1 /* 2 * Data for the log part of pow. 3 * 4 * Copyright (c) 2018-2023, 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 << V_POW_LOG_TABLE_BITS) 11 12 /* Algorithm: 13 14 x = 2^k z 15 log(x) = k ln2 + log(c) + log(z/c) 16 log(z/c) = poly(z/c - 1) 17 18 where z is in [0x1.69555p-1; 0x1.69555p0] which is split into N subintervals 19 and z falls into the ith one, then table entries are computed as 20 21 tab[i].invc = 1/c 22 tab[i].logc = round(0x1p43*log(c))/0x1p43 23 tab[i].logctail = (double)(log(c) - logc) 24 25 where c is chosen near the center of the subinterval such that 1/c has only 26 a few precision bits so z/c - 1 is exactly representible as double: 27 28 1/c = center < 1 ? round(N/center)/N : round(2*N/center)/N/2 29 30 Note: |z/c - 1| < 1/N for the chosen c, |log(c) - logc - logctail| < 31 0x1p-97, the last few bits of logc are rounded away so k*ln2hi + logc has no 32 rounding error and the interval for z is selected such that near x == 1, 33 where log(x) 34 is tiny, large cancellation error is avoided in logc + poly(z/c - 1). */ 35 const struct v_pow_log_data __v_pow_log_data = { 36 /* relative error: 0x1.11922ap-70 in [-0x1.6bp-8, 0x1.6bp-8] 37 Coefficients are scaled to match the scaling during evaluation. */ 38 .poly = { -0x1p-1, -0x1.555555555556p-1, 0x1.0000000000006p-1, 39 0x1.999999959554ep-1, -0x1.555555529a47ap-1, -0x1.2495b9b4845e9p0, 40 0x1.0002b8b263fc3p0, }, 41 .ln2_hi = 0x1.62e42fefa3800p-1, 42 .ln2_lo = 0x1.ef35793c76730p-45, 43 .invc = { 0x1.6a00000000000p+0, 0x1.6800000000000p+0, 0x1.6600000000000p+0, 44 0x1.6400000000000p+0, 0x1.6200000000000p+0, 0x1.6000000000000p+0, 45 0x1.5e00000000000p+0, 0x1.5c00000000000p+0, 0x1.5a00000000000p+0, 46 0x1.5800000000000p+0, 0x1.5600000000000p+0, 0x1.5600000000000p+0, 47 0x1.5400000000000p+0, 0x1.5200000000000p+0, 0x1.5000000000000p+0, 48 0x1.4e00000000000p+0, 0x1.4c00000000000p+0, 0x1.4a00000000000p+0, 49 0x1.4a00000000000p+0, 0x1.4800000000000p+0, 0x1.4600000000000p+0, 50 0x1.4400000000000p+0, 0x1.4200000000000p+0, 0x1.4000000000000p+0, 51 0x1.4000000000000p+0, 0x1.3e00000000000p+0, 0x1.3c00000000000p+0, 52 0x1.3a00000000000p+0, 0x1.3a00000000000p+0, 0x1.3800000000000p+0, 53 0x1.3600000000000p+0, 0x1.3400000000000p+0, 0x1.3400000000000p+0, 54 0x1.3200000000000p+0, 0x1.3000000000000p+0, 0x1.3000000000000p+0, 55 0x1.2e00000000000p+0, 0x1.2c00000000000p+0, 0x1.2c00000000000p+0, 56 0x1.2a00000000000p+0, 0x1.2800000000000p+0, 0x1.2600000000000p+0, 57 0x1.2600000000000p+0, 0x1.2400000000000p+0, 0x1.2400000000000p+0, 58 0x1.2200000000000p+0, 0x1.2000000000000p+0, 0x1.2000000000000p+0, 59 0x1.1e00000000000p+0, 0x1.1c00000000000p+0, 0x1.1c00000000000p+0, 60 0x1.1a00000000000p+0, 0x1.1a00000000000p+0, 0x1.1800000000000p+0, 61 0x1.1600000000000p+0, 0x1.1600000000000p+0, 0x1.1400000000000p+0, 62 0x1.1400000000000p+0, 0x1.1200000000000p+0, 0x1.1000000000000p+0, 63 0x1.1000000000000p+0, 0x1.0e00000000000p+0, 0x1.0e00000000000p+0, 64 0x1.0c00000000000p+0, 0x1.0c00000000000p+0, 0x1.0a00000000000p+0, 65 0x1.0a00000000000p+0, 0x1.0800000000000p+0, 0x1.0800000000000p+0, 66 0x1.0600000000000p+0, 0x1.0400000000000p+0, 0x1.0400000000000p+0, 67 0x1.0200000000000p+0, 0x1.0200000000000p+0, 0x1.0000000000000p+0, 68 0x1.0000000000000p+0, 0x1.fc00000000000p-1, 0x1.f800000000000p-1, 69 0x1.f400000000000p-1, 0x1.f000000000000p-1, 0x1.ec00000000000p-1, 70 0x1.e800000000000p-1, 0x1.e400000000000p-1, 0x1.e200000000000p-1, 71 0x1.de00000000000p-1, 0x1.da00000000000p-1, 0x1.d600000000000p-1, 72 0x1.d400000000000p-1, 0x1.d000000000000p-1, 0x1.cc00000000000p-1, 73 0x1.ca00000000000p-1, 0x1.c600000000000p-1, 0x1.c400000000000p-1, 74 0x1.c000000000000p-1, 0x1.be00000000000p-1, 0x1.ba00000000000p-1, 75 0x1.b800000000000p-1, 0x1.b400000000000p-1, 0x1.b200000000000p-1, 76 0x1.ae00000000000p-1, 0x1.ac00000000000p-1, 0x1.aa00000000000p-1, 77 0x1.a600000000000p-1, 0x1.a400000000000p-1, 0x1.a000000000000p-1, 78 0x1.9e00000000000p-1, 0x1.9c00000000000p-1, 0x1.9a00000000000p-1, 79 0x1.9600000000000p-1, 0x1.9400000000000p-1, 0x1.9200000000000p-1, 80 0x1.9000000000000p-1, 0x1.8c00000000000p-1, 0x1.8a00000000000p-1, 81 0x1.8800000000000p-1, 0x1.8600000000000p-1, 0x1.8400000000000p-1, 82 0x1.8200000000000p-1, 0x1.7e00000000000p-1, 0x1.7c00000000000p-1, 83 0x1.7a00000000000p-1, 0x1.7800000000000p-1, 0x1.7600000000000p-1, 84 0x1.7400000000000p-1, 0x1.7200000000000p-1, 0x1.7000000000000p-1, 85 0x1.6e00000000000p-1, 0x1.6c00000000000p-1, }, 86 .logc 87 = { -0x1.62c82f2b9c800p-2, -0x1.5d1bdbf580800p-2, -0x1.5767717455800p-2, 88 -0x1.51aad872df800p-2, -0x1.4be5f95777800p-2, -0x1.4618bc21c6000p-2, 89 -0x1.404308686a800p-2, -0x1.3a64c55694800p-2, -0x1.347dd9a988000p-2, 90 -0x1.2e8e2bae12000p-2, -0x1.2895a13de8800p-2, -0x1.2895a13de8800p-2, 91 -0x1.22941fbcf7800p-2, -0x1.1c898c1699800p-2, -0x1.1675cababa800p-2, 92 -0x1.1058bf9ae4800p-2, -0x1.0a324e2739000p-2, -0x1.0402594b4d000p-2, 93 -0x1.0402594b4d000p-2, -0x1.fb9186d5e4000p-3, -0x1.ef0adcbdc6000p-3, 94 -0x1.e27076e2af000p-3, -0x1.d5c216b4fc000p-3, -0x1.c8ff7c79aa000p-3, 95 -0x1.c8ff7c79aa000p-3, -0x1.bc286742d9000p-3, -0x1.af3c94e80c000p-3, 96 -0x1.a23bc1fe2b000p-3, -0x1.a23bc1fe2b000p-3, -0x1.9525a9cf45000p-3, 97 -0x1.87fa06520d000p-3, -0x1.7ab890210e000p-3, -0x1.7ab890210e000p-3, 98 -0x1.6d60fe719d000p-3, -0x1.5ff3070a79000p-3, -0x1.5ff3070a79000p-3, 99 -0x1.526e5e3a1b000p-3, -0x1.44d2b6ccb8000p-3, -0x1.44d2b6ccb8000p-3, 100 -0x1.371fc201e9000p-3, -0x1.29552f81ff000p-3, -0x1.1b72ad52f6000p-3, 101 -0x1.1b72ad52f6000p-3, -0x1.0d77e7cd09000p-3, -0x1.0d77e7cd09000p-3, 102 -0x1.fec9131dbe000p-4, -0x1.e27076e2b0000p-4, -0x1.e27076e2b0000p-4, 103 -0x1.c5e548f5bc000p-4, -0x1.a926d3a4ae000p-4, -0x1.a926d3a4ae000p-4, 104 -0x1.8c345d631a000p-4, -0x1.8c345d631a000p-4, -0x1.6f0d28ae56000p-4, 105 -0x1.51b073f062000p-4, -0x1.51b073f062000p-4, -0x1.341d7961be000p-4, 106 -0x1.341d7961be000p-4, -0x1.16536eea38000p-4, -0x1.f0a30c0118000p-5, 107 -0x1.f0a30c0118000p-5, -0x1.b42dd71198000p-5, -0x1.b42dd71198000p-5, 108 -0x1.77458f632c000p-5, -0x1.77458f632c000p-5, -0x1.39e87b9fec000p-5, 109 -0x1.39e87b9fec000p-5, -0x1.f829b0e780000p-6, -0x1.f829b0e780000p-6, 110 -0x1.7b91b07d58000p-6, -0x1.fc0a8b0fc0000p-7, -0x1.fc0a8b0fc0000p-7, 111 -0x1.fe02a6b100000p-8, -0x1.fe02a6b100000p-8, 0x0.0000000000000p+0, 112 0x0.0000000000000p+0, 0x1.0101575890000p-7, 0x1.0205658938000p-6, 113 0x1.8492528c90000p-6, 0x1.0415d89e74000p-5, 0x1.466aed42e0000p-5, 114 0x1.894aa149fc000p-5, 0x1.ccb73cdddc000p-5, 0x1.eea31c006c000p-5, 115 0x1.1973bd1466000p-4, 0x1.3bdf5a7d1e000p-4, 0x1.5e95a4d97a000p-4, 116 0x1.700d30aeac000p-4, 0x1.9335e5d594000p-4, 0x1.b6ac88dad6000p-4, 117 0x1.c885801bc4000p-4, 0x1.ec739830a2000p-4, 0x1.fe89139dbe000p-4, 118 0x1.1178e8227e000p-3, 0x1.1aa2b7e23f000p-3, 0x1.2d1610c868000p-3, 119 0x1.365fcb0159000p-3, 0x1.4913d8333b000p-3, 0x1.527e5e4a1b000p-3, 120 0x1.6574ebe8c1000p-3, 0x1.6f0128b757000p-3, 0x1.7898d85445000p-3, 121 0x1.8beafeb390000p-3, 0x1.95a5adcf70000p-3, 0x1.a93ed3c8ae000p-3, 122 0x1.b31d8575bd000p-3, 0x1.bd087383be000p-3, 0x1.c6ffbc6f01000p-3, 123 0x1.db13db0d49000p-3, 0x1.e530effe71000p-3, 0x1.ef5ade4dd0000p-3, 124 0x1.f991c6cb3b000p-3, 0x1.07138604d5800p-2, 0x1.0c42d67616000p-2, 125 0x1.1178e8227e800p-2, 0x1.16b5ccbacf800p-2, 0x1.1bf99635a6800p-2, 126 0x1.214456d0eb800p-2, 0x1.2bef07cdc9000p-2, 0x1.314f1e1d36000p-2, 127 0x1.36b6776be1000p-2, 0x1.3c25277333000p-2, 0x1.419b423d5e800p-2, 128 0x1.4718dc271c800p-2, 0x1.4c9e09e173000p-2, 0x1.522ae0738a000p-2, 129 0x1.57bf753c8d000p-2, 0x1.5d5bddf596000p-2, }, 130 .logctail 131 = { 0x1.ab42428375680p-48, -0x1.ca508d8e0f720p-46, -0x1.362a4d5b6506dp-45, 132 -0x1.684e49eb067d5p-49, -0x1.41b6993293ee0p-47, 0x1.3d82f484c84ccp-46, 133 0x1.c42f3ed820b3ap-50, 0x1.0b1c686519460p-45, 0x1.5594dd4c58092p-45, 134 0x1.67b1e99b72bd8p-45, 0x1.5ca14b6cfb03fp-46, 0x1.5ca14b6cfb03fp-46, 135 -0x1.65a242853da76p-46, -0x1.fafbc68e75404p-46, 0x1.f1fc63382a8f0p-46, 136 -0x1.6a8c4fd055a66p-45, -0x1.c6bee7ef4030ep-47, -0x1.036b89ef42d7fp-48, 137 -0x1.036b89ef42d7fp-48, 0x1.d572aab993c87p-47, 0x1.b26b79c86af24p-45, 138 -0x1.72f4f543fff10p-46, 0x1.1ba91bbca681bp-45, 0x1.7794f689f8434p-45, 139 0x1.7794f689f8434p-45, 0x1.94eb0318bb78fp-46, 0x1.a4e633fcd9066p-52, 140 -0x1.58c64dc46c1eap-45, -0x1.58c64dc46c1eap-45, -0x1.ad1d904c1d4e3p-45, 141 0x1.bbdbf7fdbfa09p-45, 0x1.bdb9072534a58p-45, 0x1.bdb9072534a58p-45, 142 -0x1.0e46aa3b2e266p-46, -0x1.e9e439f105039p-46, -0x1.e9e439f105039p-46, 143 -0x1.0de8b90075b8fp-45, 0x1.70cc16135783cp-46, 0x1.70cc16135783cp-46, 144 0x1.178864d27543ap-48, -0x1.48d301771c408p-45, -0x1.e80a41811a396p-45, 145 -0x1.e80a41811a396p-45, 0x1.a699688e85bf4p-47, 0x1.a699688e85bf4p-47, 146 -0x1.575545ca333f2p-45, 0x1.a342c2af0003cp-45, 0x1.a342c2af0003cp-45, 147 -0x1.d0c57585fbe06p-46, 0x1.53935e85baac8p-45, 0x1.53935e85baac8p-45, 148 0x1.37c294d2f5668p-46, 0x1.37c294d2f5668p-46, -0x1.69737c93373dap-45, 149 0x1.f025b61c65e57p-46, 0x1.f025b61c65e57p-46, 0x1.c5edaccf913dfp-45, 150 0x1.c5edaccf913dfp-45, 0x1.47c5e768fa309p-46, 0x1.d599e83368e91p-45, 151 0x1.d599e83368e91p-45, 0x1.c827ae5d6704cp-46, 0x1.c827ae5d6704cp-46, 152 -0x1.cfc4634f2a1eep-45, -0x1.cfc4634f2a1eep-45, 0x1.502b7f526feaap-48, 153 0x1.502b7f526feaap-48, -0x1.980267c7e09e4p-45, -0x1.980267c7e09e4p-45, 154 -0x1.88d5493faa639p-45, -0x1.f1e7cf6d3a69cp-50, -0x1.f1e7cf6d3a69cp-50, 155 -0x1.9e23f0dda40e4p-46, -0x1.9e23f0dda40e4p-46, 0x0.0000000000000p+0, 156 0x0.0000000000000p+0, -0x1.0c76b999d2be8p-46, -0x1.3dc5b06e2f7d2p-45, 157 -0x1.aa0ba325a0c34p-45, 0x1.111c05cf1d753p-47, -0x1.c167375bdfd28p-45, 158 -0x1.97995d05a267dp-46, -0x1.a68f247d82807p-46, -0x1.e113e4fc93b7bp-47, 159 -0x1.5325d560d9e9bp-45, 0x1.cc85ea5db4ed7p-45, -0x1.c69063c5d1d1ep-45, 160 0x1.c1e8da99ded32p-49, 0x1.3115c3abd47dap-45, -0x1.390802bf768e5p-46, 161 0x1.646d1c65aacd3p-45, -0x1.dc068afe645e0p-45, -0x1.534d64fa10afdp-45, 162 0x1.1ef78ce2d07f2p-45, 0x1.ca78e44389934p-45, 0x1.39d6ccb81b4a1p-47, 163 0x1.62fa8234b7289p-51, 0x1.5837954fdb678p-45, 0x1.633e8e5697dc7p-45, 164 0x1.9cf8b2c3c2e78p-46, -0x1.5118de59c21e1p-45, -0x1.c661070914305p-46, 165 -0x1.73d54aae92cd1p-47, 0x1.7f22858a0ff6fp-47, -0x1.8724350562169p-45, 166 -0x1.c358d4eace1aap-47, -0x1.d4bc4595412b6p-45, -0x1.1ec72c5962bd2p-48, 167 -0x1.aff2af715b035p-45, 0x1.212276041f430p-51, -0x1.a211565bb8e11p-51, 168 0x1.bcbecca0cdf30p-46, 0x1.89cdb16ed4e91p-48, 0x1.7188b163ceae9p-45, 169 -0x1.c210e63a5f01cp-45, 0x1.b9acdf7a51681p-45, 0x1.ca6ed5147bdb7p-45, 170 0x1.a87deba46baeap-47, 0x1.a9cfa4a5004f4p-45, -0x1.8e27ad3213cb8p-45, 171 0x1.16ecdb0f177c8p-46, 0x1.83b54b606bd5cp-46, 0x1.8e436ec90e09dp-47, 172 -0x1.f27ce0967d675p-45, -0x1.e20891b0ad8a4p-45, 0x1.ebe708164c759p-45, 173 0x1.fadedee5d40efp-46, -0x1.a0b2a08a465dcp-47, }, 174 }; 175