sysdeps/ieee754/ldbl-128/e_rem_pio2l.c

   1 /* Quad-precision floating point argument reduction.
   2    Copyright (C) 1999 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4    Contributed by Jakub Jelinek <jj@ultra.linux.cz>
   5
   6    The GNU C Library is free software; you can redistribute it and/or
   7    modify it under the terms of the GNU Lesser General Public
   8    License as published by the Free Software Foundation; either
   9    version 2.1 of the License, or (at your option) any later version.
  10
  11    The GNU C Library is distributed in the hope that it will be useful,
  12    but WITHOUT ANY WARRANTY; without even the implied warranty of
  13    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  14    Lesser General Public License for more details.
  15
  16    You should have received a copy of the GNU Lesser General Public
  17    License along with the GNU C Library; if not, see
  18    <http://www.gnu.org/licenses/>.  */
  19
  20 #include "math.h"
  21 #include "math_private.h"
  22
  23 /*
  24  * Table of constants for 2/pi, 5628 hexadecimal digits of 2/pi
  25  */
  26 static const int32_t two_over_pi[] = {
  27 0xa2f983, 0x6e4e44, 0x1529fc, 0x2757d1, 0xf534dd, 0xc0db62,
  28 0x95993c, 0x439041, 0xfe5163, 0xabdebb, 0xc561b7, 0x246e3a,
  29 0x424dd2, 0xe00649, 0x2eea09, 0xd1921c, 0xfe1deb, 0x1cb129,
  30 0xa73ee8, 0x8235f5, 0x2ebb44, 0x84e99c, 0x7026b4, 0x5f7e41,
  31 0x3991d6, 0x398353, 0x39f49c, 0x845f8b, 0xbdf928, 0x3b1ff8,
  32 0x97ffde, 0x05980f, 0xef2f11, 0x8b5a0a, 0x6d1f6d, 0x367ecf,
  33 0x27cb09, 0xb74f46, 0x3f669e, 0x5fea2d, 0x7527ba, 0xc7ebe5,
  34 0xf17b3d, 0x0739f7, 0x8a5292, 0xea6bfb, 0x5fb11f, 0x8d5d08,
  35 0x560330, 0x46fc7b, 0x6babf0, 0xcfbc20, 0x9af436, 0x1da9e3,
  36 0x91615e, 0xe61b08, 0x659985, 0x5f14a0, 0x68408d, 0xffd880,
  37 0x4d7327, 0x310606, 0x1556ca, 0x73a8c9, 0x60e27b, 0xc08c6b,
  38 0x47c419, 0xc367cd, 0xdce809, 0x2a8359, 0xc4768b, 0x961ca6,
  39 0xddaf44, 0xd15719, 0x053ea5, 0xff0705, 0x3f7e33, 0xe832c2,
  40 0xde4f98, 0x327dbb, 0xc33d26, 0xef6b1e, 0x5ef89f, 0x3a1f35,
  41 0xcaf27f, 0x1d87f1, 0x21907c, 0x7c246a, 0xfa6ed5, 0x772d30,
  42 0x433b15, 0xc614b5, 0x9d19c3, 0xc2c4ad, 0x414d2c, 0x5d000c,
  43 0x467d86, 0x2d71e3, 0x9ac69b, 0x006233, 0x7cd2b4, 0x97a7b4,
  44 0xd55537, 0xf63ed7, 0x1810a3, 0xfc764d, 0x2a9d64, 0xabd770,
  45 0xf87c63, 0x57b07a, 0xe71517, 0x5649c0, 0xd9d63b, 0x3884a7,
  46 0xcb2324, 0x778ad6, 0x23545a, 0xb91f00, 0x1b0af1, 0xdfce19,
  47 0xff319f, 0x6a1e66, 0x615799, 0x47fbac, 0xd87f7e, 0xb76522,
  48 0x89e832, 0x60bfe6, 0xcdc4ef, 0x09366c, 0xd43f5d, 0xd7de16,
  49 0xde3b58, 0x929bde, 0x2822d2, 0xe88628, 0x4d58e2, 0x32cac6,
  50 0x16e308, 0xcb7de0, 0x50c017, 0xa71df3, 0x5be018, 0x34132e,
  51 0x621283, 0x014883, 0x5b8ef5, 0x7fb0ad, 0xf2e91e, 0x434a48,
  52 0xd36710, 0xd8ddaa, 0x425fae, 0xce616a, 0xa4280a, 0xb499d3,
  53 0xf2a606, 0x7f775c, 0x83c2a3, 0x883c61, 0x78738a, 0x5a8caf,
  54 0xbdd76f, 0x63a62d, 0xcbbff4, 0xef818d, 0x67c126, 0x45ca55,
  55 0x36d9ca, 0xd2a828, 0x8d61c2, 0x77c912, 0x142604, 0x9b4612,
  56 0xc459c4, 0x44c5c8, 0x91b24d, 0xf31700, 0xad43d4, 0xe54929,
  57 0x10d5fd, 0xfcbe00, 0xcc941e, 0xeece70, 0xf53e13, 0x80f1ec,
  58 0xc3e7b3, 0x28f8c7, 0x940593, 0x3e71c1, 0xb3092e, 0xf3450b,
  59 0x9c1288, 0x7b20ab, 0x9fb52e, 0xc29247, 0x2f327b, 0x6d550c,
  60 0x90a772, 0x1fe76b, 0x96cb31, 0x4a1679, 0xe27941, 0x89dff4,
  61 0x9794e8, 0x84e6e2, 0x973199, 0x6bed88, 0x365f5f, 0x0efdbb,
  62 0xb49a48, 0x6ca467, 0x427271, 0x325d8d, 0xb8159f, 0x09e5bc,
  63 0x25318d, 0x3974f7, 0x1c0530, 0x010c0d, 0x68084b, 0x58ee2c,
  64 0x90aa47, 0x02e774, 0x24d6bd, 0xa67df7, 0x72486e, 0xef169f,
  65 0xa6948e, 0xf691b4, 0x5153d1, 0xf20acf, 0x339820, 0x7e4bf5,
  66 0x6863b2, 0x5f3edd, 0x035d40, 0x7f8985, 0x295255, 0xc06437,
  67 0x10d86d, 0x324832, 0x754c5b, 0xd4714e, 0x6e5445, 0xc1090b,
  68 0x69f52a, 0xd56614, 0x9d0727, 0x50045d, 0xdb3bb4, 0xc576ea,
  69 0x17f987, 0x7d6b49, 0xba271d, 0x296996, 0xacccc6, 0x5414ad,
  70 0x6ae290, 0x89d988, 0x50722c, 0xbea404, 0x940777, 0x7030f3,
  71 0x27fc00, 0xa871ea, 0x49c266, 0x3de064, 0x83dd97, 0x973fa3,
  72 0xfd9443, 0x8c860d, 0xde4131, 0x9d3992, 0x8c70dd, 0xe7b717,
  73 0x3bdf08, 0x2b3715, 0xa0805c, 0x93805a, 0x921110, 0xd8e80f,
  74 0xaf806c, 0x4bffdb, 0x0f9038, 0x761859, 0x15a562, 0xbbcb61,
  75 0xb989c7, 0xbd4010, 0x04f2d2, 0x277549, 0xf6b6eb, 0xbb22db,
  76 0xaa140a, 0x2f2689, 0x768364, 0x333b09, 0x1a940e, 0xaa3a51,
  77 0xc2a31d, 0xaeedaf, 0x12265c, 0x4dc26d, 0x9c7a2d, 0x9756c0,
  78 0x833f03, 0xf6f009, 0x8c402b, 0x99316d, 0x07b439, 0x15200c,
  79 0x5bc3d8, 0xc492f5, 0x4badc6, 0xa5ca4e, 0xcd37a7, 0x36a9e6,
  80 0x9492ab, 0x6842dd, 0xde6319, 0xef8c76, 0x528b68, 0x37dbfc,
  81 0xaba1ae, 0x3115df, 0xa1ae00, 0xdafb0c, 0x664d64, 0xb705ed,
  82 0x306529, 0xbf5657, 0x3aff47, 0xb9f96a, 0xf3be75, 0xdf9328,
  83 0x3080ab, 0xf68c66, 0x15cb04, 0x0622fa, 0x1de4d9, 0xa4b33d,
  84 0x8f1b57, 0x09cd36, 0xe9424e, 0xa4be13, 0xb52333, 0x1aaaf0,
  85 0xa8654f, 0xa5c1d2, 0x0f3f0b, 0xcd785b, 0x76f923, 0x048b7b,
  86 0x721789, 0x53a6c6, 0xe26e6f, 0x00ebef, 0x584a9b, 0xb7dac4,
  87 0xba66aa, 0xcfcf76, 0x1d02d1, 0x2df1b1, 0xc1998c, 0x77adc3,
  88 0xda4886, 0xa05df7, 0xf480c6, 0x2ff0ac, 0x9aecdd, 0xbc5c3f,
  89 0x6dded0, 0x1fc790, 0xb6db2a, 0x3a25a3, 0x9aaf00, 0x9353ad,
  90 0x0457b6, 0xb42d29, 0x7e804b, 0xa707da, 0x0eaa76, 0xa1597b,
  91 0x2a1216, 0x2db7dc, 0xfde5fa, 0xfedb89, 0xfdbe89, 0x6c76e4,
  92 0xfca906, 0x70803e, 0x156e85, 0xff87fd, 0x073e28, 0x336761,
  93 0x86182a, 0xeabd4d, 0xafe7b3, 0x6e6d8f, 0x396795, 0x5bbf31,
  94 0x48d784, 0x16df30, 0x432dc7, 0x356125, 0xce70c9, 0xb8cb30,
  95 0xfd6cbf, 0xa200a4, 0xe46c05, 0xa0dd5a, 0x476f21, 0xd21262,
  96 0x845cb9, 0x496170, 0xe0566b, 0x015299, 0x375550, 0xb7d51e,
  97 0xc4f133, 0x5f6e13, 0xe4305d, 0xa92e85, 0xc3b21d, 0x3632a1,
  98 0xa4b708, 0xd4b1ea, 0x21f716, 0xe4698f, 0x77ff27, 0x80030c,
  99 0x2d408d, 0xa0cd4f, 0x99a520, 0xd3a2b3, 0x0a5d2f, 0x42f9b4,
 100 0xcbda11, 0xd0be7d, 0xc1db9b, 0xbd17ab, 0x81a2ca, 0x5c6a08,
 101 0x17552e, 0x550027, 0xf0147f, 0x8607e1, 0x640b14, 0x8d4196,
 102 0xdebe87, 0x2afdda, 0xb6256b, 0x34897b, 0xfef305, 0x9ebfb9,
 103 0x4f6a68, 0xa82a4a, 0x5ac44f, 0xbcf82d, 0x985ad7, 0x95c7f4,
 104 0x8d4d0d, 0xa63a20, 0x5f57a4, 0xb13f14, 0x953880, 0x0120cc,
 105 0x86dd71, 0xb6dec9, 0xf560bf, 0x11654d, 0x6b0701, 0xacb08c,
 106 0xd0c0b2, 0x485551, 0x0efb1e, 0xc37295, 0x3b06a3, 0x3540c0,
 107 0x7bdc06, 0xcc45e0, 0xfa294e, 0xc8cad6, 0x41f3e8, 0xde647c,
 108 0xd8649b, 0x31bed9, 0xc397a4, 0xd45877, 0xc5e369, 0x13daf0,
 109 0x3c3aba, 0x461846, 0x5f7555, 0xf5bdd2, 0xc6926e, 0x5d2eac,
 110 0xed440e, 0x423e1c, 0x87c461, 0xe9fd29, 0xf3d6e7, 0xca7c22,
 111 0x35916f, 0xc5e008, 0x8dd7ff, 0xe26a6e, 0xc6fdb0, 0xc10893,
 112 0x745d7c, 0xb2ad6b, 0x9d6ecd, 0x7b723e, 0x6a11c6, 0xa9cff7,
 113 0xdf7329, 0xbac9b5, 0x5100b7, 0x0db2e2, 0x24ba74, 0x607de5,
 114 0x8ad874, 0x2c150d, 0x0c1881, 0x94667e, 0x162901, 0x767a9f,
 115 0xbefdfd, 0xef4556, 0x367ed9, 0x13d9ec, 0xb9ba8b, 0xfc97c4,
 116 0x27a831, 0xc36ef1, 0x36c594, 0x56a8d8, 0xb5a8b4, 0x0ecccf,
 117 0x2d8912, 0x34576f, 0x89562c, 0xe3ce99, 0xb920d6, 0xaa5e6b,
 118 0x9c2a3e, 0xcc5f11, 0x4a0bfd, 0xfbf4e1, 0x6d3b8e, 0x2c86e2,
 119 0x84d4e9, 0xa9b4fc, 0xd1eeef, 0xc9352e, 0x61392f, 0x442138,
 120 0xc8d91b, 0x0afc81, 0x6a4afb, 0xd81c2f, 0x84b453, 0x8c994e,
 121 0xcc2254, 0xdc552a, 0xd6c6c0, 0x96190b, 0xb8701a, 0x649569,
 122 0x605a26, 0xee523f, 0x0f117f, 0x11b5f4, 0xf5cbfc, 0x2dbc34,
 123 0xeebc34, 0xcc5de8, 0x605edd, 0x9b8e67, 0xef3392, 0xb817c9,
 124 0x9b5861, 0xbc57e1, 0xc68351, 0x103ed8, 0x4871dd, 0xdd1c2d,
 125 0xa118af, 0x462c21, 0xd7f359, 0x987ad9, 0xc0549e, 0xfa864f,
 126 0xfc0656, 0xae79e5, 0x362289, 0x22ad38, 0xdc9367, 0xaae855,
 127 0x382682, 0x9be7ca, 0xa40d51, 0xb13399, 0x0ed7a9, 0x480569,
 128 0xf0b265, 0xa7887f, 0x974c88, 0x36d1f9, 0xb39221, 0x4a827b,
 129 0x21cf98, 0xdc9f40, 0x5547dc, 0x3a74e1, 0x42eb67, 0xdf9dfe,
 130 0x5fd45e, 0xa4677b, 0x7aacba, 0xa2f655, 0x23882b, 0x55ba41,
 131 0x086e59, 0x862a21, 0x834739, 0xe6e389, 0xd49ee5, 0x40fb49,
 132 0xe956ff, 0xca0f1c, 0x8a59c5, 0x2bfa94, 0xc5c1d3, 0xcfc50f,
 133 0xae5adb, 0x86c547, 0x624385, 0x3b8621, 0x94792c, 0x876110,
 134 0x7b4c2a, 0x1a2c80, 0x12bf43, 0x902688, 0x893c78, 0xe4c4a8,
 135 0x7bdbe5, 0xc23ac4, 0xeaf426, 0x8a67f7, 0xbf920d, 0x2ba365,
 136 0xb1933d, 0x0b7cbd, 0xdc51a4, 0x63dd27, 0xdde169, 0x19949a,
 137 0x9529a8, 0x28ce68, 0xb4ed09, 0x209f44, 0xca984e, 0x638270,
 138 0x237c7e, 0x32b90f, 0x8ef5a7, 0xe75614, 0x08f121, 0x2a9db5,
 139 0x4d7e6f, 0x5119a5, 0xabf9b5, 0xd6df82, 0x61dd96, 0x023616,
 140 0x9f3ac4, 0xa1a283, 0x6ded72, 0x7a8d39, 0xa9b882, 0x5c326b,
 141 0x5b2746, 0xed3400, 0x7700d2, 0x55f4fc, 0x4d5901, 0x8071e0,
 142 0xe13f89, 0xb295f3, 0x64a8f1, 0xaea74b, 0x38fc4c, 0xeab2bb,
 143 0x47270b, 0xabc3a7, 0x34ba60, 0x52dd34, 0xf8563a, 0xeb7e8a,
 144 0x31bb36, 0x5895b7, 0x47f7a9, 0x94c3aa, 0xd39225, 0x1e7f3e,
 145 0xd8974e, 0xbba94f, 0xd8ae01, 0xe661b4, 0x393d8e, 0xa523aa,
 146 0x33068e, 0x1633b5, 0x3bb188, 0x1d3a9d, 0x4013d0, 0xcc1be5,
 147 0xf862e7, 0x3bf28f, 0x39b5bf, 0x0bc235, 0x22747e, 0xa247c0,
 148 0xd52d1f, 0x19add3, 0x9094df, 0x9311d0, 0xb42b25, 0x496db2,
 149 0xe264b2, 0x5ef135, 0x3bc6a4, 0x1a4ad0, 0xaac92e, 0x64e886,
 150 0x573091, 0x982cfb, 0x311b1a, 0x08728b, 0xbdcee1, 0x60e142,
 151 0xeb641d, 0xd0bba3, 0xe559d4, 0x597b8c, 0x2a4483, 0xf332ba,
 152 0xf84867, 0x2c8d1b, 0x2fa9b0, 0x50f3dd, 0xf9f573, 0xdb61b4,
 153 0xfe233e, 0x6c41a6, 0xeea318, 0x775a26, 0xbc5e5c, 0xcea708,
 154 0x94dc57, 0xe20196, 0xf1e839, 0xbe4851, 0x5d2d2f, 0x4e9555,
 155 0xd96ec2, 0xe7d755, 0x6304e0, 0xc02e0e, 0xfc40a0, 0xbbf9b3,
 156 0x7125a7, 0x222dfb, 0xf619d8, 0x838c1c, 0x6619e6, 0xb20d55,
 157 0xbb5137, 0x79e809, 0xaf9149, 0x0d73de, 0x0b0da5, 0xce7f58,
 158 0xac1934, 0x724667, 0x7a1a13, 0x9e26bc, 0x4555e7, 0x585cb5,
 159 0x711d14, 0x486991, 0x480d60, 0x56adab, 0xd62f64, 0x96ee0c,
 160 0x212ff3, 0x5d6d88, 0xa67684, 0x95651e, 0xab9e0a, 0x4ddefe,
 161 0x571010, 0x836a39, 0xf8ea31, 0x9e381d, 0xeac8b1, 0xcac96b,
 162 0x37f21e, 0xd505e9, 0x984743, 0x9fc56c, 0x0331b7, 0x3b8bf8,
 163 0x86e56a, 0x8dc343, 0x6230e7, 0x93cfd5, 0x6a8f2d, 0x733005,
 164 0x1af021, 0xa09fcb, 0x7415a1, 0xd56b23, 0x6ff725, 0x2f4bc7,
 165 0xb8a591, 0x7fac59, 0x5c55de, 0x212c38, 0xb13296, 0x5cff50,
 166 0x366262, 0xfa7b16, 0xf4d9a6, 0x2acfe7, 0xf07403, 0xd4d604,
 167 0x6fd916, 0x31b1bf, 0xcbb450, 0x5bd7c8, 0x0ce194, 0x6bd643,
 168 0x4fd91c, 0xdf4543, 0x5f3453, 0xe2b5aa, 0xc9aec8, 0x131485,
 169 0xf9d2bf, 0xbadb9e, 0x76f5b9, 0xaf15cf, 0xca3182, 0x14b56d,
 170 0xe9fe4d, 0x50fc35, 0xf5aed5, 0xa2d0c1, 0xc96057, 0x192eb6,
 171 0xe91d92, 0x07d144, 0xaea3c6, 0x343566, 0x26d5b4, 0x3161e2,
 172 0x37f1a2, 0x209eff, 0x958e23, 0x493798, 0x35f4a6, 0x4bdc02,
 173 0xc2be13, 0xbe80a0, 0x0b72a3, 0x115c5f, 0x1e1bd1, 0x0db4d3,
 174 0x869e85, 0x96976b, 0x2ac91f, 0x8a26c2, 0x3070f0, 0x041412,
 175 0xfc9fa5, 0xf72a38, 0x9c6878, 0xe2aa76, 0x50cfe1, 0x559274,
 176 0x934e38, 0x0a92f7, 0x5533f0, 0xa63db4, 0x399971, 0xe2b755,
 177 0xa98a7c, 0x008f19, 0xac54d2, 0x2ea0b4, 0xf5f3e0, 0x60c849,
 178 0xffd269, 0xae52ce, 0x7a5fdd, 0xe9ce06, 0xfb0ae8, 0xa50cce,
 179 0xea9d3e, 0x3766dd, 0xb834f5, 0x0da090, 0x846f88, 0x4ae3d5,
 180 0x099a03, 0x2eae2d, 0xfcb40a, 0xfb9b33, 0xe281dd, 0x1b16ba,
 181 0xd8c0af, 0xd96b97, 0xb52dc9, 0x9c277f, 0x5951d5, 0x21ccd6,
 182 0xb6496b, 0x584562, 0xb3baf2, 0xa1a5c4, 0x7ca2cf, 0xa9b93d,
 183 0x7b7b89, 0x483d38,
 184 };
 185
 186 static const long double c[] = {
 187 /* 93 bits of pi/2 */
 188 #define PI_2_1 c[0]
 189  1.57079632679489661923132169155131424e+00L, /* 3fff921fb54442d18469898cc5100000 */
 190
 191 /* pi/2 - PI_2_1 */
 192 #define PI_2_1t c[1]
 193  8.84372056613570112025531863263659260e-29L, /* 3fa1c06e0e68948127044533e63a0106 */
 194 };
 195
 196 int32_t __ieee754_rem_pio2l(long double x, long double *y)
 197 {
 198   long double z, w, t;
 199   double tx[8];
 200   int64_t exp, n, ix, hx;
 201   u_int64_t lx;
 202
 203   GET_LDOUBLE_WORDS64 (hx, lx, x);
 204   ix = hx & 0x7fffffffffffffffLL;
 205   if (ix <= 0x3ffe921fb54442d1LL)       /* x in <-pi/4, pi/4> */
 206     {
 207       y[0] = x;
 208       y[1] = 0;
 209       return 0;
 210     }
 211
 212   if (ix < 0x40002d97c7f3321dLL)        /* |x| in <pi/4, 3pi/4) */
 213     {
 214       if (hx > 0)
 215         {
 216           /* 113 + 93 bit PI is ok */
 217           z = x - PI_2_1;
 218           y[0] = z - PI_2_1t;
 219           y[1] = (z - y[0]) - PI_2_1t;
 220           return 1;
 221         }
 222       else
 223         {
 224           /* 113 + 93 bit PI is ok */
 225           z = x + PI_2_1;
 226           y[0] = z + PI_2_1t;
 227           y[1] = (z - y[0]) + PI_2_1t;
 228           return -1;
 229         }
 230     }
 231
 232   if (ix >= 0x7fff000000000000LL)       /* x is +=oo or NaN */
 233     {
 234       y[0] = x - x;
 235       y[1] = y[0];
 236       return 0;
 237     }
 238
 239   /* Handle large arguments.
 240      We split the 113 bits of the mantissa into 5 24bit integers
 241      stored in a double array.  */
 242   exp = (ix >> 48) - 16383 - 23;
 243
 244   /* This is faster than doing this in floating point, because we
 245      have to convert it to integers anyway and like this we can keep
 246      both integer and floating point units busy.  */
 247   tx [0] = (double)(((ix >> 25) & 0x7fffff) | 0x800000);
 248   tx [1] = (double)((ix >> 1) & 0xffffff);
 249   tx [2] = (double)(((ix << 23) | (lx >> 41)) & 0xffffff);
 250   tx [3] = (double)((lx >> 17) & 0xffffff);
 251   tx [4] = (double)((lx << 7) & 0xffffff);
 252
 253   n = __kernel_rem_pio2 (tx, tx + 5, exp, ((lx << 7) & 0xffffff) ? 5 : 4,
 254                          3, two_over_pi);
 255
 256   /* The result is now stored in 3 double values, we need to convert it into
 257      two long double values.  */
 258   t = (long double) tx [6] + (long double) tx [7];
 259   w = (long double) tx [5];
 260
 261   if (hx >= 0)
 262     {
 263       y[0] = w + t;
 264       y[1] = t - (y[0] - w);
 265       return n;
 266     }
 267   else
 268     {
 269       y[0] = -(w + t);
 270       y[1] = -t - (y[0] + w);
 271       return -n;
 272     }
 273 }