/* 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 This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2, or (at your option) any later version. This program 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 General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. */ #include #include #include "mathl.h" /* 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) { long double z, w, t; double tx[8]; int exp, n; if (x >= -0.78539816339744830961566084581987572104929234984377 && x < 0.78539816339744830961566084581987572104929234984377) /* x in <-pi/4, pi/4> */ { y[0] = x; y[1] = 0; return 0; } if (x >= 2.35619449019234492884698253745962716314787704953131 && x < 2.35619449019234492884698253745962716314787704953131) if (x > 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 (x + x == x || x != x) /* 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. */ z = frexp (x, &exp); 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); 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. */ t = (long double) tx[6] + (long double) tx[7]; 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(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[] ouput 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 */ int kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec, const int *ipio2) { int jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; double z, fw, f[20], fq[20], q[20]; /* initialize jk */ jk = init_jk[prec]; jp = jk; /* determine jx,jv,q0, note that 3>q0 */ jx = nx - 1; jv = (e0 - 3) / 24; if (jv < 0) jv = 0; q0 = e0 - 24 * (jv + 1); /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ j = jv - jx; m = jx + jk; for (i = 0; i <= m; i++, j++) f[i] = (j < 0) ? zero : (double) ipio2[j]; /* compute q[0],q[1],...q[jk] */ for (i = 0; i <= jk; i++) { for (j = 0, fw = 0.0; j <= jx; j++) fw += x[j] * f[jx + i - j]; q[i] = fw; } jz = jk; recompute: /* distill q[] into iq[] reversingly */ for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) { 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 */ n = (int) z; z -= (double) n; ih = 0; if (q0 > 0) { /* need iq[jz-1] to determine n */ 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; carry = 0; for (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 (i = jz - 1; i >= jk; i--) j |= iq[i]; if (j == 0) { /* need recomputation */ for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */ for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */ f[jx + i] = (double) ipio2[jv + i]; 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) { 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 */ fw = ldexp (one, q0); for (i = jz; i >= 0; i--) { q[i] = fw * (double) iq[i]; fw *= twon24; } /* compute PIo2[0,...,jp]*q[jz,...,0] */ for (i = jz; i >= 0; i--) { 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[] */ switch (prec) { case 0: fw = 0.0; for (i = jz; i >= 0; i--) fw += fq[i]; y[0] = (ih == 0) ? fw : -fw; break; case 1: case 2: fw = 0.0; for (i = jz; i >= 0; i--) fw += fq[i]; y[0] = (ih == 0) ? fw : -fw; fw = fq[0] - fw; for (i = 1; i <= jz; i++) fw += fq[i]; y[1] = (ih == 0) ? fw : -fw; break; case 3: /* painful */ for (i = jz; i > 0; i--) { fw = fq[i - 1] + fq[i]; fq[i] += fq[i - 1] - fw; fq[i - 1] = fw; } for (i = jz; i > 1; i--) { fw = fq[i - 1] + fq[i]; fq[i] += fq[i - 1] - fw; fq[i - 1] = fw; } 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; }