lib/trigl.c

/* Quad-precision floating point argument reduction.  -*- coding: utf-8 -*-
   Copyright (C) 1999, 2007, 2009-2026 Free Software Foundation, Inc.
   This file is part of the GNU C Library.
   Contributed by Jakub Jelinek <jj@ultra.linux.cz>

   This file 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 3 of the
   License, or (at your option) any later version.

   This file 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 this program.  If not, see <https://www.gnu.org/licenses/>.  */

#include <config.h>

/* Specification.  */
#include "trigl.h"

#include <float.h>
#include <math.h>

/* Code based on glibc/sysdeps/ieee754/ldbl-128/e_rem_pio2l.c
   and           glibc/sysdeps/ieee754/dbl-64/k_rem_pio2.c.  */

/* Table of constants for 2/pi, 5628 hexadecimal digits of 2/pi */
static const int 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 */
};

static int kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec,
                            const int *ipio2);

int
ieee754_rem_pio2l (long double x, long double *y)
{
  if (x >= -0.78539816339744830961566084581987572104929234984377
      && x <= 0.78539816339744830961566084581987572104929234984377)
    /* x in <-pi/4, pi/4> */
    {
      y[0] = x;
      y[1] = 0;
      return 0;
    }

  if (x > 0 && x < 2.35619449019234492884698253745962716314787704953131)
      {
        /* 113 + 93 bit PI is ok */
        long double z = x - PI_2_1;
        y[0] = z - PI_2_1t;
        y[1] = (z - y[0]) - PI_2_1t;
        return 1;
      }

  if (x < 0 && x > -2.35619449019234492884698253745962716314787704953131)
      {
        /* 113 + 93 bit PI is ok */
        long double z = x + PI_2_1;
        y[0] = z + PI_2_1t;
        y[1] = (z - y[0]) + PI_2_1t;
        return -1;
      }

  if (x + x == x)       /* x is ±oo */
    {
      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.  */
  int exp;
  long double z = frexp (x, &exp);

  double tx[8];
  tx[0] = floorl (z *= 16777216.0);
  z -= tx[0];
  tx[1] = floorl (z *= 16777216.0);
  z -= tx[1];
  tx[2] = floorl (z *= 16777216.0);
  z -= tx[2];
  tx[3] = floorl (z *= 16777216.0);
  z -= tx[3];
  tx[4] = floorl (z *= 16777216.0);

  int n = kernel_rem_pio2 (tx, tx + 5, exp - 24, tx[4] ? 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.  */
  long double t = (long double) tx[6] + (long double) tx[7];
  long double w = (long double) tx[5];

  if (x > 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;
    }
}

/* @(#)k_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 GCC_LINT && !defined lint
static char rcsid[] =
  "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
#endif

/*
 * kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
 * double x[],y[]; int e0,nx,prec; int ipio2[];
 *
 * kernel_rem_pio2 return the last three digits of N with
 *              y = x - N*pi/2
 * so that |y| < pi/2.
 *
 * The method is to compute the integer (mod 8) and fraction parts of
 * (2/pi)*x without doing the full multiplication. In general we
 * skip the part of the product that are known to be a huge integer (
 * more accurately, = 0 mod 8 ). Thus the number of operations are
 * independent of the exponent of the input.
 *
 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
 *
 * Input parameters:
 *      x[]     The input value (must be positive) is broken into nx
 *              pieces of 24-bit integers in double precision format.
 *              x[i] will be the i-th 24 bit of x. The scaled exponent
 *              of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
 *              match x's up to 24 bits.
 *
 *              Example of breaking a double positive z into x[0]+x[1]+x[2]:
 *                      e0 = ilogb(z)-23
 *                      z  = scalbn(z,-e0)
 *              for i = 0,1,2
 *                      x[i] = floor(z)
 *                      z    = (z-x[i])*2**24
 *
 *
 *      y[]     output result in an array of double precision numbers.
 *              The dimension of y[] is:
 *                      24-bit  precision       1
 *                      53-bit  precision       2
 *                      64-bit  precision       2
 *                      113-bit precision       3
 *              The actual value is the sum of them. Thus for 113-bit
 *              precision, one may have to do something like:
 *
 *              long double t,w,r_head, r_tail;
 *              t = (long double)y[2] + (long double)y[1];
 *              w = (long double)y[0];
 *              r_head = t+w;
 *              r_tail = w - (r_head - t);
 *
 *      e0      The exponent of x[0]
 *
 *      nx      dimension of x[]
 *
 *      prec    an integer indicating the precision:
 *                      0       24  bits (single)
 *                      1       53  bits (double)
 *                      2       64  bits (extended)
 *                      3       113 bits (quad)
 *
 *      ipio2[]
 *              integer array, contains the (24*i)-th to (24*i+23)-th
 *              bit of 2/pi after binary point. The corresponding
 *              floating value is
 *
 *                      ipio2[i] * 2^(-24(i+1)).
 *
 * External function:
 *      double scalbn(), floor();
 *
 *
 * Here is the description of some local variables:
 *
 *      jk      jk+1 is the initial number of terms of ipio2[] needed
 *              in the computation. The recommended value is 2,3,4,
 *              6 for single, double, extended,and quad.
 *
 *      jz      local integer variable indicating the number of
 *              terms of ipio2[] used.
 *
 *      jx      nx - 1
 *
 *      jv      index for pointing to the suitable ipio2[] for the
 *              computation. In general, we want
 *                      ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
 *              is an integer. Thus
 *                      e0-3-24*jv >= 0 or (e0-3)/24 >= jv
 *              Hence jv = max(0,(e0-3)/24).
 *
 *      jp      jp+1 is the number of terms in PIo2[] needed, jp = jk.
 *
 *      q[]     double array with integral value, representing the
 *              24-bits chunk of the product of x and 2/pi.
 *
 *      q0      the corresponding exponent of q[0]. Note that the
 *              exponent for q[i] would be q0-24*i.
 *
 *      PIo2[]  double precision array, obtained by cutting pi/2
 *              into 24 bits chunks.
 *
 *      f[]     ipio2[] in floating point
 *
 *      iq[]    integer array by breaking up q[] in 24-bits chunk.
 *
 *      fq[]    final product of x*(2/pi) in fq[0],..,fq[jk]
 *
 *      ih      integer. If >0 it indicates q[] is >= 0.5, hence
 *              it also indicates the *sign* of the result.
 *
 */


/*
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

static const int init_jk[] = { 2, 3, 4, 6 };    /* initial value for jk */
static const double PIo2[] = {
  1.57079625129699707031e+00,   /* 0x3FF921FB, 0x40000000 */
  7.54978941586159635335e-08,   /* 0x3E74442D, 0x00000000 */
  5.39030252995776476554e-15,   /* 0x3CF84698, 0x80000000 */
  3.28200341580791294123e-22,   /* 0x3B78CC51, 0x60000000 */
  1.27065575308067607349e-29,   /* 0x39F01B83, 0x80000000 */
  1.22933308981111328932e-36,   /* 0x387A2520, 0x40000000 */
  2.73370053816464559624e-44,   /* 0x36E38222, 0x80000000 */
  2.16741683877804819444e-51,   /* 0x3569F31D, 0x00000000 */
};

static const double zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07,  /* 0x41700000, 0x00000000 */
  twon24 = 5.96046447753906250000e-08;  /* 0x3E700000, 0x00000000 */

static int
kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec,
                 const int *ipio2)
{
  /* initialize jk */
  int jk = init_jk[prec];
  int jp = jk;

  /* determine jx,jv,q0, note that 3>q0 */
  int jx = nx - 1;
  int jv = (e0 - 3) / 24;
  if (jv < 0)
    jv = 0;
  int q0 = e0 - 24 * (jv + 1);

  /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
  int j = jv - jx;
  int m = jx + jk;
  double f[20];
  for (int i = 0; i <= m; i++, j++)
    f[i] = (j < 0) ? zero : (double) ipio2[j];

  /* compute q[0],q[1],...q[jk] */
  double q[20];
  for (int i = 0; i <= jk; i++)
    {
      double fw;
      for (j = 0, fw = 0.0; j <= jx; j++)
        fw += x[j] * f[jx + i - j];
      q[i] = fw;
    }

  int jz = jk;
recompute: ;
  double z;

  /* distill q[] into iq[] in reverse order */
  int iq[20];
  {
    int i;
    for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--)
      {
        double fw = (double) ((int) (twon24 * z));
        iq[i] = (int) (z - two24 * fw);
        z = q[j - 1] + fw;
      }
  }

  /* compute n */
  z = ldexp (z, q0);            /* actual value of z */
  z -= 8.0 * floor (z * 0.125); /* trim off integer >= 8 */
  int n = (int) z;
  z -= (double) n;
  int ih = 0;
  if (q0 > 0)
    {                           /* need iq[jz-1] to determine n */
      int i = (iq[jz - 1] >> (24 - q0));
      n += i;
      iq[jz - 1] -= i << (24 - q0);
      ih = iq[jz - 1] >> (23 - q0);
    }
  else if (q0 == 0)
    ih = iq[jz - 1] >> 23;
  else if (z >= 0.5)
    ih = 2;

  if (ih > 0)
    {                           /* q > 0.5 */
      n += 1;
      int carry = 0;
      for (int i = 0; i < jz; i++)
        {                       /* compute 1-q */
          j = iq[i];
          if (carry == 0)
            {
              if (j != 0)
                {
                  carry = 1;
                  iq[i] = 0x1000000 - j;
                }
            }
          else
            iq[i] = 0xffffff - j;
        }
      if (q0 > 0)
        {                       /* rare case: chance is 1 in 12 */
          switch (q0)
            {
            case 1:
              iq[jz - 1] &= 0x7fffff;
              break;
            case 2:
              iq[jz - 1] &= 0x3fffff;
              break;
            }
        }
      if (ih == 2)
        {
          z = one - z;
          if (carry != 0)
            z -= ldexp (one, q0);
        }
    }

  /* check if recomputation is needed */
  if (z == zero)
    {
      j = 0;
      for (int i = jz - 1; i >= jk; i--)
        j |= iq[i];
      if (j == 0)
        {                       /* need recomputation */
          int k;
          for (k = 1; iq[jk - k] == 0; k++);    /* k = no. of terms needed */

          for (int i = jz + 1; i <= jz + k; i++)
            {                   /* add q[jz+1] to q[jz+k] */
              f[jx + i] = (double) ipio2[jv + i];
              double fw;
              for (j = 0, fw = 0.0; j <= jx; j++)
                fw += x[j] * f[jx + i - j];
              q[i] = fw;
            }
          jz += k;
          goto recompute;
        }
    }

  /* chop off zero terms */
  if (z == 0.0)
    {
      jz -= 1;
      q0 -= 24;
      while (iq[jz] == 0)
        {
          jz--;
          q0 -= 24;
        }
    }
  else
    {                           /* break z into 24-bit if necessary */
      z = ldexp (z, -q0);
      if (z >= two24)
        {
          double fw = (double) ((int) (twon24 * z));
          iq[jz] = (int) (z - two24 * fw);
          jz += 1;
          q0 += 24;
          iq[jz] = (int) fw;
        }
      else
        iq[jz] = (int) z;
    }

  /* convert integer "bit" chunk to floating-point value */
  {
    double fw = ldexp (one, q0);
    for (int i = jz; i >= 0; i--)
      {
        q[i] = fw * (double) iq[i];
        fw *= twon24;
      }
  }

  /* compute PIo2[0,...,jp]*q[jz,...,0] */
  double fq[20];
  for (int i = jz; i >= 0; i--)
    {
      double fw;
      int k;
      for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
        fw += PIo2[k] * q[i + k];
      fq[jz - i] = fw;
    }

  /* compress fq[] into y[] */
  double fw;
  switch (prec)
    {
    case 0:
      fw = 0.0;
      for (int i = jz; i >= 0; i--)
        fw += fq[i];
      y[0] = (ih == 0) ? fw : -fw;
      break;
    case 1:
    case 2:
      fw = 0.0;
      for (int i = jz; i >= 0; i--)
        fw += fq[i];
      y[0] = (ih == 0) ? fw : -fw;
      fw = fq[0] - fw;
      for (int i = 1; i <= jz; i++)
        fw += fq[i];
      y[1] = (ih == 0) ? fw : -fw;
      break;
    case 3:                     /* painful */
      for (int i = jz; i > 0; i--)
        {
          fw = fq[i - 1] + fq[i];
          fq[i] += fq[i - 1] - fw;
          fq[i - 1] = fw;
        }
      for (int i = jz; i > 1; i--)
        {
          fw = fq[i - 1] + fq[i];
          fq[i] += fq[i - 1] - fw;
          fq[i - 1] = fw;
        }
      int i;
      for (fw = 0.0, i = jz; i >= 2; i--)
        fw += fq[i];
      if (ih == 0)
        {
          y[0] = fq[0];
          y[1] = fq[1];
          y[2] = fw;
        }
      else
        {
          y[0] = -fq[0];
          y[1] = -fq[1];
          y[2] = -fw;
        }
    }
  return n & 7;
}