8, 0x91b24d, 0xf31700, 0xad43d4, 0xe54929, 
0x10d5fd, 0xfcbe00, 0xcc941e, 0xeece70, 0xf53e13, 0x80f1ec, 
0xc3e7b3, 0x28f8c7, 0x940593, 0x3e71c1, 0xb3092e, 0xf3450b, 
0x9c1288, 0x7b20ab, 0x9fb52e, 0xc29247, 0x2f327b, 0x6d550c, 
0x90a772, 0x1fe76b, 0x96cb31, 0x4a1679, 0xe27941, 0x89dff4, 
0x9794e8, 0x84e6e2, 0x973199, 0x6bed88, 0x365f5f, 0x0efdbb, 
0xb49a48, 0x6ca467, 0x427271, 0x325d8d, 0xb8159f, 0x09e5bc, 
0x25318d, 0x3974f7, 0x1c0530, 0x010c0d, 0x68084b, 0x58ee2c, 
0x90aa47, 0x02e774, 0x24d6bd, 0xa67df7, 0x72486e, 0xef169f, 
0xa6948e, 0xf691b4, 0x5153d1, 0xf20acf, 0x339820, 0x7e4bf5, 
0x6863b2, 0x5f3edd, 0x035d40, 0x7f8985, 0x295255, 0xc06437, 
0x10d86d, 0x324832, 0x754c5b, 0xd4714e, 0x6e5445, 0xc1090b, 
0x69f52a, 0xd56614, 0x9d0727, 0x50045d, 0xdb3bb4, 0xc576ea, 
0x17f987, 0x7d6b49, 0xba271d, 0x296996, 0xacccc6, 0x5414ad, 
0x6ae290, 0x89d988, 0x50722c, 0xbea404, 0x940777, 0x7030f3, 
0x27fc00, 0xa871ea, 0x49c266, 0x3de064, 0x83dd97, 0x973fa3, 
0xfd9443, 0x8c860d, 0xde4131, 0x9d3992, 0x8c70dd, 0xe7b717, 
0x3bdf08, 0x2b3715, 0xa0805c, 0x93805a, 0x921110, 0xd8e80f, 
0xaf806c, 0x4bffdb, 0x0f9038, 0x761859, 0x15a562, 0xbbcb61, 
0xb989c7, 0xbd4010, 0x04f2d2, 0x277549, 0xf6b6eb, 0xbb22db, 
0xaa140a, 0x2f2689, 0x768364, 0x333b09, 0x1a940e, 0xaa3a51, 
0xc2a31d, 0xaeedaf, 0x12265c, 0x4dc26d, 0x9c7a2d, 0x9756c0, 
0x833f03, 0xf6f009, 0x8c402b, 0x99316d, 0x07b439, 0x15200c, 
0x5bc3d8, 0xc492f5, 0x4badc6, 0xa5ca4e, 0xcd37a7, 0x36a9e6, 
0x9492ab, 0x6842dd, 0xde6319, 0xef8c76, 0x528b68, 0x37dbfc, 
0xaba1ae, 0x3115df, 0xa1ae00, 0xdafb0c, 0x664d64, 0xb705ed, 
0x306529, 0xbf5657, 0x3aff47, 0xb9f96a, 0xf3be75, 0xdf9328, 
0x3080ab, 0xf68c66, 0x15cb04, 0x0622fa, 0x1de4d9, 0xa4b33d, 
0x8f1b57, 0x09cd36, 0xe9424e, 0xa4be13, 0xb52333, 0x1aaaf0, 
0xa8654f, 0xa5c1d2, 0x0f3f0b, 0xcd785b, 0x76f923, 0x048b7b, 
0x721789, 0x53a6c6, 0xe26e6f, 0x00ebef, 0x584a9b, 0xb7dac4, 
0xba66aa, 0xcfcf76, 0x1d02d1, 0x2df1b1, 0xc1998c, 0x77adc3, 
0xda4886, 0xa05df7, 0xf480c6, 0x2ff0ac, 0x9aecdd, 0xbc5c3f, 
0x6dded0, 0x1fc790, 0xb6db2a, 0x3a25a3, 0x9aaf00, 0x9353ad, 
0x0457b6, 0xb42d29, 0x7e804b, 0xa707da, 0x0eaa76, 0xa1597b, 
0x2a1216, 0x2db7dc, 0xfde5fa, 0xfedb89, 0xfdbe89, 0x6c76e4, 
0xfca906, 0x70803e, 0x156e85, 0xff87fd, 0x073e28, 0x336761, 
0x86182a, 0xeabd4d, 0xafe7b3, 0x6e6d8f, 0x396795, 0x5bbf31, 
0x48d784, 0x16df30, 0x432dc7, 0x356125, 0xce70c9, 0xb8cb30, 
0xfd6cbf, 0xa200a4, 0xe46c05, 0xa0dd5a, 0x476f21, 0xd21262, 
0x845cb9, 0x496170, 0xe0566b, 0x015299, 0x375550, 0xb7d51e, 
0xc4f133, 0x5f6e13, 0xe4305d, 0xa92e85, 0xc3b21d, 0x3632a1, 
0xa4b708, 0xd4b1ea, 0x21f716, 0xe4698f, 0x77ff27, 0x80030c, 
0x2d408d, 0xa0cd4f, 0x99a520, 0xd3a2b3, 0x0a5d2f, 0x42f9b4, 
0xcbda11, 0xd0be7d, 0xc1db9b, 0xbd17ab, 0x81a2ca, 0x5c6a08, 
0x17552e, 0x550027, 0xf0147f, 0x8607e1, 0x640b14, 0x8d4196, 
0xdebe87, 0x2afdda, 0xb6256b, 0x34897b, 0xfef305, 0x9ebfb9, 
0x4f6a68, 0xa82a4a, 0x5ac44f, 0xbcf82d, 0x985ad7, 0x95c7f4, 
0x8d4d0d, 0xa63a20, 0x5f57a4, 0xb13f14, 0x953880, 0x0120cc, 
0x86dd71, 0xb6dec9, 0xf560bf, 0x11654d, 0x6b0701, 0xacb08c, 
0xd0c0b2, 0x485551, 0x0efb1e, 0xc37295, 0x3b06a3, 0x3540c0, 
0x7bdc06, 0xcc45e0, 0xfa294e, 0xc8cad6, 0x41f3e8, 0xde647c, 
0xd8649b, 0x31bed9, 0xc397a4, 0xd45877, 0xc5e369, 0x13daf0, 
0x3c3aba, 0x461846, 0x5f7555, 0xf5bdd2, 0xc6926e, 0x5d2eac, 
0xed440e, 0x423e1c, 0x87c461, 0xe9fd29, 0xf3d6e7, 0xca7c22, 
0x35916f, 0xc5e008, 0x8dd7ff, 0xe26a6e, 0xc6fdb0, 0xc10893, 
0x745d7c, 0xb2ad6b, 0x9d6ecd, 0x7b723e, 0x6a11c6, 0xa9cff7, 
0xdf7329, 0xbac9b5, 0x5100b7, 0x0db2e2, 0x24ba74, 0x607de5, 
0x8ad874, 0x2c150d, 0x0c1881, 0x94667e, 0x162901, 0x767a9f, 
0xbefdfd, 0xef4556, 0x367ed9, 0x13d9ec, 0xb9ba8b, 0xfc97c4, 
0x27a831, 0xc36ef1, 0x36c594, 0x56a8d8, 0xb5a8b4, 0x0ecccf, 
0x2d8912, 0x34576f, 0x89562c, 0xe3ce99, 0xb920d6, 0xaa5e6b, 
0x9c2a3e, 0xcc5f11, 0x4a0bfd, 0xfbf4e1, 0x6d3b8e, 0x2c86e2, 
0x84d4e9, 0xa9b4fc, 0xd1eeef, 0xc9352e, 0x61392f, 0x442138, 
0xc8d91b, 0x0afc81, 0x6a4afb, 0xd81c2f, 0x84b453, 0x8c994e, 
0xcc2254, 0xdc552a, 0xd6c6c0, 0x96190b, 0xb8701a, 0x649569, 
0x605a26, 0xee523f, 0x0f117f, 0x11b5f4, 0xf5cbfc, 0x2dbc34, 
0xeebc34, 0xcc5de8, 0x605edd, 0x9b8e67, 0xef3392, 0xb817c9, 
0x9b5861, 0xbc57e1, 0xc68351, 0x103ed8, 0x4871dd, 0xdd1c2d, 
0xa118af, 0x462c21, 0xd7f359, 0x987ad9, 0xc0549e, 0xfa864f, 
0xfc0656, 0xae79e5, 0x362289, 0x22ad38, 0xdc9367, 0xaae855, 
0x382682, 0x9be7ca, 0xa40d51, 0xb13399, 0x0ed7a9, 0x480569, 
0xf0b265, 0xa7887f, 0x974c88, 0x36d1f9, 0xb39221, 0x4a827b, 
0x21cf98, 0xdc9f40, 0x5547dc, 0x3a74e1, 0x42eb67, 0xdf9dfe, 
0x5fd45e, 0xa4677b, 0x7aacba, 0xa2f655, 0x23882b, 0x55ba41, 
0x086e59, 0x862a21, 0x834739, 0xe6e389, 0xd49ee5, 0x40fb49, 
0xe956ff, 0xca0f1c, 0x8a59c5, 0x2bfa94, 0xc5c1d3, 0xcfc50f, 
0xae5adb, 0x86c547, 0x624385, 0x3b8621, 0x94792c, 0x876110, 
0x7b4c2a, 0x1a2c80, 0x12bf43, 0x902688, 0x893c78, 0xe4c4a8, 
0x7bdbe5, 0xc23ac4, 0xeaf426, 0x8a67f7, 0xbf920d, 0x2ba365, 
0xb1933d, 0x0b7cbd, 0xdc51a4, 0x63dd27, 0xdde169, 0x19949a, 
0x9529a8, 0x28ce68, 0xb4ed09, 0x209f44, 0xca984e, 0x638270, 
0x237c7e, 0x32b90f, 0x8ef5a7, 0xe75614, 0x08f121, 0x2a9db5, 
0x4d7e6f, 0x5119a5, 0xabf9b5, 0xd6df82, 0x61dd96, 0x023616, 
0x9f3ac4, 0xa1a283, 0x6ded72, 0x7a8d39, 0xa9b882, 0x5c326b, 
0x5b2746, 0xed3400, 0x7700d2, 0x55f4fc, 0x4d5901, 0x8071e0, 
0xe13f89, 0xb295f3, 0x64a8f1, 0xaea74b, 0x38fc4c, 0xeab2bb, 
0x47270b, 0xabc3a7, 0x34ba60, 0x52dd34, 0xf8563a, 0xeb7e8a, 
0x31bb36, 0x5895b7, 0x47f7a9, 0x94c3aa, 0xd39225, 0x1e7f3e, 
0xd8974e, 0xbba94f, 0xd8ae01, 0xe661b4, 0x393d8e, 0xa523aa, 
0x33068e, 0x1633b5, 0x3bb188, 0x1d3a9d, 0x4013d0, 0xcc1be5, 
0xf862e7, 0x3bf28f, 0x39b5bf, 0x0bc235, 0x22747e, 0xa247c0, 
0xd52d1f, 0x19add3, 0x9094df, 0x9311d0, 0xb42b25, 0x496db2, 
0xe264b2, 0x5ef135, 0x3bc6a4, 0x1a4ad0, 0xaac92e, 0x64e886, 
0x573091, 0x982cfb, 0x311b1a, 0x08728b, 0xbdcee1, 0x60e142, 
0xeb641d, 0xd0bba3, 0xe559d4, 0x597b8c, 0x2a4483, 0xf332ba, 
0xf84867, 0x2c8d1b, 0x2fa9b0, 0x50f3dd, 0xf9f573, 0xdb61b4, 
0xfe233e, 0x6c41a6, 0xeea318, 0x775a26, 0xbc5e5c, 0xcea708, 
0x94dc57, 0xe20196, 0xf1e839, 0xbe4851, 0x5d2d2f, 0x4e9555, 
0xd96ec2, 0xe7d755, 0x6304e0, 0xc02e0e, 0xfc40a0, 0xbbf9b3, 
0x7125a7, 0x222dfb, 0xf619d8, 0x838c1c, 0x6619e6, 0xb20d55, 
0xbb5137, 0x79e809, 0xaf9149, 0x0d73de, 0x0b0da5, 0xce7f58, 
0xac1934, 0x724667, 0x7a1a13, 0x9e26bc, 0x4555e7, 0x585cb5, 
0x711d14, 0x486991, 0x480d60, 0x56adab, 0xd62f64, 0x96ee0c, 
0x212ff3, 0x5d6d88, 0xa67684, 0x95651e, 0xab9e0a, 0x4ddefe, 
0x571010, 0x836a39, 0xf8ea31, 0x9e381d, 0xeac8b1, 0xcac96b, 
0x37f21e, 0xd505e9, 0x984743, 0x9fc56c, 0x0331b7, 0x3b8bf8, 
0x86e56a, 0x8dc343, 0x6230e7, 0x93cfd5, 0x6a8f2d, 0x733005, 
0x1af021, 0xa09fcb, 0x7415a1, 0xd56b23, 0x6ff725, 0x2f4bc7, 
0xb8a591, 0x7fac59, 0x5c55de, 0x212c38, 0xb13296, 0x5cff50, 
0x366262, 0xfa7b16, 0xf4d9a6, 0x2acfe7, 0xf07403, 0xd4d604, 
0x6fd916, 0x31b1bf, 0xcbb450, 0x5bd7c8, 0x0ce194, 0x6bd643, 
0x4fd91c, 0xdf4543, 0x5f3453, 0xe2b5aa, 0xc9aec8, 0x131485, 
0xf9d2bf, 0xbadb9e, 0x76f5b9, 0xaf15cf, 0xca3182, 0x14b56d, 
0xe9fe4d, 0x50fc35, 0xf5aed5, 0xa2d0c1, 0xc96057, 0x192eb6, 
0xe91d92, 0x07d144, 0xaea3c6, 0x343566, 0x26d5b4, 0x3161e2, 
0x37f1a2, 0x209eff, 0x958e23, 0x493798, 0x35f4a6, 0x4bdc02, 
0xc2be13, 0xbe80a0, 0x0b72a3, 0x115c5f, 0x1e1bd1, 0x0db4d3, 
0x869e85, 0x96976b, 0x2ac91f, 0x8a26c2, 0x3070f0, 0x041412, 
0xfc9fa5, 0xf72a38, 0x9c6878, 0xe2aa76, 0x50cfe1, 0x559274, 
0x934e38, 0x0a92f7, 0x5533f0, 0xa63db4, 0x399971, 0xe2b755, 
0xa98a7c, 0x008f19, 0xac54d2, 0x2ea0b4, 0xf5f3e0, 0x60c849, 
0xffd269, 0xae52ce, 0x7a5fdd, 0xe9ce06, 0xfb0ae8, 0xa50cce, 
0xea9d3e, 0x3766dd, 0xb834f5, 0x0da090, 0x846f88, 0x4ae3d5, 
0x099a03, 0x2eae2d, 0xfcb40a, 0xfb9b33, 0xe281dd, 0x1b16ba, 
0xd8c0af, 0xd96b97, 0xb52dc9, 0x9c277f, 0x5951d5, 0x21ccd6, 
0xb6496b, 0x584562, 0xb3baf2, 0xa1a5c4, 0x7ca2cf, 0xa9b93d, 
0x7b7b89, 0x483d38, 
};

static const long double c[] = {
/* 93 bits of pi/2 */
#define PI_2_1 c[0]
 1.57079632679489661923132169155131424e+00L, /* 3fff921fb54442d18469898cc5100000 */

/* pi/2 - PI_2_1 */
#define PI_2_1t c[1]
 8.84372056613570112025531863263659260e-29L, /* 3fa1c06e0e68948127044533e63a0106 */
};

int32_t __ieee754_rem_pio2l(long double x, long double *y)
{
  long double z, w, t;
  double tx[8];
  int64_t exp, n, ix, hx;
  u_int64_t lx;

  GET_LDOUBLE_WORDS64 (hx, lx, x);
  ix = hx & 0x7fffffffffffffffLL;
  if (ix <= 0x3ffe921fb54442d1LL)   /* x in <-pi/4, pi/4> */
    {
      y[0] = x;
      y[1] = 0;
      return 0;
    }

  if (ix < 0x40002d97c7f3321dLL)    /* |x| in <pi/4, 3pi/4) */
    {
      if (hx > 0)
      { 
        /* 113 + 93 bit PI is ok */
        z = x - PI_2_1;
        y[0] = z - PI_2_1t;
        y[1] = (z - y[0]) - PI_2_1t;
        return 1;
      }
      else
        {
        /* 113 + 93 bit PI is ok */
        z = x + PI_2_1;
        y[0] = z + PI_2_1t;
        y[1] = (z - y[0]) + PI_2_1t;
        return -1;
      }
    }

  if (ix >= 0x7fff000000000000LL)   /* x is +=oo or NaN */
    {
      y[0] = x - x;
      y[1] = y[0];
      return 0;
    }

  /* Handle large arguments.
     We split the 113 bits of the mantissa into 5 24bit integers
     stored in a double array.  */
  exp = (ix >> 48) - 16383 - 23;

  /* This is faster than doing this in floating point, because we
     have to convert it to integers anyway and like this we can keep
     both integer and floating point units busy.  */
  tx [0] = (double)(((ix >> 25) & 0x7fffff) | 0x800000);
  tx [1] = (double)((ix >> 1) & 0xffffff);
  tx [2] = (double)(((ix << 23) | (lx >> 41)) & 0xffffff);
  tx [3] = (double)((lx >> 17) & 0xffffff);
  tx [4] = (double)((lx << 7) & 0xffffff);

  n = __kernel_rem_pio2 (tx, tx + 5, exp, ((lx << 7) & 0xffffff) ? 5 : 4,
                   3, two_over_pi);

  /* The result is now stored in 3 double values, we need to convert it into
     two long double values.  */
  t = (long double) tx [6] + (long double) tx [7];
  w = (long double) tx [5];

  if (hx >= 0)
    {
      y[0] = w + t;
      y[1] = t - (y[0] - w);
      return n;
    }
  else
    {
      y[0] = -(w + t);
      y[1] = -t - (y[0] + w);
      return -n;
    }
}

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