1 /* 2 * Data for log2. 3 * 4 * Copyright (c) 2018, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8 #include "math_config.h" 9 10 #define N (1 << LOG2_TABLE_BITS) 11 12 const struct log2_data __log2_data = { 13 // First coefficient: 0x1.71547652b82fe1777d0ffda0d24p0 14 .invln2hi = 0x1.7154765200000p+0, 15 .invln2lo = 0x1.705fc2eefa200p-33, 16 .poly1 = { 17 #if LOG2_POLY1_ORDER == 11 18 // relative error: 0x1.2fad8188p-63 19 // in -0x1.5b51p-5 0x1.6ab2p-5 20 -0x1.71547652b82fep-1, 21 0x1.ec709dc3a03f7p-2, 22 -0x1.71547652b7c3fp-2, 23 0x1.2776c50f05be4p-2, 24 -0x1.ec709dd768fe5p-3, 25 0x1.a61761ec4e736p-3, 26 -0x1.7153fbc64a79bp-3, 27 0x1.484d154f01b4ap-3, 28 -0x1.289e4a72c383cp-3, 29 0x1.0b32f285aee66p-3, 30 #endif 31 }, 32 .poly = { 33 #if N == 64 && LOG2_POLY_ORDER == 7 34 // relative error: 0x1.a72c2bf8p-58 35 // abs error: 0x1.67a552c8p-66 36 // in -0x1.f45p-8 0x1.f45p-8 37 -0x1.71547652b8339p-1, 38 0x1.ec709dc3a04bep-2, 39 -0x1.7154764702ffbp-2, 40 0x1.2776c50034c48p-2, 41 -0x1.ec7b328ea92bcp-3, 42 0x1.a6225e117f92ep-3, 43 #endif 44 }, 45 /* Algorithm: 46 47 x = 2^k z 48 log2(x) = k + log2(c) + log2(z/c) 49 log2(z/c) = poly(z/c - 1) 50 51 where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls 52 into the ith one, then table entries are computed as 53 54 tab[i].invc = 1/c 55 tab[i].logc = (double)log2(c) 56 tab2[i].chi = (double)c 57 tab2[i].clo = (double)(c - (double)c) 58 59 where c is near the center of the subinterval and is chosen by trying +-2^29 60 floating point invc candidates around 1/center and selecting one for which 61 62 1) the rounding error in 0x1.8p10 + logc is 0, 63 2) the rounding error in z - chi - clo is < 0x1p-64 and 64 3) the rounding error in (double)log2(c) is minimized (< 0x1p-68). 65 66 Note: 1) ensures that k + logc can be computed without rounding error, 2) 67 ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to a 68 single rounding error when there is no fast fma for z*invc - 1, 3) ensures 69 that logc + poly(z/c - 1) has small error, however near x == 1 when 70 |log2(x)| < 0x1p-4, this is not enough so that is special cased. */ 71 .tab = { 72 #if N == 64 73 {0x1.724286bb1acf8p+0, -0x1.1095feecdb000p-1}, 74 {0x1.6e1f766d2cca1p+0, -0x1.08494bd76d000p-1}, 75 {0x1.6a13d0e30d48ap+0, -0x1.00143aee8f800p-1}, 76 {0x1.661ec32d06c85p+0, -0x1.efec5360b4000p-2}, 77 {0x1.623fa951198f8p+0, -0x1.dfdd91ab7e000p-2}, 78 {0x1.5e75ba4cf026cp+0, -0x1.cffae0cc79000p-2}, 79 {0x1.5ac055a214fb8p+0, -0x1.c043811fda000p-2}, 80 {0x1.571ed0f166e1ep+0, -0x1.b0b67323ae000p-2}, 81 {0x1.53909590bf835p+0, -0x1.a152f5a2db000p-2}, 82 {0x1.5014fed61adddp+0, -0x1.9217f5af86000p-2}, 83 {0x1.4cab88e487bd0p+0, -0x1.8304db0719000p-2}, 84 {0x1.49539b4334feep+0, -0x1.74189f9a9e000p-2}, 85 {0x1.460cbdfafd569p+0, -0x1.6552bb5199000p-2}, 86 {0x1.42d664ee4b953p+0, -0x1.56b23a29b1000p-2}, 87 {0x1.3fb01111dd8a6p+0, -0x1.483650f5fa000p-2}, 88 {0x1.3c995b70c5836p+0, -0x1.39de937f6a000p-2}, 89 {0x1.3991c4ab6fd4ap+0, -0x1.2baa1538d6000p-2}, 90 {0x1.3698e0ce099b5p+0, -0x1.1d98340ca4000p-2}, 91 {0x1.33ae48213e7b2p+0, -0x1.0fa853a40e000p-2}, 92 {0x1.30d191985bdb1p+0, -0x1.01d9c32e73000p-2}, 93 {0x1.2e025cab271d7p+0, -0x1.e857da2fa6000p-3}, 94 {0x1.2b404cf13cd82p+0, -0x1.cd3c8633d8000p-3}, 95 {0x1.288b02c7ccb50p+0, -0x1.b26034c14a000p-3}, 96 {0x1.25e2263944de5p+0, -0x1.97c1c2f4fe000p-3}, 97 {0x1.234563d8615b1p+0, -0x1.7d6023f800000p-3}, 98 {0x1.20b46e33eaf38p+0, -0x1.633a71a05e000p-3}, 99 {0x1.1e2eefdcda3ddp+0, -0x1.494f5e9570000p-3}, 100 {0x1.1bb4a580b3930p+0, -0x1.2f9e424e0a000p-3}, 101 {0x1.19453847f2200p+0, -0x1.162595afdc000p-3}, 102 {0x1.16e06c0d5d73cp+0, -0x1.f9c9a75bd8000p-4}, 103 {0x1.1485f47b7e4c2p+0, -0x1.c7b575bf9c000p-4}, 104 {0x1.12358ad0085d1p+0, -0x1.960c60ff48000p-4}, 105 {0x1.0fef00f532227p+0, -0x1.64ce247b60000p-4}, 106 {0x1.0db2077d03a8fp+0, -0x1.33f78b2014000p-4}, 107 {0x1.0b7e6d65980d9p+0, -0x1.0387d1a42c000p-4}, 108 {0x1.0953efe7b408dp+0, -0x1.a6f9208b50000p-5}, 109 {0x1.07325cac53b83p+0, -0x1.47a954f770000p-5}, 110 {0x1.05197e40d1b5cp+0, -0x1.d23a8c50c0000p-6}, 111 {0x1.03091c1208ea2p+0, -0x1.16a2629780000p-6}, 112 {0x1.0101025b37e21p+0, -0x1.720f8d8e80000p-8}, 113 {0x1.fc07ef9caa76bp-1, 0x1.6fe53b1500000p-7}, 114 {0x1.f4465d3f6f184p-1, 0x1.11ccce10f8000p-5}, 115 {0x1.ecc079f84107fp-1, 0x1.c4dfc8c8b8000p-5}, 116 {0x1.e573a99975ae8p-1, 0x1.3aa321e574000p-4}, 117 {0x1.de5d6f0bd3de6p-1, 0x1.918a0d08b8000p-4}, 118 {0x1.d77b681ff38b3p-1, 0x1.e72e9da044000p-4}, 119 {0x1.d0cb5724de943p-1, 0x1.1dcd2507f6000p-3}, 120 {0x1.ca4b2dc0e7563p-1, 0x1.476ab03dea000p-3}, 121 {0x1.c3f8ee8d6cb51p-1, 0x1.7074377e22000p-3}, 122 {0x1.bdd2b4f020c4cp-1, 0x1.98ede8ba94000p-3}, 123 {0x1.b7d6c006015cap-1, 0x1.c0db86ad2e000p-3}, 124 {0x1.b20366e2e338fp-1, 0x1.e840aafcee000p-3}, 125 {0x1.ac57026295039p-1, 0x1.0790ab4678000p-2}, 126 {0x1.a6d01bc2731ddp-1, 0x1.1ac056801c000p-2}, 127 {0x1.a16d3bc3ff18bp-1, 0x1.2db11d4fee000p-2}, 128 {0x1.9c2d14967feadp-1, 0x1.406464ec58000p-2}, 129 {0x1.970e4f47c9902p-1, 0x1.52dbe093af000p-2}, 130 {0x1.920fb3982bcf2p-1, 0x1.651902050d000p-2}, 131 {0x1.8d30187f759f1p-1, 0x1.771d2cdeaf000p-2}, 132 {0x1.886e5ebb9f66dp-1, 0x1.88e9c857d9000p-2}, 133 {0x1.83c97b658b994p-1, 0x1.9a80155e16000p-2}, 134 {0x1.7f405ffc61022p-1, 0x1.abe186ed3d000p-2}, 135 {0x1.7ad22181415cap-1, 0x1.bd0f2aea0e000p-2}, 136 {0x1.767dcf99eff8cp-1, 0x1.ce0a43dbf4000p-2}, 137 #endif 138 }, 139 #if !HAVE_FAST_FMA 140 .tab2 = { 141 # if N == 64 142 {0x1.6200012b90a8ep-1, 0x1.904ab0644b605p-55}, 143 {0x1.66000045734a6p-1, 0x1.1ff9bea62f7a9p-57}, 144 {0x1.69fffc325f2c5p-1, 0x1.27ecfcb3c90bap-55}, 145 {0x1.6e00038b95a04p-1, 0x1.8ff8856739326p-55}, 146 {0x1.71fffe09994e3p-1, 0x1.afd40275f82b1p-55}, 147 {0x1.7600015590e1p-1, -0x1.2fd75b4238341p-56}, 148 {0x1.7a00012655bd5p-1, 0x1.808e67c242b76p-56}, 149 {0x1.7e0003259e9a6p-1, -0x1.208e426f622b7p-57}, 150 {0x1.81fffedb4b2d2p-1, -0x1.402461ea5c92fp-55}, 151 {0x1.860002dfafcc3p-1, 0x1.df7f4a2f29a1fp-57}, 152 {0x1.89ffff78c6b5p-1, -0x1.e0453094995fdp-55}, 153 {0x1.8e00039671566p-1, -0x1.a04f3bec77b45p-55}, 154 {0x1.91fffe2bf1745p-1, -0x1.7fa34400e203cp-56}, 155 {0x1.95fffcc5c9fd1p-1, -0x1.6ff8005a0695dp-56}, 156 {0x1.9a0003bba4767p-1, 0x1.0f8c4c4ec7e03p-56}, 157 {0x1.9dfffe7b92da5p-1, 0x1.e7fd9478c4602p-55}, 158 {0x1.a1fffd72efdafp-1, -0x1.a0c554dcdae7ep-57}, 159 {0x1.a5fffde04ff95p-1, 0x1.67da98ce9b26bp-55}, 160 {0x1.a9fffca5e8d2bp-1, -0x1.284c9b54c13dep-55}, 161 {0x1.adfffddad03eap-1, 0x1.812c8ea602e3cp-58}, 162 {0x1.b1ffff10d3d4dp-1, -0x1.efaddad27789cp-55}, 163 {0x1.b5fffce21165ap-1, 0x1.3cb1719c61237p-58}, 164 {0x1.b9fffd950e674p-1, 0x1.3f7d94194cep-56}, 165 {0x1.be000139ca8afp-1, 0x1.50ac4215d9bcp-56}, 166 {0x1.c20005b46df99p-1, 0x1.beea653e9c1c9p-57}, 167 {0x1.c600040b9f7aep-1, -0x1.c079f274a70d6p-56}, 168 {0x1.ca0006255fd8ap-1, -0x1.a0b4076e84c1fp-56}, 169 {0x1.cdfffd94c095dp-1, 0x1.8f933f99ab5d7p-55}, 170 {0x1.d1ffff975d6cfp-1, -0x1.82c08665fe1bep-58}, 171 {0x1.d5fffa2561c93p-1, -0x1.b04289bd295f3p-56}, 172 {0x1.d9fff9d228b0cp-1, 0x1.70251340fa236p-55}, 173 {0x1.de00065bc7e16p-1, -0x1.5011e16a4d80cp-56}, 174 {0x1.e200002f64791p-1, 0x1.9802f09ef62ep-55}, 175 {0x1.e600057d7a6d8p-1, -0x1.e0b75580cf7fap-56}, 176 {0x1.ea00027edc00cp-1, -0x1.c848309459811p-55}, 177 {0x1.ee0006cf5cb7cp-1, -0x1.f8027951576f4p-55}, 178 {0x1.f2000782b7dccp-1, -0x1.f81d97274538fp-55}, 179 {0x1.f6000260c450ap-1, -0x1.071002727ffdcp-59}, 180 {0x1.f9fffe88cd533p-1, -0x1.81bdce1fda8bp-58}, 181 {0x1.fdfffd50f8689p-1, 0x1.7f91acb918e6ep-55}, 182 {0x1.0200004292367p+0, 0x1.b7ff365324681p-54}, 183 {0x1.05fffe3e3d668p+0, 0x1.6fa08ddae957bp-55}, 184 {0x1.0a0000a85a757p+0, -0x1.7e2de80d3fb91p-58}, 185 {0x1.0e0001a5f3fccp+0, -0x1.1823305c5f014p-54}, 186 {0x1.11ffff8afbaf5p+0, -0x1.bfabb6680bac2p-55}, 187 {0x1.15fffe54d91adp+0, -0x1.d7f121737e7efp-54}, 188 {0x1.1a00011ac36e1p+0, 0x1.c000a0516f5ffp-54}, 189 {0x1.1e00019c84248p+0, -0x1.082fbe4da5dap-54}, 190 {0x1.220000ffe5e6ep+0, -0x1.8fdd04c9cfb43p-55}, 191 {0x1.26000269fd891p+0, 0x1.cfe2a7994d182p-55}, 192 {0x1.2a00029a6e6dap+0, -0x1.00273715e8bc5p-56}, 193 {0x1.2dfffe0293e39p+0, 0x1.b7c39dab2a6f9p-54}, 194 {0x1.31ffff7dcf082p+0, 0x1.df1336edc5254p-56}, 195 {0x1.35ffff05a8b6p+0, -0x1.e03564ccd31ebp-54}, 196 {0x1.3a0002e0eaeccp+0, 0x1.5f0e74bd3a477p-56}, 197 {0x1.3e000043bb236p+0, 0x1.c7dcb149d8833p-54}, 198 {0x1.4200002d187ffp+0, 0x1.e08afcf2d3d28p-56}, 199 {0x1.460000d387cb1p+0, 0x1.20837856599a6p-55}, 200 {0x1.4a00004569f89p+0, -0x1.9fa5c904fbcd2p-55}, 201 {0x1.4e000043543f3p+0, -0x1.81125ed175329p-56}, 202 {0x1.51fffcc027f0fp+0, 0x1.883d8847754dcp-54}, 203 {0x1.55ffffd87b36fp+0, -0x1.709e731d02807p-55}, 204 {0x1.59ffff21df7bap+0, 0x1.7f79f68727b02p-55}, 205 {0x1.5dfffebfc3481p+0, -0x1.180902e30e93ep-54}, 206 # endif 207 }, 208 #endif /* !HAVE_FAST_FMA */ 209 }; 210