1 /* 2 * Data for the log part of pow. 3 * 4 * Copyright (c) 2018, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "math_config.h" 9 10 #define N (1 << POW_LOG_TABLE_BITS) 11 12 const struct pow_log_data __pow_log_data = { 13 .ln2hi = 0x1.62e42fefa3800p-1, 14 .ln2lo = 0x1.ef35793c76730p-45, 15 .poly = { 16 #if N == 128 && POW_LOG_POLY_ORDER == 8 17 // relative error: 0x1.11922ap-70 18 // in -0x1.6bp-8 0x1.6bp-8 19 // Coefficients are scaled to match the scaling during evaluation. 20 -0x1p-1, 21 0x1.555555555556p-2 * -2, 22 -0x1.0000000000006p-2 * -2, 23 0x1.999999959554ep-3 * 4, 24 -0x1.555555529a47ap-3 * 4, 25 0x1.2495b9b4845e9p-3 * -8, 26 -0x1.0002b8b263fc3p-3 * -8, 27 #endif 28 }, 29 /* Algorithm: 30 31 x = 2^k z 32 log(x) = k ln2 + log(c) + log(z/c) 33 log(z/c) = poly(z/c - 1) 34 35 where z is in [0x1.69555p-1; 0x1.69555p0] which is split into N subintervals 36 and z falls into the ith one, then table entries are computed as 37 38 tab[i].invc = 1/c 39 tab[i].logc = round(0x1p43*log(c))/0x1p43 40 tab[i].logctail = (double)(log(c) - logc) 41 42 where c is chosen near the center of the subinterval such that 1/c has only a 43 few precision bits so z/c - 1 is exactly representible as double: 44 45 1/c = center < 1 ? round(N/center)/N : round(2*N/center)/N/2 46 47 Note: |z/c - 1| < 1/N for the chosen c, |log(c) - logc - logctail| < 0x1p-97, 48 the last few bits of logc are rounded away so k*ln2hi + logc has no rounding 49 error and the interval for z is selected such that near x == 1, where log(x) 50 is tiny, large cancellation error is avoided in logc + poly(z/c - 1). */ 51 .tab = { 52 #if N == 128 53 #define A(a, b, c) {a, 0, b, c}, 54 A(0x1.6a00000000000p+0, -0x1.62c82f2b9c800p-2, 0x1.ab42428375680p-48) 55 A(0x1.6800000000000p+0, -0x1.5d1bdbf580800p-2, -0x1.ca508d8e0f720p-46) 56 A(0x1.6600000000000p+0, -0x1.5767717455800p-2, -0x1.362a4d5b6506dp-45) 57 A(0x1.6400000000000p+0, -0x1.51aad872df800p-2, -0x1.684e49eb067d5p-49) 58 A(0x1.6200000000000p+0, -0x1.4be5f95777800p-2, -0x1.41b6993293ee0p-47) 59 A(0x1.6000000000000p+0, -0x1.4618bc21c6000p-2, 0x1.3d82f484c84ccp-46) 60 A(0x1.5e00000000000p+0, -0x1.404308686a800p-2, 0x1.c42f3ed820b3ap-50) 61 A(0x1.5c00000000000p+0, -0x1.3a64c55694800p-2, 0x1.0b1c686519460p-45) 62 A(0x1.5a00000000000p+0, -0x1.347dd9a988000p-2, 0x1.5594dd4c58092p-45) 63 A(0x1.5800000000000p+0, -0x1.2e8e2bae12000p-2, 0x1.67b1e99b72bd8p-45) 64 A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46) 65 A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46) 66 A(0x1.5400000000000p+0, -0x1.22941fbcf7800p-2, -0x1.65a242853da76p-46) 67 A(0x1.5200000000000p+0, -0x1.1c898c1699800p-2, -0x1.fafbc68e75404p-46) 68 A(0x1.5000000000000p+0, -0x1.1675cababa800p-2, 0x1.f1fc63382a8f0p-46) 69 A(0x1.4e00000000000p+0, -0x1.1058bf9ae4800p-2, -0x1.6a8c4fd055a66p-45) 70 A(0x1.4c00000000000p+0, -0x1.0a324e2739000p-2, -0x1.c6bee7ef4030ep-47) 71 A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48) 72 A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48) 73 A(0x1.4800000000000p+0, -0x1.fb9186d5e4000p-3, 0x1.d572aab993c87p-47) 74 A(0x1.4600000000000p+0, -0x1.ef0adcbdc6000p-3, 0x1.b26b79c86af24p-45) 75 A(0x1.4400000000000p+0, -0x1.e27076e2af000p-3, -0x1.72f4f543fff10p-46) 76 A(0x1.4200000000000p+0, -0x1.d5c216b4fc000p-3, 0x1.1ba91bbca681bp-45) 77 A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45) 78 A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45) 79 A(0x1.3e00000000000p+0, -0x1.bc286742d9000p-3, 0x1.94eb0318bb78fp-46) 80 A(0x1.3c00000000000p+0, -0x1.af3c94e80c000p-3, 0x1.a4e633fcd9066p-52) 81 A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45) 82 A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45) 83 A(0x1.3800000000000p+0, -0x1.9525a9cf45000p-3, -0x1.ad1d904c1d4e3p-45) 84 A(0x1.3600000000000p+0, -0x1.87fa06520d000p-3, 0x1.bbdbf7fdbfa09p-45) 85 A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45) 86 A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45) 87 A(0x1.3200000000000p+0, -0x1.6d60fe719d000p-3, -0x1.0e46aa3b2e266p-46) 88 A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46) 89 A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46) 90 A(0x1.2e00000000000p+0, -0x1.526e5e3a1b000p-3, -0x1.0de8b90075b8fp-45) 91 A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46) 92 A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46) 93 A(0x1.2a00000000000p+0, -0x1.371fc201e9000p-3, 0x1.178864d27543ap-48) 94 A(0x1.2800000000000p+0, -0x1.29552f81ff000p-3, -0x1.48d301771c408p-45) 95 A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45) 96 A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45) 97 A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47) 98 A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47) 99 A(0x1.2200000000000p+0, -0x1.fec9131dbe000p-4, -0x1.575545ca333f2p-45) 100 A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45) 101 A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45) 102 A(0x1.1e00000000000p+0, -0x1.c5e548f5bc000p-4, -0x1.d0c57585fbe06p-46) 103 A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45) 104 A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45) 105 A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46) 106 A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46) 107 A(0x1.1800000000000p+0, -0x1.6f0d28ae56000p-4, -0x1.69737c93373dap-45) 108 A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46) 109 A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46) 110 A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45) 111 A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45) 112 A(0x1.1200000000000p+0, -0x1.16536eea38000p-4, 0x1.47c5e768fa309p-46) 113 A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45) 114 A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45) 115 A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46) 116 A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46) 117 A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45) 118 A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45) 119 A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48) 120 A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48) 121 A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45) 122 A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45) 123 A(0x1.0600000000000p+0, -0x1.7b91b07d58000p-6, -0x1.88d5493faa639p-45) 124 A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50) 125 A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50) 126 A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46) 127 A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46) 128 A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0) 129 A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0) 130 A(0x1.fc00000000000p-1, 0x1.0101575890000p-7, -0x1.0c76b999d2be8p-46) 131 A(0x1.f800000000000p-1, 0x1.0205658938000p-6, -0x1.3dc5b06e2f7d2p-45) 132 A(0x1.f400000000000p-1, 0x1.8492528c90000p-6, -0x1.aa0ba325a0c34p-45) 133 A(0x1.f000000000000p-1, 0x1.0415d89e74000p-5, 0x1.111c05cf1d753p-47) 134 A(0x1.ec00000000000p-1, 0x1.466aed42e0000p-5, -0x1.c167375bdfd28p-45) 135 A(0x1.e800000000000p-1, 0x1.894aa149fc000p-5, -0x1.97995d05a267dp-46) 136 A(0x1.e400000000000p-1, 0x1.ccb73cdddc000p-5, -0x1.a68f247d82807p-46) 137 A(0x1.e200000000000p-1, 0x1.eea31c006c000p-5, -0x1.e113e4fc93b7bp-47) 138 A(0x1.de00000000000p-1, 0x1.1973bd1466000p-4, -0x1.5325d560d9e9bp-45) 139 A(0x1.da00000000000p-1, 0x1.3bdf5a7d1e000p-4, 0x1.cc85ea5db4ed7p-45) 140 A(0x1.d600000000000p-1, 0x1.5e95a4d97a000p-4, -0x1.c69063c5d1d1ep-45) 141 A(0x1.d400000000000p-1, 0x1.700d30aeac000p-4, 0x1.c1e8da99ded32p-49) 142 A(0x1.d000000000000p-1, 0x1.9335e5d594000p-4, 0x1.3115c3abd47dap-45) 143 A(0x1.cc00000000000p-1, 0x1.b6ac88dad6000p-4, -0x1.390802bf768e5p-46) 144 A(0x1.ca00000000000p-1, 0x1.c885801bc4000p-4, 0x1.646d1c65aacd3p-45) 145 A(0x1.c600000000000p-1, 0x1.ec739830a2000p-4, -0x1.dc068afe645e0p-45) 146 A(0x1.c400000000000p-1, 0x1.fe89139dbe000p-4, -0x1.534d64fa10afdp-45) 147 A(0x1.c000000000000p-1, 0x1.1178e8227e000p-3, 0x1.1ef78ce2d07f2p-45) 148 A(0x1.be00000000000p-1, 0x1.1aa2b7e23f000p-3, 0x1.ca78e44389934p-45) 149 A(0x1.ba00000000000p-1, 0x1.2d1610c868000p-3, 0x1.39d6ccb81b4a1p-47) 150 A(0x1.b800000000000p-1, 0x1.365fcb0159000p-3, 0x1.62fa8234b7289p-51) 151 A(0x1.b400000000000p-1, 0x1.4913d8333b000p-3, 0x1.5837954fdb678p-45) 152 A(0x1.b200000000000p-1, 0x1.527e5e4a1b000p-3, 0x1.633e8e5697dc7p-45) 153 A(0x1.ae00000000000p-1, 0x1.6574ebe8c1000p-3, 0x1.9cf8b2c3c2e78p-46) 154 A(0x1.ac00000000000p-1, 0x1.6f0128b757000p-3, -0x1.5118de59c21e1p-45) 155 A(0x1.aa00000000000p-1, 0x1.7898d85445000p-3, -0x1.c661070914305p-46) 156 A(0x1.a600000000000p-1, 0x1.8beafeb390000p-3, -0x1.73d54aae92cd1p-47) 157 A(0x1.a400000000000p-1, 0x1.95a5adcf70000p-3, 0x1.7f22858a0ff6fp-47) 158 A(0x1.a000000000000p-1, 0x1.a93ed3c8ae000p-3, -0x1.8724350562169p-45) 159 A(0x1.9e00000000000p-1, 0x1.b31d8575bd000p-3, -0x1.c358d4eace1aap-47) 160 A(0x1.9c00000000000p-1, 0x1.bd087383be000p-3, -0x1.d4bc4595412b6p-45) 161 A(0x1.9a00000000000p-1, 0x1.c6ffbc6f01000p-3, -0x1.1ec72c5962bd2p-48) 162 A(0x1.9600000000000p-1, 0x1.db13db0d49000p-3, -0x1.aff2af715b035p-45) 163 A(0x1.9400000000000p-1, 0x1.e530effe71000p-3, 0x1.212276041f430p-51) 164 A(0x1.9200000000000p-1, 0x1.ef5ade4dd0000p-3, -0x1.a211565bb8e11p-51) 165 A(0x1.9000000000000p-1, 0x1.f991c6cb3b000p-3, 0x1.bcbecca0cdf30p-46) 166 A(0x1.8c00000000000p-1, 0x1.07138604d5800p-2, 0x1.89cdb16ed4e91p-48) 167 A(0x1.8a00000000000p-1, 0x1.0c42d67616000p-2, 0x1.7188b163ceae9p-45) 168 A(0x1.8800000000000p-1, 0x1.1178e8227e800p-2, -0x1.c210e63a5f01cp-45) 169 A(0x1.8600000000000p-1, 0x1.16b5ccbacf800p-2, 0x1.b9acdf7a51681p-45) 170 A(0x1.8400000000000p-1, 0x1.1bf99635a6800p-2, 0x1.ca6ed5147bdb7p-45) 171 A(0x1.8200000000000p-1, 0x1.214456d0eb800p-2, 0x1.a87deba46baeap-47) 172 A(0x1.7e00000000000p-1, 0x1.2bef07cdc9000p-2, 0x1.a9cfa4a5004f4p-45) 173 A(0x1.7c00000000000p-1, 0x1.314f1e1d36000p-2, -0x1.8e27ad3213cb8p-45) 174 A(0x1.7a00000000000p-1, 0x1.36b6776be1000p-2, 0x1.16ecdb0f177c8p-46) 175 A(0x1.7800000000000p-1, 0x1.3c25277333000p-2, 0x1.83b54b606bd5cp-46) 176 A(0x1.7600000000000p-1, 0x1.419b423d5e800p-2, 0x1.8e436ec90e09dp-47) 177 A(0x1.7400000000000p-1, 0x1.4718dc271c800p-2, -0x1.f27ce0967d675p-45) 178 A(0x1.7200000000000p-1, 0x1.4c9e09e173000p-2, -0x1.e20891b0ad8a4p-45) 179 A(0x1.7000000000000p-1, 0x1.522ae0738a000p-2, 0x1.ebe708164c759p-45) 180 A(0x1.6e00000000000p-1, 0x1.57bf753c8d000p-2, 0x1.fadedee5d40efp-46) 181 A(0x1.6c00000000000p-1, 0x1.5d5bddf596000p-2, -0x1.a0b2a08a465dcp-47) 182 #endif 183 }, 184 }; 185