1 //===-- Common constants for math functions ---------------------*- C++ -*-===// 2 // 3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4 // See https://llvm.org/LICENSE.txt for license information. 5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6 // 7 //===----------------------------------------------------------------------===// 8 9 #include "common_constants.h" 10 #include "src/__support/FPUtil/triple_double.h" 11 #include "src/__support/macros/config.h" 12 #include "src/__support/number_pair.h" 13 14 namespace LIBC_NAMESPACE_DECL { 15 16 // Lookup table for (1/f) where f = 1 + n*2^(-7), n = 0..127. 17 const double ONE_OVER_F[128] = { 18 0x1.0000000000000p+0, 0x1.fc07f01fc07f0p-1, 0x1.f81f81f81f820p-1, 19 0x1.f44659e4a4271p-1, 0x1.f07c1f07c1f08p-1, 0x1.ecc07b301ecc0p-1, 20 0x1.e9131abf0b767p-1, 0x1.e573ac901e574p-1, 0x1.e1e1e1e1e1e1ep-1, 21 0x1.de5d6e3f8868ap-1, 0x1.dae6076b981dbp-1, 0x1.d77b654b82c34p-1, 22 0x1.d41d41d41d41dp-1, 0x1.d0cb58f6ec074p-1, 0x1.cd85689039b0bp-1, 23 0x1.ca4b3055ee191p-1, 0x1.c71c71c71c71cp-1, 0x1.c3f8f01c3f8f0p-1, 24 0x1.c0e070381c0e0p-1, 0x1.bdd2b899406f7p-1, 0x1.bacf914c1bad0p-1, 25 0x1.b7d6c3dda338bp-1, 0x1.b4e81b4e81b4fp-1, 0x1.b2036406c80d9p-1, 26 0x1.af286bca1af28p-1, 0x1.ac5701ac5701bp-1, 0x1.a98ef606a63bep-1, 27 0x1.a6d01a6d01a6dp-1, 0x1.a41a41a41a41ap-1, 0x1.a16d3f97a4b02p-1, 28 0x1.9ec8e951033d9p-1, 0x1.9c2d14ee4a102p-1, 0x1.999999999999ap-1, 29 0x1.970e4f80cb872p-1, 0x1.948b0fcd6e9e0p-1, 0x1.920fb49d0e229p-1, 30 0x1.8f9c18f9c18fap-1, 0x1.8d3018d3018d3p-1, 0x1.8acb90f6bf3aap-1, 31 0x1.886e5f0abb04ap-1, 0x1.8618618618618p-1, 0x1.83c977ab2beddp-1, 32 0x1.8181818181818p-1, 0x1.7f405fd017f40p-1, 0x1.7d05f417d05f4p-1, 33 0x1.7ad2208e0ecc3p-1, 0x1.78a4c8178a4c8p-1, 0x1.767dce434a9b1p-1, 34 0x1.745d1745d1746p-1, 0x1.724287f46debcp-1, 0x1.702e05c0b8170p-1, 35 0x1.6e1f76b4337c7p-1, 0x1.6c16c16c16c17p-1, 0x1.6a13cd1537290p-1, 36 0x1.6816816816817p-1, 0x1.661ec6a5122f9p-1, 0x1.642c8590b2164p-1, 37 0x1.623fa77016240p-1, 0x1.6058160581606p-1, 0x1.5e75bb8d015e7p-1, 38 0x1.5c9882b931057p-1, 0x1.5ac056b015ac0p-1, 0x1.58ed2308158edp-1, 39 0x1.571ed3c506b3ap-1, 0x1.5555555555555p-1, 0x1.5390948f40febp-1, 40 0x1.51d07eae2f815p-1, 0x1.5015015015015p-1, 0x1.4e5e0a72f0539p-1, 41 0x1.4cab88725af6ep-1, 0x1.4afd6a052bf5bp-1, 0x1.49539e3b2d067p-1, 42 0x1.47ae147ae147bp-1, 0x1.460cbc7f5cf9ap-1, 0x1.446f86562d9fbp-1, 43 0x1.42d6625d51f87p-1, 0x1.4141414141414p-1, 0x1.3fb013fb013fbp-1, 44 0x1.3e22cbce4a902p-1, 0x1.3c995a47babe7p-1, 0x1.3b13b13b13b14p-1, 45 0x1.3991c2c187f63p-1, 0x1.3813813813814p-1, 0x1.3698df3de0748p-1, 46 0x1.3521cfb2b78c1p-1, 0x1.33ae45b57bcb2p-1, 0x1.323e34a2b10bfp-1, 47 0x1.30d190130d190p-1, 0x1.2f684bda12f68p-1, 0x1.2e025c04b8097p-1, 48 0x1.2c9fb4d812ca0p-1, 0x1.2b404ad012b40p-1, 0x1.29e4129e4129ep-1, 49 0x1.288b01288b013p-1, 0x1.27350b8812735p-1, 0x1.25e22708092f1p-1, 50 0x1.2492492492492p-1, 0x1.23456789abcdfp-1, 0x1.21fb78121fb78p-1, 51 0x1.20b470c67c0d9p-1, 0x1.1f7047dc11f70p-1, 0x1.1e2ef3b3fb874p-1, 52 0x1.1cf06ada2811dp-1, 0x1.1bb4a4046ed29p-1, 0x1.1a7b9611a7b96p-1, 53 0x1.19453808ca29cp-1, 0x1.1811811811812p-1, 0x1.16e0689427379p-1, 54 0x1.15b1e5f75270dp-1, 0x1.1485f0e0acd3bp-1, 0x1.135c81135c811p-1, 55 0x1.12358e75d3033p-1, 0x1.1111111111111p-1, 0x1.0fef010fef011p-1, 56 0x1.0ecf56be69c90p-1, 0x1.0db20a88f4696p-1, 0x1.0c9714fbcda3bp-1, 57 0x1.0b7e6ec259dc8p-1, 0x1.0a6810a6810a7p-1, 0x1.0953f39010954p-1, 58 0x1.0842108421084p-1, 0x1.073260a47f7c6p-1, 0x1.0624dd2f1a9fcp-1, 59 0x1.05197f7d73404p-1, 0x1.0410410410410p-1, 0x1.03091b51f5e1ap-1, 60 0x1.0204081020408p-1, 0x1.0101010101010p-1}; 61 62 // Lookup table for log(f) = log(1 + n*2^(-7)) where n = 0..127. 63 const double LOG_F[128] = { 64 0x0.0000000000000p+0, 0x1.fe02a6b106788p-8, 0x1.fc0a8b0fc03e3p-7, 65 0x1.7b91b07d5b11ap-6, 0x1.f829b0e783300p-6, 0x1.39e87b9febd5fp-5, 66 0x1.77458f632dcfcp-5, 0x1.b42dd711971bep-5, 0x1.f0a30c01162a6p-5, 67 0x1.16536eea37ae0p-4, 0x1.341d7961bd1d0p-4, 0x1.51b073f06183fp-4, 68 0x1.6f0d28ae56b4bp-4, 0x1.8c345d6319b20p-4, 0x1.a926d3a4ad563p-4, 69 0x1.c5e548f5bc743p-4, 0x1.e27076e2af2e5p-4, 0x1.fec9131dbeabap-4, 70 0x1.0d77e7cd08e59p-3, 0x1.1b72ad52f67a0p-3, 0x1.29552f81ff523p-3, 71 0x1.371fc201e8f74p-3, 0x1.44d2b6ccb7d1ep-3, 0x1.526e5e3a1b437p-3, 72 0x1.5ff3070a793d3p-3, 0x1.6d60fe719d21cp-3, 0x1.7ab890210d909p-3, 73 0x1.87fa06520c910p-3, 0x1.9525a9cf456b4p-3, 0x1.a23bc1fe2b563p-3, 74 0x1.af3c94e80bff2p-3, 0x1.bc286742d8cd6p-3, 0x1.c8ff7c79a9a21p-3, 75 0x1.d5c216b4fbb91p-3, 0x1.e27076e2af2e5p-3, 0x1.ef0adcbdc5936p-3, 76 0x1.fb9186d5e3e2ap-3, 0x1.0402594b4d040p-2, 0x1.0a324e27390e3p-2, 77 0x1.1058bf9ae4ad5p-2, 0x1.1675cababa60ep-2, 0x1.1c898c16999fap-2, 78 0x1.22941fbcf7965p-2, 0x1.2895a13de86a3p-2, 0x1.2e8e2bae11d30p-2, 79 0x1.347dd9a987d54p-2, 0x1.3a64c556945e9p-2, 0x1.404308686a7e3p-2, 80 0x1.4618bc21c5ec2p-2, 0x1.4be5f957778a0p-2, 0x1.51aad872df82dp-2, 81 0x1.5767717455a6cp-2, 0x1.5d1bdbf5809cap-2, 0x1.62c82f2b9c795p-2, 82 0x1.686c81e9b14aep-2, 0x1.6e08eaa2ba1e3p-2, 0x1.739d7f6bbd006p-2, 83 0x1.792a55fdd47a2p-2, 0x1.7eaf83b82afc3p-2, 0x1.842d1da1e8b17p-2, 84 0x1.89a3386c1425ap-2, 0x1.8f11e873662c7p-2, 0x1.947941c2116fap-2, 85 0x1.99d958117e08ap-2, 0x1.9f323ecbf984bp-2, 0x1.a484090e5bb0ap-2, 86 0x1.a9cec9a9a0849p-2, 0x1.af1293247786bp-2, 0x1.b44f77bcc8f62p-2, 87 0x1.b9858969310fbp-2, 0x1.beb4d9da71b7bp-2, 0x1.c3dd7a7cdad4dp-2, 88 0x1.c8ff7c79a9a21p-2, 0x1.ce1af0b85f3ebp-2, 0x1.d32fe7e00ebd5p-2, 89 0x1.d83e7258a2f3ep-2, 0x1.dd46a04c1c4a0p-2, 0x1.e24881a7c6c26p-2, 90 0x1.e744261d68787p-2, 0x1.ec399d2468cc0p-2, 0x1.f128f5faf06ecp-2, 91 0x1.f6123fa7028acp-2, 0x1.faf588f78f31ep-2, 0x1.ffd2e0857f498p-2, 92 0x1.02552a5a5d0fep-1, 0x1.04bdf9da926d2p-1, 0x1.0723e5c1cdf40p-1, 93 0x1.0986f4f573520p-1, 0x1.0be72e4252a82p-1, 0x1.0e44985d1cc8bp-1, 94 0x1.109f39e2d4c96p-1, 0x1.12f719593efbcp-1, 0x1.154c3d2f4d5e9p-1, 95 0x1.179eabbd899a0p-1, 0x1.19ee6b467c96ep-1, 0x1.1c3b81f713c24p-1, 96 0x1.1e85f5e7040d0p-1, 0x1.20cdcd192ab6dp-1, 0x1.23130d7bebf42p-1, 97 0x1.2555bce98f7cbp-1, 0x1.2795e1289b11ap-1, 0x1.29d37fec2b08ap-1, 98 0x1.2c0e9ed448e8bp-1, 0x1.2e47436e40268p-1, 0x1.307d7334f10bep-1, 99 0x1.32b1339121d71p-1, 0x1.34e289d9ce1d3p-1, 0x1.37117b54747b5p-1, 100 0x1.393e0d3562a19p-1, 0x1.3b68449fffc22p-1, 0x1.3d9026a7156fap-1, 101 0x1.3fb5b84d16f42p-1, 0x1.41d8fe84672aep-1, 0x1.43f9fe2f9ce67p-1, 102 0x1.4618bc21c5ec2p-1, 0x1.48353d1ea88dfp-1, 0x1.4a4f85db03ebbp-1, 103 0x1.4c679afccee39p-1, 0x1.4e7d811b75bb0p-1, 0x1.50913cc01686bp-1, 104 0x1.52a2d265bc5aap-1, 0x1.54b2467999497p-1, 0x1.56bf9d5b3f399p-1, 105 0x1.58cadb5cd7989p-1, 0x1.5ad404c359f2cp-1, 0x1.5cdb1dc6c1764p-1, 106 0x1.5ee02a9241675p-1, 0x1.60e32f44788d8p-1}; 107 108 // Range reduction constants for logarithms. 109 // r(0) = 1, r(127) = 0.5 110 // r(k) = 2^-8 * ceil(2^8 * (1 - 2^-8) / (1 + k*2^-7)) 111 // The constants are chosen so that v = fma(r, m_x, -1) is exact in single 112 // precision, and -2^-8 <= v < 2^-7. 113 // TODO(lntue): Add reference to how the constants are derived after the 114 // resulting paper is ready. 115 alignas(32) const float R[128] = { 116 0x1p0, 0x1.fcp-1, 0x1.f8p-1, 0x1.f4p-1, 0x1.fp-1, 0x1.ecp-1, 0x1.e8p-1, 117 0x1.e4p-1, 0x1.ep-1, 0x1.dep-1, 0x1.dap-1, 0x1.d6p-1, 0x1.d4p-1, 0x1.dp-1, 118 0x1.ccp-1, 0x1.cap-1, 0x1.c6p-1, 0x1.c4p-1, 0x1.cp-1, 0x1.bep-1, 0x1.bap-1, 119 0x1.b8p-1, 0x1.b4p-1, 0x1.b2p-1, 0x1.aep-1, 0x1.acp-1, 0x1.a8p-1, 0x1.a6p-1, 120 0x1.a4p-1, 0x1.ap-1, 0x1.9ep-1, 0x1.9cp-1, 0x1.98p-1, 0x1.96p-1, 0x1.94p-1, 121 0x1.92p-1, 0x1.9p-1, 0x1.8cp-1, 0x1.8ap-1, 0x1.88p-1, 0x1.86p-1, 0x1.84p-1, 122 0x1.8p-1, 0x1.7ep-1, 0x1.7cp-1, 0x1.7ap-1, 0x1.78p-1, 0x1.76p-1, 0x1.74p-1, 123 0x1.72p-1, 0x1.7p-1, 0x1.6ep-1, 0x1.6cp-1, 0x1.6ap-1, 0x1.68p-1, 0x1.66p-1, 124 0x1.64p-1, 0x1.62p-1, 0x1.6p-1, 0x1.5ep-1, 0x1.5cp-1, 0x1.5ap-1, 0x1.58p-1, 125 0x1.56p-1, 0x1.54p-1, 0x1.54p-1, 0x1.52p-1, 0x1.5p-1, 0x1.4ep-1, 0x1.4cp-1, 126 0x1.4ap-1, 0x1.4ap-1, 0x1.48p-1, 0x1.46p-1, 0x1.44p-1, 0x1.42p-1, 0x1.4p-1, 127 0x1.4p-1, 0x1.3ep-1, 0x1.3cp-1, 0x1.3ap-1, 0x1.3ap-1, 0x1.38p-1, 0x1.36p-1, 128 0x1.34p-1, 0x1.34p-1, 0x1.32p-1, 0x1.3p-1, 0x1.3p-1, 0x1.2ep-1, 0x1.2cp-1, 129 0x1.2cp-1, 0x1.2ap-1, 0x1.28p-1, 0x1.28p-1, 0x1.26p-1, 0x1.24p-1, 0x1.24p-1, 130 0x1.22p-1, 0x1.2p-1, 0x1.2p-1, 0x1.1ep-1, 0x1.1cp-1, 0x1.1cp-1, 0x1.1ap-1, 131 0x1.1ap-1, 0x1.18p-1, 0x1.16p-1, 0x1.16p-1, 0x1.14p-1, 0x1.14p-1, 0x1.12p-1, 132 0x1.1p-1, 0x1.1p-1, 0x1.0ep-1, 0x1.0ep-1, 0x1.0cp-1, 0x1.0cp-1, 0x1.0ap-1, 133 0x1.0ap-1, 0x1.08p-1, 0x1.08p-1, 0x1.06p-1, 0x1.06p-1, 0x1.04p-1, 0x1.04p-1, 134 0x1.02p-1, 0x1.0p-1}; 135 136 alignas(64) const double RD[128] = { 137 0x1p0, 0x1.fcp-1, 0x1.f8p-1, 0x1.f4p-1, 0x1.fp-1, 0x1.ecp-1, 0x1.e8p-1, 138 0x1.e4p-1, 0x1.ep-1, 0x1.dep-1, 0x1.dap-1, 0x1.d6p-1, 0x1.d4p-1, 0x1.dp-1, 139 0x1.ccp-1, 0x1.cap-1, 0x1.c6p-1, 0x1.c4p-1, 0x1.cp-1, 0x1.bep-1, 0x1.bap-1, 140 0x1.b8p-1, 0x1.b4p-1, 0x1.b2p-1, 0x1.aep-1, 0x1.acp-1, 0x1.a8p-1, 0x1.a6p-1, 141 0x1.a4p-1, 0x1.ap-1, 0x1.9ep-1, 0x1.9cp-1, 0x1.98p-1, 0x1.96p-1, 0x1.94p-1, 142 0x1.92p-1, 0x1.9p-1, 0x1.8cp-1, 0x1.8ap-1, 0x1.88p-1, 0x1.86p-1, 0x1.84p-1, 143 0x1.8p-1, 0x1.7ep-1, 0x1.7cp-1, 0x1.7ap-1, 0x1.78p-1, 0x1.76p-1, 0x1.74p-1, 144 0x1.72p-1, 0x1.7p-1, 0x1.6ep-1, 0x1.6cp-1, 0x1.6ap-1, 0x1.68p-1, 0x1.66p-1, 145 0x1.64p-1, 0x1.62p-1, 0x1.6p-1, 0x1.5ep-1, 0x1.5cp-1, 0x1.5ap-1, 0x1.58p-1, 146 0x1.56p-1, 0x1.54p-1, 0x1.54p-1, 0x1.52p-1, 0x1.5p-1, 0x1.4ep-1, 0x1.4cp-1, 147 0x1.4ap-1, 0x1.4ap-1, 0x1.48p-1, 0x1.46p-1, 0x1.44p-1, 0x1.42p-1, 0x1.4p-1, 148 0x1.4p-1, 0x1.3ep-1, 0x1.3cp-1, 0x1.3ap-1, 0x1.3ap-1, 0x1.38p-1, 0x1.36p-1, 149 0x1.34p-1, 0x1.34p-1, 0x1.32p-1, 0x1.3p-1, 0x1.3p-1, 0x1.2ep-1, 0x1.2cp-1, 150 0x1.2cp-1, 0x1.2ap-1, 0x1.28p-1, 0x1.28p-1, 0x1.26p-1, 0x1.24p-1, 0x1.24p-1, 151 0x1.22p-1, 0x1.2p-1, 0x1.2p-1, 0x1.1ep-1, 0x1.1cp-1, 0x1.1cp-1, 0x1.1ap-1, 152 0x1.1ap-1, 0x1.18p-1, 0x1.16p-1, 0x1.16p-1, 0x1.14p-1, 0x1.14p-1, 0x1.12p-1, 153 0x1.1p-1, 0x1.1p-1, 0x1.0ep-1, 0x1.0ep-1, 0x1.0cp-1, 0x1.0cp-1, 0x1.0ap-1, 154 0x1.0ap-1, 0x1.08p-1, 0x1.08p-1, 0x1.06p-1, 0x1.06p-1, 0x1.04p-1, 0x1.04p-1, 155 0x1.02p-1, 0x1.0p-1}; 156 157 // Compensated constants for exact logarithm range reduction when FMA is not 158 // available. 159 // Generated by Sollya with the formula: CD[i] = RD[i]*(1 + i*2^-7) - 1 160 // for RD[i] defined on the table above. 161 alignas(64) const double CD[128] = { 162 0.0, -0x1p-14, -0x1p-12, -0x1.2p-11, -0x1p-10, -0x1.9p-10, 163 -0x1.2p-9, -0x1.88p-9, -0x1p-8, -0x1.9p-11, -0x1.fp-10, -0x1.9cp-9, 164 -0x1p-12, -0x1.cp-10, -0x1.bp-9, -0x1.5p-11, -0x1.4p-9, 0x1p-14, 165 -0x1p-9, 0x1.ap-12, -0x1.ep-10, 0x1.8p-12, -0x1.1p-9, -0x1p-15, 166 -0x1.6p-9, -0x1.ap-11, -0x1.ep-9, -0x1.f8p-10, -0x1p-12, -0x1.cp-9, 167 -0x1.fp-10, -0x1.cp-12, -0x1p-8, -0x1.54p-9, -0x1.6p-10, -0x1.4p-13, 168 0x1p-10, -0x1.88p-9, -0x1.08p-9, -0x1.2p-10, -0x1p-12, 0x1.2p-11, 169 -0x1p-8, -0x1.acp-9, -0x1.6p-9, -0x1.1cp-9, -0x1.cp-10, -0x1.58p-10, 170 -0x1p-10, -0x1.7p-11, -0x1p-11, -0x1.6p-12, -0x1p-12, -0x1.cp-13, 171 -0x1p-12, -0x1.6p-12, -0x1p-11, -0x1.7p-11, -0x1p-10, -0x1.58p-10, 172 -0x1.cp-10, -0x1.1cp-9, -0x1.6p-9, -0x1.acp-9, -0x1p-8, 0x1.5p-10, 173 0x1.2p-11, -0x1p-12, -0x1.2p-10, -0x1.08p-9, -0x1.88p-9, 0x1.0cp-9, 174 0x1p-10, -0x1.4p-13, -0x1.6p-10, -0x1.54p-9, -0x1p-8, 0x1p-10, 175 -0x1.cp-12, -0x1.fp-10, -0x1.cp-9, 0x1.68p-10, -0x1p-12, -0x1.f8p-10, 176 -0x1.ep-9, 0x1.1p-10, -0x1.ap-11, -0x1.6p-9, 0x1p-9, -0x1p-15, 177 -0x1.1p-9, 0x1.48p-9, 0x1.8p-12, -0x1.ep-10, 0x1.6p-9, 0x1.ap-12, 178 -0x1p-9, 0x1.48p-9, 0x1p-14, -0x1.4p-9, 0x1p-9, -0x1.5p-11, 179 -0x1.bp-9, 0x1.1p-10, -0x1.cp-10, 0x1.54p-9, -0x1p-12, -0x1.9cp-9, 180 0x1.2p-10, -0x1.fp-10, 0x1.3p-9, -0x1.9p-11, -0x1p-8, 0x1p-12, 181 -0x1.88p-9, 0x1.28p-10, -0x1.2p-9, 0x1.fp-10, -0x1.9p-10, 0x1.4cp-9, 182 -0x1p-10, 0x1.9p-9, -0x1.2p-11, 0x1.c4p-9, -0x1p-12, 0x1.e8p-9, 183 -0x1p-14, -0x1p-8, 184 }; 185 186 alignas(64) const double LOG_R[128] = { 187 0x0.0000000000000p0, 0x1.010157588de71p-7, 0x1.0205658935847p-6, 188 0x1.8492528c8cabfp-6, 0x1.0415d89e74444p-5, 0x1.466aed42de3eap-5, 189 0x1.894aa149fb343p-5, 0x1.ccb73cdddb2ccp-5, 0x1.08598b59e3a07p-4, 190 0x1.1973bd1465567p-4, 0x1.3bdf5a7d1ee64p-4, 0x1.5e95a4d9791cbp-4, 191 0x1.700d30aeac0e1p-4, 0x1.9335e5d594989p-4, 0x1.b6ac88dad5b1cp-4, 192 0x1.c885801bc4b23p-4, 0x1.ec739830a112p-4, 0x1.fe89139dbd566p-4, 193 0x1.1178e8227e47cp-3, 0x1.1aa2b7e23f72ap-3, 0x1.2d1610c86813ap-3, 194 0x1.365fcb0159016p-3, 0x1.4913d8333b561p-3, 0x1.527e5e4a1b58dp-3, 195 0x1.6574ebe8c133ap-3, 0x1.6f0128b756abcp-3, 0x1.823c16551a3c2p-3, 196 0x1.8beafeb38fe8cp-3, 0x1.95a5adcf7017fp-3, 0x1.a93ed3c8ad9e3p-3, 197 0x1.b31d8575bce3dp-3, 0x1.bd087383bd8adp-3, 0x1.d1037f2655e7bp-3, 198 0x1.db13db0d4894p-3, 0x1.e530effe71012p-3, 0x1.ef5ade4dcffe6p-3, 199 0x1.f991c6cb3b379p-3, 0x1.07138604d5862p-2, 0x1.0c42d676162e3p-2, 200 0x1.1178e8227e47cp-2, 0x1.16b5ccbacfb73p-2, 0x1.1bf99635a6b95p-2, 201 0x1.269621134db92p-2, 0x1.2bef07cdc9354p-2, 0x1.314f1e1d35ce4p-2, 202 0x1.36b6776be1117p-2, 0x1.3c25277333184p-2, 0x1.419b423d5e8c7p-2, 203 0x1.4718dc271c41bp-2, 0x1.4c9e09e172c3cp-2, 0x1.522ae0738a3d8p-2, 204 0x1.57bf753c8d1fbp-2, 0x1.5d5bddf595f3p-2, 0x1.630030b3aac49p-2, 205 0x1.68ac83e9c6a14p-2, 0x1.6e60ee6af1972p-2, 0x1.741d876c67bb1p-2, 206 0x1.79e26687cfb3ep-2, 0x1.7fafa3bd8151cp-2, 0x1.85855776dcbfbp-2, 207 0x1.8b639a88b2df5p-2, 0x1.914a8635bf68ap-2, 0x1.973a3431356aep-2, 208 0x1.9d32bea15ed3bp-2, 0x1.a33440224fa79p-2, 0x1.a33440224fa79p-2, 209 0x1.a93ed3c8ad9e3p-2, 0x1.af5295248cddp-2, 0x1.b56fa04462909p-2, 210 0x1.bb9611b80e2fbp-2, 0x1.c1c60693fa39ep-2, 0x1.c1c60693fa39ep-2, 211 0x1.c7ff9c74554c9p-2, 0x1.ce42f18064743p-2, 0x1.d490246defa6bp-2, 212 0x1.dae75484c9616p-2, 0x1.e148a1a2726cep-2, 0x1.e148a1a2726cep-2, 213 0x1.e7b42c3ddad73p-2, 0x1.ee2a156b413e5p-2, 0x1.f4aa7ee03192dp-2, 214 0x1.f4aa7ee03192dp-2, 0x1.fb358af7a4884p-2, 0x1.00e5ae5b207abp-1, 215 0x1.04360be7603adp-1, 0x1.04360be7603adp-1, 0x1.078bf0533c568p-1, 216 0x1.0ae76e2d054fap-1, 0x1.0ae76e2d054fap-1, 0x1.0e4898611cce1p-1, 217 0x1.11af823c75aa8p-1, 0x1.11af823c75aa8p-1, 0x1.151c3f6f29612p-1, 218 0x1.188ee40f23ca6p-1, 0x1.188ee40f23ca6p-1, 0x1.1c07849ae6007p-1, 219 0x1.1f8635fc61659p-1, 0x1.1f8635fc61659p-1, 0x1.230b0d8bebc98p-1, 220 0x1.269621134db92p-1, 0x1.269621134db92p-1, 0x1.2a2786d0ec107p-1, 221 0x1.2dbf557b0df43p-1, 0x1.2dbf557b0df43p-1, 0x1.315da4434068bp-1, 222 0x1.315da4434068bp-1, 0x1.35028ad9d8c86p-1, 0x1.38ae2171976e7p-1, 223 0x1.38ae2171976e7p-1, 0x1.3c6080c36bfb5p-1, 0x1.3c6080c36bfb5p-1, 224 0x1.4019c2125ca93p-1, 0x1.43d9ff2f923c5p-1, 0x1.43d9ff2f923c5p-1, 225 0x1.47a1527e8a2d3p-1, 0x1.47a1527e8a2d3p-1, 0x1.4b6fd6f970c1fp-1, 226 0x1.4b6fd6f970c1fp-1, 0x1.4f45a835a4e19p-1, 0x1.4f45a835a4e19p-1, 227 0x1.5322e26867857p-1, 0x1.5322e26867857p-1, 0x1.5707a26bb8c66p-1, 228 0x1.5707a26bb8c66p-1, 0x1.5af405c3649ep-1, 0x1.5af405c3649ep-1, 229 0x1.5ee82aa24192p-1, 0x0.000000000000p0}; 230 231 alignas(64) const double LOG2_R[128] = { 232 0x0.0000000000000p+0, 0x1.72c7ba20f7327p-7, 0x1.743ee861f3556p-6, 233 0x1.184b8e4c56af8p-5, 0x1.77394c9d958d5p-5, 0x1.d6ebd1f1febfep-5, 234 0x1.1bb32a600549dp-4, 0x1.4c560fe68af88p-4, 0x1.7d60496cfbb4cp-4, 235 0x1.960caf9abb7cap-4, 0x1.c7b528b70f1c5p-4, 0x1.f9c95dc1d1165p-4, 236 0x1.097e38ce60649p-3, 0x1.22dadc2ab3497p-3, 0x1.3c6fb650cde51p-3, 237 0x1.494f863b8df35p-3, 0x1.633a8bf437ce1p-3, 0x1.7046031c79f85p-3, 238 0x1.8a8980abfbd32p-3, 0x1.97c1cb13c7ec1p-3, 0x1.b2602497d5346p-3, 239 0x1.bfc67a7fff4ccp-3, 0x1.dac22d3e441d3p-3, 0x1.e857d3d361368p-3, 240 0x1.01d9bbcfa61d4p-2, 0x1.08bce0d95fa38p-2, 0x1.169c05363f158p-2, 241 0x1.1d982c9d52708p-2, 0x1.249cd2b13cd6cp-2, 0x1.32bfee370ee68p-2, 242 0x1.39de8e1559f6fp-2, 0x1.4106017c3eca3p-2, 0x1.4f6fbb2cec598p-2, 243 0x1.56b22e6b578e5p-2, 0x1.5dfdcf1eeae0ep-2, 0x1.6552b49986277p-2, 244 0x1.6cb0f6865c8eap-2, 0x1.7b89f02cf2aadp-2, 0x1.8304d90c11fd3p-2, 245 0x1.8a8980abfbd32p-2, 0x1.921800924dd3bp-2, 0x1.99b072a96c6b2p-2, 246 0x1.a8ff971810a5ep-2, 0x1.b0b67f4f4681p-2, 0x1.b877c57b1b07p-2, 247 0x1.c043859e2fdb3p-2, 0x1.c819dc2d45fe4p-2, 0x1.cffae611ad12bp-2, 248 0x1.d7e6c0abc3579p-2, 0x1.dfdd89d586e2bp-2, 0x1.e7df5fe538ab3p-2, 249 0x1.efec61b011f85p-2, 0x1.f804ae8d0cd02p-2, 0x1.0014332be0033p-1, 250 0x1.042bd4b9a7c99p-1, 0x1.08494c66b8efp-1, 0x1.0c6caaf0c5597p-1, 251 0x1.1096015dee4dap-1, 0x1.14c560fe68af9p-1, 0x1.18fadb6e2d3c2p-1, 252 0x1.1d368296b5255p-1, 0x1.217868b0c37e8p-1, 0x1.25c0a0463bebp-1, 253 0x1.2a0f3c340705cp-1, 0x1.2e644fac04fd8p-1, 0x1.2e644fac04fd8p-1, 254 0x1.32bfee370ee68p-1, 0x1.37222bb70747cp-1, 0x1.3b8b1c68fa6edp-1, 255 0x1.3ffad4e74f1d6p-1, 0x1.44716a2c08262p-1, 0x1.44716a2c08262p-1, 256 0x1.48eef19317991p-1, 0x1.4d7380dcc422dp-1, 0x1.51ff2e30214bcp-1, 257 0x1.5692101d9b4a6p-1, 0x1.5b2c3da19723bp-1, 0x1.5b2c3da19723bp-1, 258 0x1.5fcdce2727ddbp-1, 0x1.6476d98ad990ap-1, 0x1.6927781d932a8p-1, 259 0x1.6927781d932a8p-1, 0x1.6ddfc2a78fc63p-1, 0x1.729fd26b707c8p-1, 260 0x1.7767c12967a45p-1, 0x1.7767c12967a45p-1, 0x1.7c37a9227e7fbp-1, 261 0x1.810fa51bf65fdp-1, 0x1.810fa51bf65fdp-1, 0x1.85efd062c656dp-1, 262 0x1.8ad846cf369a4p-1, 0x1.8ad846cf369a4p-1, 0x1.8fc924c89ac84p-1, 263 0x1.94c287492c4dbp-1, 0x1.94c287492c4dbp-1, 0x1.99c48be2063c8p-1, 264 0x1.9ecf50bf43f13p-1, 0x1.9ecf50bf43f13p-1, 0x1.a3e2f4ac43f6p-1, 265 0x1.a8ff971810a5ep-1, 0x1.a8ff971810a5ep-1, 0x1.ae255819f022dp-1, 266 0x1.b35458761d479p-1, 0x1.b35458761d479p-1, 0x1.b88cb9a2ab521p-1, 267 0x1.b88cb9a2ab521p-1, 0x1.bdce9dcc96187p-1, 0x1.c31a27dd00b4ap-1, 268 0x1.c31a27dd00b4ap-1, 0x1.c86f7b7ea4a89p-1, 0x1.c86f7b7ea4a89p-1, 269 0x1.cdcebd2373995p-1, 0x1.d338120a6dd9dp-1, 0x1.d338120a6dd9dp-1, 270 0x1.d8aba045b01c8p-1, 0x1.d8aba045b01c8p-1, 0x1.de298ec0bac0dp-1, 271 0x1.de298ec0bac0dp-1, 0x1.e3b20546f554ap-1, 0x1.e3b20546f554ap-1, 272 0x1.e9452c8a71028p-1, 0x1.e9452c8a71028p-1, 0x1.eee32e2aeccbfp-1, 273 0x1.eee32e2aeccbfp-1, 0x1.f48c34bd1e96fp-1, 0x1.f48c34bd1e96fp-1, 274 0x1.fa406bd2443dfp-1, 0x1.0000000000000p0}; 275 276 // Generated by Sollya with: 277 // for i from 0 to 127 do { 278 // r = 2^-8 * ceil( 2^8 * (1 - 2^(-8)) / (1 + i*2^-7) ); 279 // b = nearestint(log(r)*2^43) * 2^-43; 280 // c = round(log(r) - b, D, RN); 281 // print("{", -c, ",", -b, "},"); 282 // }; 283 // We replace LOG_R[0] with log10(1.0) == 0.0 284 alignas(64) const NumberPair<double> LOG_R_DD[128] = { 285 {0.0, 0.0}, 286 {-0x1.0c76b999d2be8p-46, 0x1.010157589p-7}, 287 {-0x1.3dc5b06e2f7d2p-45, 0x1.0205658938p-6}, 288 {-0x1.aa0ba325a0c34p-45, 0x1.8492528c9p-6}, 289 {0x1.111c05cf1d753p-47, 0x1.0415d89e74p-5}, 290 {-0x1.c167375bdfd28p-45, 0x1.466aed42ep-5}, 291 {-0x1.97995d05a267dp-46, 0x1.894aa149fcp-5}, 292 {-0x1.a68f247d82807p-46, 0x1.ccb73cdddcp-5}, 293 {-0x1.7e5dd7009902cp-46, 0x1.08598b59e4p-4}, 294 {-0x1.5325d560d9e9bp-45, 0x1.1973bd1466p-4}, 295 {0x1.cc85ea5db4ed7p-45, 0x1.3bdf5a7d1ep-4}, 296 {-0x1.c69063c5d1d1ep-45, 0x1.5e95a4d97ap-4}, 297 {0x1.c1e8da99ded32p-49, 0x1.700d30aeacp-4}, 298 {0x1.3115c3abd47dap-45, 0x1.9335e5d594p-4}, 299 {-0x1.390802bf768e5p-46, 0x1.b6ac88dad6p-4}, 300 {0x1.646d1c65aacd3p-45, 0x1.c885801bc4p-4}, 301 {-0x1.dc068afe645ep-45, 0x1.ec739830a2p-4}, 302 {-0x1.534d64fa10afdp-45, 0x1.fe89139dbep-4}, 303 {0x1.1ef78ce2d07f2p-45, 0x1.1178e8227ep-3}, 304 {0x1.ca78e44389934p-45, 0x1.1aa2b7e23fp-3}, 305 {0x1.39d6ccb81b4a1p-47, 0x1.2d1610c868p-3}, 306 {0x1.62fa8234b7289p-51, 0x1.365fcb0159p-3}, 307 {0x1.5837954fdb678p-45, 0x1.4913d8333bp-3}, 308 {0x1.633e8e5697dc7p-45, 0x1.527e5e4a1bp-3}, 309 {0x1.9cf8b2c3c2e78p-46, 0x1.6574ebe8c1p-3}, 310 {-0x1.5118de59c21e1p-45, 0x1.6f0128b757p-3}, 311 {0x1.e0ddb9a631e83p-46, 0x1.823c16551ap-3}, 312 {-0x1.73d54aae92cd1p-47, 0x1.8beafeb39p-3}, 313 {0x1.7f22858a0ff6fp-47, 0x1.95a5adcf7p-3}, 314 {-0x1.8724350562169p-45, 0x1.a93ed3c8aep-3}, 315 {-0x1.c358d4eace1aap-47, 0x1.b31d8575bdp-3}, 316 {-0x1.d4bc4595412b6p-45, 0x1.bd087383bep-3}, 317 {-0x1.84a7e75b6f6e4p-47, 0x1.d1037f2656p-3}, 318 {-0x1.aff2af715b035p-45, 0x1.db13db0d49p-3}, 319 {0x1.212276041f43p-51, 0x1.e530effe71p-3}, 320 {-0x1.a211565bb8e11p-51, 0x1.ef5ade4ddp-3}, 321 {0x1.bcbecca0cdf3p-46, 0x1.f991c6cb3bp-3}, 322 {0x1.89cdb16ed4e91p-48, 0x1.07138604d58p-2}, 323 {0x1.7188b163ceae9p-45, 0x1.0c42d67616p-2}, 324 {-0x1.c210e63a5f01cp-45, 0x1.1178e8227e8p-2}, 325 {0x1.b9acdf7a51681p-45, 0x1.16b5ccbacf8p-2}, 326 {0x1.ca6ed5147bdb7p-45, 0x1.1bf99635a68p-2}, 327 {0x1.c93c1df5bb3b6p-45, 0x1.269621134d8p-2}, 328 {0x1.a9cfa4a5004f4p-45, 0x1.2bef07cdc9p-2}, 329 {-0x1.8e27ad3213cb8p-45, 0x1.314f1e1d36p-2}, 330 {0x1.16ecdb0f177c8p-46, 0x1.36b6776be1p-2}, 331 {0x1.83b54b606bd5cp-46, 0x1.3c25277333p-2}, 332 {0x1.8e436ec90e09dp-47, 0x1.419b423d5e8p-2}, 333 {-0x1.f27ce0967d675p-45, 0x1.4718dc271c8p-2}, 334 {-0x1.e20891b0ad8a4p-45, 0x1.4c9e09e173p-2}, 335 {0x1.ebe708164c759p-45, 0x1.522ae0738ap-2}, 336 {0x1.fadedee5d40efp-46, 0x1.57bf753c8dp-2}, 337 {-0x1.a0b2a08a465dcp-47, 0x1.5d5bddf596p-2}, 338 {-0x1.db623e731aep-45, 0x1.630030b3abp-2}, 339 {0x1.0a0d32756ebap-45, 0x1.68ac83e9c68p-2}, 340 {0x1.721657c222d87p-46, 0x1.6e60ee6af18p-2}, 341 {0x1.d8b0949dc60b3p-45, 0x1.741d876c678p-2}, 342 {0x1.9ec7d2efd1778p-45, 0x1.79e26687cf8p-2}, 343 {-0x1.72090c812566ap-45, 0x1.7fafa3bd818p-2}, 344 {0x1.fd56f3333778ap-45, 0x1.85855776dc8p-2}, 345 {-0x1.05ae1e5e7047p-45, 0x1.8b639a88b3p-2}, 346 {-0x1.766b52ee6307dp-46, 0x1.914a8635bf8p-2}, 347 {-0x1.52313a502d9fp-46, 0x1.973a3431358p-2}, 348 {-0x1.6279e10d0c0bp-45, 0x1.9d32bea15fp-2}, 349 {0x1.3c6457f9d79f5p-45, 0x1.a33440224f8p-2}, 350 {0x1.3c6457f9d79f5p-45, 0x1.a33440224f8p-2}, 351 {0x1.e36f2bea77a5dp-46, 0x1.a93ed3c8ad8p-2}, 352 {-0x1.17cc552774458p-45, 0x1.af5295248dp-2}, 353 {0x1.095252d841995p-46, 0x1.b56fa044628p-2}, 354 {0x1.7d85bf40a666dp-45, 0x1.bb9611b80ep-2}, 355 {0x1.cec807fe8e18p-45, 0x1.c1c60693fap-2}, 356 {0x1.cec807fe8e18p-45, 0x1.c1c60693fap-2}, 357 {-0x1.9b6ddc15249aep-45, 0x1.c7ff9c74558p-2}, 358 {-0x1.797c33ec7a6bp-47, 0x1.ce42f180648p-2}, 359 {0x1.35bafe9a767a8p-45, 0x1.d490246def8p-2}, 360 {-0x1.ea42d60dc616ap-46, 0x1.dae75484c98p-2}, 361 {-0x1.326b207322938p-46, 0x1.e148a1a2728p-2}, 362 {-0x1.326b207322938p-46, 0x1.e148a1a2728p-2}, 363 {-0x1.465505372bd08p-45, 0x1.e7b42c3ddbp-2}, 364 {0x1.f27f45a470251p-45, 0x1.ee2a156b41p-2}, 365 {0x1.2cde56f014a8bp-46, 0x1.f4aa7ee0318p-2}, 366 {0x1.2cde56f014a8bp-46, 0x1.f4aa7ee0318p-2}, 367 {0x1.085fa3c164935p-47, 0x1.fb358af7a48p-2}, 368 {-0x1.53ba3b1727b1cp-47, 0x1.00e5ae5b208p-1}, 369 {-0x1.4c45fe79539ep-47, 0x1.04360be7604p-1}, 370 {-0x1.4c45fe79539ep-47, 0x1.04360be7604p-1}, 371 {0x1.6812241edf5fdp-45, 0x1.078bf0533c4p-1}, 372 {0x1.f486b887e7e27p-46, 0x1.0ae76e2d054p-1}, 373 {0x1.f486b887e7e27p-46, 0x1.0ae76e2d054p-1}, 374 {0x1.c299807801742p-46, 0x1.0e4898611ccp-1}, 375 {-0x1.58647bb9ddcb2p-45, 0x1.11af823c75cp-1}, 376 {-0x1.58647bb9ddcb2p-45, 0x1.11af823c75cp-1}, 377 {-0x1.edd97a293ae49p-45, 0x1.151c3f6f298p-1}, 378 {0x1.4cc4ef8ab465p-46, 0x1.188ee40f23cp-1}, 379 {0x1.4cc4ef8ab465p-46, 0x1.188ee40f23cp-1}, 380 {0x1.cacdeed70e667p-51, 0x1.1c07849ae6p-1}, 381 {-0x1.a7242c9fe81d3p-45, 0x1.1f8635fc618p-1}, 382 {-0x1.a7242c9fe81d3p-45, 0x1.1f8635fc618p-1}, 383 {0x1.2fc066e48667bp-46, 0x1.230b0d8bebcp-1}, 384 {-0x1.b61f10522625p-47, 0x1.269621134dcp-1}, 385 {-0x1.b61f10522625p-47, 0x1.269621134dcp-1}, 386 {0x1.06d2be797882dp-45, 0x1.2a2786d0ecp-1}, 387 {-0x1.7a6e507b9dc11p-46, 0x1.2dbf557b0ep-1}, 388 {-0x1.7a6e507b9dc11p-46, 0x1.2dbf557b0ep-1}, 389 {-0x1.74e93c5a0ed9cp-45, 0x1.315da443408p-1}, 390 {-0x1.74e93c5a0ed9cp-45, 0x1.315da443408p-1}, 391 {0x1.0b83f9527e6acp-46, 0x1.35028ad9d8cp-1}, 392 {-0x1.18b7abb5569a4p-45, 0x1.38ae2171978p-1}, 393 {-0x1.18b7abb5569a4p-45, 0x1.38ae2171978p-1}, 394 {-0x1.2b7367cfe13c2p-47, 0x1.3c6080c36cp-1}, 395 {-0x1.2b7367cfe13c2p-47, 0x1.3c6080c36cp-1}, 396 {-0x1.6ce7930f0c74cp-45, 0x1.4019c2125ccp-1}, 397 {-0x1.d984f481051f7p-48, 0x1.43d9ff2f924p-1}, 398 {-0x1.d984f481051f7p-48, 0x1.43d9ff2f924p-1}, 399 {-0x1.2cb6af94d60aap-45, 0x1.47a1527e8a4p-1}, 400 {-0x1.2cb6af94d60aap-45, 0x1.47a1527e8a4p-1}, 401 {0x1.f7115ed4c541cp-49, 0x1.4b6fd6f970cp-1}, 402 {0x1.f7115ed4c541cp-49, 0x1.4b6fd6f970cp-1}, 403 {-0x1.e6c516d93b8fbp-45, 0x1.4f45a835a5p-1}, 404 {-0x1.e6c516d93b8fbp-45, 0x1.4f45a835a5p-1}, 405 {0x1.5ccc45d257531p-47, 0x1.5322e268678p-1}, 406 {0x1.5ccc45d257531p-47, 0x1.5322e268678p-1}, 407 {0x1.9980bff3303ddp-47, 0x1.5707a26bb8cp-1}, 408 {0x1.9980bff3303ddp-47, 0x1.5707a26bb8cp-1}, 409 {0x1.dfa63ac10c9fbp-45, 0x1.5af405c3648p-1}, 410 {0x1.dfa63ac10c9fbp-45, 0x1.5af405c3648p-1}, 411 {0x1.202380cda46bep-45, 0x1.5ee82aa2418p-1}, 412 {0.0, 0.0}, 413 }; 414 415 // Logarithm range reduction - Step 2: 416 // r(k) = 2^-16 round(2^16 / (1 + k*2^-14)) for k = -2^6 .. 2^7. 417 // Output range: 418 // [-0x1.3ffcp-15, 0x1.3e3dp-15] 419 // We store S2[i] = 2^16 (r(i - 2^6) - 1). 420 alignas(64) const int S2[193] = { 421 0x101, 0xfd, 0xf9, 0xf5, 0xf1, 0xed, 0xe9, 0xe5, 0xe1, 422 0xdd, 0xd9, 0xd5, 0xd1, 0xcd, 0xc9, 0xc5, 0xc1, 0xbd, 423 0xb9, 0xb4, 0xb0, 0xac, 0xa8, 0xa4, 0xa0, 0x9c, 0x98, 424 0x94, 0x90, 0x8c, 0x88, 0x84, 0x80, 0x7c, 0x78, 0x74, 425 0x70, 0x6c, 0x68, 0x64, 0x60, 0x5c, 0x58, 0x54, 0x50, 426 0x4c, 0x48, 0x44, 0x40, 0x3c, 0x38, 0x34, 0x30, 0x2c, 427 0x28, 0x24, 0x20, 0x1c, 0x18, 0x14, 0x10, 0xc, 0x8, 428 0x4, 0x0, -0x4, -0x8, -0xc, -0x10, -0x14, -0x18, -0x1c, 429 -0x20, -0x24, -0x28, -0x2c, -0x30, -0x34, -0x38, -0x3c, -0x40, 430 -0x44, -0x48, -0x4c, -0x50, -0x54, -0x58, -0x5c, -0x60, -0x64, 431 -0x68, -0x6c, -0x70, -0x74, -0x78, -0x7c, -0x80, -0x84, -0x88, 432 -0x8c, -0x90, -0x94, -0x98, -0x9c, -0xa0, -0xa4, -0xa8, -0xac, 433 -0xb0, -0xb4, -0xb7, -0xbb, -0xbf, -0xc3, -0xc7, -0xcb, -0xcf, 434 -0xd3, -0xd7, -0xdb, -0xdf, -0xe3, -0xe7, -0xeb, -0xef, -0xf3, 435 -0xf7, -0xfb, -0xff, -0x103, -0x107, -0x10b, -0x10f, -0x113, -0x117, 436 -0x11b, -0x11f, -0x123, -0x127, -0x12b, -0x12f, -0x133, -0x137, -0x13a, 437 -0x13e, -0x142, -0x146, -0x14a, -0x14e, -0x152, -0x156, -0x15a, -0x15e, 438 -0x162, -0x166, -0x16a, -0x16e, -0x172, -0x176, -0x17a, -0x17e, -0x182, 439 -0x186, -0x18a, -0x18e, -0x192, -0x195, -0x199, -0x19d, -0x1a1, -0x1a5, 440 -0x1a9, -0x1ad, -0x1b1, -0x1b5, -0x1b9, -0x1bd, -0x1c1, -0x1c5, -0x1c9, 441 -0x1cd, -0x1d1, -0x1d5, -0x1d9, -0x1dd, -0x1e0, -0x1e4, -0x1e8, -0x1ec, 442 -0x1f0, -0x1f4, -0x1f8, -0x1fc}; 443 444 alignas(64) const double R2[193] = { 445 0x1.0101p0, 0x1.00fdp0, 0x1.00f9p0, 0x1.00f5p0, 0x1.00f1p0, 446 0x1.00edp0, 0x1.00e9p0, 0x1.00e5p0, 0x1.00e1p0, 0x1.00ddp0, 447 0x1.00d9p0, 0x1.00d5p0, 0x1.00d1p0, 0x1.00cdp0, 0x1.00c9p0, 448 0x1.00c5p0, 0x1.00c1p0, 0x1.00bdp0, 0x1.00b9p0, 0x1.00b4p0, 449 0x1.00bp0, 0x1.00acp0, 0x1.00a8p0, 0x1.00a4p0, 0x1.00ap0, 450 0x1.009cp0, 0x1.0098p0, 0x1.0094p0, 0x1.009p0, 0x1.008cp0, 451 0x1.0088p0, 0x1.0084p0, 0x1.008p0, 0x1.007cp0, 0x1.0078p0, 452 0x1.0074p0, 0x1.007p0, 0x1.006cp0, 0x1.0068p0, 0x1.0064p0, 453 0x1.006p0, 0x1.005cp0, 0x1.0058p0, 0x1.0054p0, 0x1.005p0, 454 0x1.004cp0, 0x1.0048p0, 0x1.0044p0, 0x1.004p0, 0x1.003cp0, 455 0x1.0038p0, 0x1.0034p0, 0x1.003p0, 0x1.002cp0, 0x1.0028p0, 456 0x1.0024p0, 0x1.002p0, 0x1.001cp0, 0x1.0018p0, 0x1.0014p0, 457 0x1.001p0, 0x1.000cp0, 0x1.0008p0, 0x1.0004p0, 0x1p0, 458 0x1.fff8p-1, 0x1.fffp-1, 0x1.ffe8p-1, 0x1.ffep-1, 0x1.ffd8p-1, 459 0x1.ffdp-1, 0x1.ffc8p-1, 0x1.ffcp-1, 0x1.ffb8p-1, 0x1.ffbp-1, 460 0x1.ffa8p-1, 0x1.ffap-1, 0x1.ff98p-1, 0x1.ff9p-1, 0x1.ff88p-1, 461 0x1.ff8p-1, 0x1.ff78p-1, 0x1.ff7p-1, 0x1.ff68p-1, 0x1.ff6p-1, 462 0x1.ff58p-1, 0x1.ff5p-1, 0x1.ff48p-1, 0x1.ff4p-1, 0x1.ff38p-1, 463 0x1.ff3p-1, 0x1.ff28p-1, 0x1.ff2p-1, 0x1.ff18p-1, 0x1.ff1p-1, 464 0x1.ff08p-1, 0x1.ffp-1, 0x1.fef8p-1, 0x1.fefp-1, 0x1.fee8p-1, 465 0x1.feep-1, 0x1.fed8p-1, 0x1.fedp-1, 0x1.fec8p-1, 0x1.fecp-1, 466 0x1.feb8p-1, 0x1.febp-1, 0x1.fea8p-1, 0x1.feap-1, 0x1.fe98p-1, 467 0x1.fe92p-1, 0x1.fe8ap-1, 0x1.fe82p-1, 0x1.fe7ap-1, 0x1.fe72p-1, 468 0x1.fe6ap-1, 0x1.fe62p-1, 0x1.fe5ap-1, 0x1.fe52p-1, 0x1.fe4ap-1, 469 0x1.fe42p-1, 0x1.fe3ap-1, 0x1.fe32p-1, 0x1.fe2ap-1, 0x1.fe22p-1, 470 0x1.fe1ap-1, 0x1.fe12p-1, 0x1.fe0ap-1, 0x1.fe02p-1, 0x1.fdfap-1, 471 0x1.fdf2p-1, 0x1.fdeap-1, 0x1.fde2p-1, 0x1.fddap-1, 0x1.fdd2p-1, 472 0x1.fdcap-1, 0x1.fdc2p-1, 0x1.fdbap-1, 0x1.fdb2p-1, 0x1.fdaap-1, 473 0x1.fda2p-1, 0x1.fd9ap-1, 0x1.fd92p-1, 0x1.fd8cp-1, 0x1.fd84p-1, 474 0x1.fd7cp-1, 0x1.fd74p-1, 0x1.fd6cp-1, 0x1.fd64p-1, 0x1.fd5cp-1, 475 0x1.fd54p-1, 0x1.fd4cp-1, 0x1.fd44p-1, 0x1.fd3cp-1, 0x1.fd34p-1, 476 0x1.fd2cp-1, 0x1.fd24p-1, 0x1.fd1cp-1, 0x1.fd14p-1, 0x1.fd0cp-1, 477 0x1.fd04p-1, 0x1.fcfcp-1, 0x1.fcf4p-1, 0x1.fcecp-1, 0x1.fce4p-1, 478 0x1.fcdcp-1, 0x1.fcd6p-1, 0x1.fccep-1, 0x1.fcc6p-1, 0x1.fcbep-1, 479 0x1.fcb6p-1, 0x1.fcaep-1, 0x1.fca6p-1, 0x1.fc9ep-1, 0x1.fc96p-1, 480 0x1.fc8ep-1, 0x1.fc86p-1, 0x1.fc7ep-1, 0x1.fc76p-1, 0x1.fc6ep-1, 481 0x1.fc66p-1, 0x1.fc5ep-1, 0x1.fc56p-1, 0x1.fc4ep-1, 0x1.fc46p-1, 482 0x1.fc4p-1, 0x1.fc38p-1, 0x1.fc3p-1, 0x1.fc28p-1, 0x1.fc2p-1, 483 0x1.fc18p-1, 0x1.fc1p-1, 0x1.fc08p-1, 484 }; 485 486 // Logarithm range reduction - Step 3: 487 // r(k) = 2^-21 round(2^21 / (1 + k*2^-21)) for k = -80 .. 80. 488 // Output range: 489 // [-0x1.01928p-22 , 0x1p-22] 490 // We store S[i] = 2^21 (r(i - 80) - 1). 491 alignas(64) const int S3[161] = { 492 0x50, 0x4f, 0x4e, 0x4d, 0x4c, 0x4b, 0x4a, 0x49, 0x48, 0x47, 0x46, 493 0x45, 0x44, 0x43, 0x42, 0x41, 0x40, 0x3f, 0x3e, 0x3d, 0x3c, 0x3b, 494 0x3a, 0x39, 0x38, 0x37, 0x36, 0x35, 0x34, 0x33, 0x32, 0x31, 0x30, 495 0x2f, 0x2e, 0x2d, 0x2c, 0x2b, 0x2a, 0x29, 0x28, 0x27, 0x26, 0x25, 496 0x24, 0x23, 0x22, 0x21, 0x20, 0x1f, 0x1e, 0x1d, 0x1c, 0x1b, 0x1a, 497 0x19, 0x18, 0x17, 0x16, 0x15, 0x14, 0x13, 0x12, 0x11, 0x10, 0xf, 498 0xe, 0xd, 0xc, 0xb, 0xa, 0x9, 0x8, 0x7, 0x6, 0x5, 0x4, 499 0x3, 0x2, 0x1, 0x0, -0x1, -0x2, -0x3, -0x4, -0x5, -0x6, -0x7, 500 -0x8, -0x9, -0xa, -0xb, -0xc, -0xd, -0xe, -0xf, -0x10, -0x11, -0x12, 501 -0x13, -0x14, -0x15, -0x16, -0x17, -0x18, -0x19, -0x1a, -0x1b, -0x1c, -0x1d, 502 -0x1e, -0x1f, -0x20, -0x21, -0x22, -0x23, -0x24, -0x25, -0x26, -0x27, -0x28, 503 -0x29, -0x2a, -0x2b, -0x2c, -0x2d, -0x2e, -0x2f, -0x30, -0x31, -0x32, -0x33, 504 -0x34, -0x35, -0x36, -0x37, -0x38, -0x39, -0x3a, -0x3b, -0x3c, -0x3d, -0x3e, 505 -0x3f, -0x40, -0x41, -0x42, -0x43, -0x44, -0x45, -0x46, -0x47, -0x48, -0x49, 506 -0x4a, -0x4b, -0x4c, -0x4d, -0x4e, -0x4f, -0x50, 507 }; 508 509 // Logarithm range reduction - Step 4 510 // r(k) = 2^-28 round(2^28 / (1 + k*2^-28)) for k = -65 .. 64. 511 // Output range: 512 // [-0x1.0002143p-29 , 0x1p-29] 513 // We store S[i] = 2^28 (r(i - 65) - 1). 514 alignas(64) const int S4[130] = { 515 0x41, 0x40, 0x3f, 0x3e, 0x3d, 0x3c, 0x3b, 0x3a, 0x39, 0x38, 0x37, 516 0x36, 0x35, 0x34, 0x33, 0x32, 0x31, 0x30, 0x2f, 0x2e, 0x2d, 0x2c, 517 0x2b, 0x2a, 0x29, 0x28, 0x27, 0x26, 0x25, 0x24, 0x23, 0x22, 0x21, 518 0x20, 0x1f, 0x1e, 0x1d, 0x1c, 0x1b, 0x1a, 0x19, 0x18, 0x17, 0x16, 519 0x15, 0x14, 0x13, 0x12, 0x11, 0x10, 0xf, 0xe, 0xd, 0xc, 0xb, 520 0xa, 0x9, 0x8, 0x7, 0x6, 0x5, 0x4, 0x3, 0x2, 0x1, 0x0, 521 -0x1, -0x2, -0x3, -0x4, -0x5, -0x6, -0x7, -0x8, -0x9, -0xa, -0xb, 522 -0xc, -0xd, -0xe, -0xf, -0x10, -0x11, -0x12, -0x13, -0x14, -0x15, -0x16, 523 -0x17, -0x18, -0x19, -0x1a, -0x1b, -0x1c, -0x1d, -0x1e, -0x1f, -0x20, -0x21, 524 -0x22, -0x23, -0x24, -0x25, -0x26, -0x27, -0x28, -0x29, -0x2a, -0x2b, -0x2c, 525 -0x2d, -0x2e, -0x2f, -0x30, -0x31, -0x32, -0x33, -0x34, -0x35, -0x36, -0x37, 526 -0x38, -0x39, -0x3a, -0x3b, -0x3c, -0x3d, -0x3e, -0x3f, -0x40, 527 }; 528 529 // Lookup table for exp(m) with m = -104, ..., 89. 530 // -104 = floor(log(single precision's min denormal)) 531 // 89 = ceil(log(single precision's max normal)) 532 // Table is generated with Sollya as follow: 533 // > display = hexadecimal; 534 // > for i from -104 to 89 do { D(exp(i)); }; 535 const double EXP_M1[195] = { 536 0x1.f1e6b68529e33p-151, 0x1.525be4e4e601dp-149, 0x1.cbe0a45f75eb1p-148, 537 0x1.3884e838aea68p-146, 0x1.a8c1f14e2af5dp-145, 0x1.20a717e64a9bdp-143, 538 0x1.8851d84118908p-142, 0x1.0a9bdfb02d240p-140, 0x1.6a5bea046b42ep-139, 539 0x1.ec7f3b269efa8p-138, 0x1.4eafb87eab0f2p-136, 0x1.c6e2d05bbc000p-135, 540 0x1.35208867c2683p-133, 0x1.a425b317eeacdp-132, 0x1.1d8508fa8246ap-130, 541 0x1.840fbc08fdc8ap-129, 0x1.07b7112bc1ffep-127, 0x1.666d0dad2961dp-126, 542 0x1.e726c3f64d0fep-125, 0x1.4b0dc07cabf98p-123, 0x1.c1f2daf3b6a46p-122, 543 0x1.31c5957a47de2p-120, 0x1.9f96445648b9fp-119, 0x1.1a6baeadb4fd1p-117, 544 0x1.7fd974d372e45p-116, 0x1.04da4d1452919p-114, 0x1.62891f06b3450p-113, 545 0x1.e1dd273aa8a4ap-112, 0x1.4775e0840bfddp-110, 0x1.bd109d9d94bdap-109, 546 0x1.2e73f53fba844p-107, 0x1.9b138170d6bfep-106, 0x1.175af0cf60ec5p-104, 547 0x1.7baee1bffa80bp-103, 0x1.02057d1245cebp-101, 0x1.5eafffb34ba31p-100, 548 0x1.dca23bae16424p-99, 0x1.43e7fc88b8056p-97, 0x1.b83bf23a9a9ebp-96, 549 0x1.2b2b8dd05b318p-94, 0x1.969d47321e4ccp-93, 0x1.1452b7723aed2p-91, 550 0x1.778fe2497184cp-90, 0x1.fe7116182e9ccp-89, 0x1.5ae191a99585ap-87, 551 0x1.d775d87da854dp-86, 0x1.4063f8cc8bb98p-84, 0x1.b374b315f87c1p-83, 552 0x1.27ec458c65e3cp-81, 0x1.923372c67a074p-80, 0x1.1152eaeb73c08p-78, 553 0x1.737c5645114b5p-77, 0x1.f8e6c24b5592ep-76, 0x1.571db733a9d61p-74, 554 0x1.d257d547e083fp-73, 0x1.3ce9b9de78f85p-71, 0x1.aebabae3a41b5p-70, 555 0x1.24b6031b49bdap-68, 0x1.8dd5e1bb09d7ep-67, 0x1.0e5b73d1ff53dp-65, 556 0x1.6f741de1748ecp-64, 0x1.f36bd37f42f3ep-63, 0x1.536452ee2f75cp-61, 557 0x1.cd480a1b74820p-60, 0x1.39792499b1a24p-58, 0x1.aa0de4bf35b38p-57, 558 0x1.2188ad6ae3303p-55, 0x1.898471fca6055p-54, 0x1.0b6c3afdde064p-52, 559 0x1.6b7719a59f0e0p-51, 0x1.ee001eed62aa0p-50, 0x1.4fb547c775da8p-48, 560 0x1.c8464f7616468p-47, 0x1.36121e24d3bbap-45, 0x1.a56e0c2ac7f75p-44, 561 0x1.1e642baeb84a0p-42, 0x1.853f01d6d53bap-41, 0x1.0885298767e9ap-39, 562 0x1.67852a7007e42p-38, 0x1.e8a37a45fc32ep-37, 0x1.4c1078fe9228ap-35, 563 0x1.c3527e433fab1p-34, 0x1.32b48bf117da2p-32, 0x1.a0db0d0ddb3ecp-31, 564 0x1.1b48655f37267p-29, 0x1.81056ff2c5772p-28, 0x1.05a628c699fa1p-26, 565 0x1.639e3175a689dp-25, 0x1.e355bbaee85cbp-24, 0x1.4875ca227ec38p-22, 566 0x1.be6c6fdb01612p-21, 0x1.2f6053b981d98p-19, 0x1.9c54c3b43bc8bp-18, 567 0x1.18354238f6764p-16, 0x1.7cd79b5647c9bp-15, 0x1.02cf22526545ap-13, 568 0x1.5fc21041027adp-12, 0x1.de16b9c24a98fp-11, 0x1.44e51f113d4d6p-9, 569 0x1.b993fe00d5376p-8, 0x1.2c155b8213cf4p-6, 0x1.97db0ccceb0afp-5, 570 0x1.152aaa3bf81ccp-3, 0x1.78b56362cef38p-2, 0x1.0000000000000p+0, 571 0x1.5bf0a8b145769p+1, 0x1.d8e64b8d4ddaep+2, 0x1.415e5bf6fb106p+4, 572 0x1.b4c902e273a58p+5, 0x1.28d389970338fp+7, 0x1.936dc5690c08fp+8, 573 0x1.122885aaeddaap+10, 0x1.749ea7d470c6ep+11, 0x1.fa7157c470f82p+12, 574 0x1.5829dcf950560p+14, 0x1.d3c4488ee4f7fp+15, 0x1.3de1654d37c9ap+17, 575 0x1.b00b5916ac955p+18, 0x1.259ac48bf05d7p+20, 0x1.8f0ccafad2a87p+21, 576 0x1.0f2ebd0a80020p+23, 0x1.709348c0ea4f9p+24, 0x1.f4f22091940bdp+25, 577 0x1.546d8f9ed26e1p+27, 0x1.ceb088b68e804p+28, 0x1.3a6e1fd9eecfdp+30, 578 0x1.ab5adb9c43600p+31, 0x1.226af33b1fdc1p+33, 0x1.8ab7fb5475fb7p+34, 579 0x1.0c3d3920962c9p+36, 0x1.6c932696a6b5dp+37, 0x1.ef822f7f6731dp+38, 580 0x1.50bba3796379ap+40, 0x1.c9aae4631c056p+41, 0x1.370470aec28edp+43, 581 0x1.a6b765d8cdf6dp+44, 0x1.1f43fcc4b662cp+46, 0x1.866f34a725782p+47, 582 0x1.0953e2f3a1ef7p+49, 0x1.689e221bc8d5bp+50, 0x1.ea215a1d20d76p+51, 583 0x1.4d13fbb1a001ap+53, 0x1.c4b334617cc67p+54, 0x1.33a43d282a519p+56, 584 0x1.a220d397972ebp+57, 0x1.1c25c88df6862p+59, 0x1.8232558201159p+60, 585 0x1.0672a3c9eb871p+62, 0x1.64b41c6d37832p+63, 0x1.e4cf766fe49bep+64, 586 0x1.49767bc0483e3p+66, 0x1.bfc951eb8bb76p+67, 0x1.304d6aeca254bp+69, 587 0x1.9d97010884251p+70, 0x1.19103e4080b45p+72, 0x1.7e013cd114461p+73, 588 0x1.03996528e074cp+75, 0x1.60d4f6fdac731p+76, 0x1.df8c5af17ba3bp+77, 589 0x1.45e3076d61699p+79, 0x1.baed16a6e0da7p+80, 0x1.2cffdfebde1a1p+82, 590 0x1.9919cabefcb69p+83, 0x1.160345c9953e3p+85, 0x1.79dbc9dc53c66p+86, 591 0x1.00c810d464097p+88, 0x1.5d009394c5c27p+89, 0x1.da57de8f107a8p+90, 592 0x1.425982cf597cdp+92, 0x1.b61e5ca3a5e31p+93, 0x1.29bb825dfcf87p+95, 593 0x1.94a90db0d6fe2p+96, 0x1.12fec759586fdp+98, 0x1.75c1dc469e3afp+99, 594 0x1.fbfd219c43b04p+100, 0x1.5936d44e1a146p+102, 0x1.d531d8a7ee79cp+103, 595 0x1.3ed9d24a2d51bp+105, 0x1.b15cfe5b6e17bp+106, 0x1.268038c2c0e00p+108, 596 0x1.9044a73545d48p+109, 0x1.1002ab6218b38p+111, 0x1.71b3540cbf921p+112, 597 0x1.f6799ea9c414ap+113, 0x1.55779b984f3ebp+115, 0x1.d01a210c44aa4p+116, 598 0x1.3b63da8e91210p+118, 0x1.aca8d6b0116b8p+119, 0x1.234de9e0c74e9p+121, 599 0x1.8bec7503ca477p+122, 0x1.0d0eda9796b90p+124, 0x1.6db0118477245p+125, 600 0x1.f1056dc7bf22dp+126, 0x1.51c2cc3433801p+128, 0x1.cb108ffbec164p+129, 601 }; 602 603 // Lookup table for exp(m * 2^(-7)) with m = 0, ..., 127. 604 // Table is generated with Sollya as follow: 605 // > display = hexadecimal; 606 // > for i from 0 to 127 do { D(exp(i / 128)); }; 607 const double EXP_M2[128] = { 608 0x1.0000000000000p0, 0x1.0202015600446p0, 0x1.04080ab55de39p0, 609 0x1.06122436410ddp0, 0x1.08205601127edp0, 0x1.0a32a84e9c1f6p0, 610 0x1.0c49236829e8cp0, 0x1.0e63cfa7ab09dp0, 0x1.1082b577d34edp0, 611 0x1.12a5dd543ccc5p0, 0x1.14cd4fc989cd6p0, 0x1.16f9157587069p0, 612 0x1.192937074e0cdp0, 0x1.1b5dbd3f68122p0, 0x1.1d96b0eff0e79p0, 613 0x1.1fd41afcba45ep0, 0x1.2216045b6f5cdp0, 0x1.245c7613b8a9bp0, 614 0x1.26a7793f60164p0, 0x1.28f7170a755fdp0, 0x1.2b4b58b372c79p0, 615 0x1.2da4478b620c7p0, 0x1.3001ecf601af7p0, 0x1.32645269ea829p0, 616 0x1.34cb8170b5835p0, 0x1.373783a722012p0, 0x1.39a862bd3c106p0, 617 0x1.3c1e2876834aap0, 0x1.3e98deaa11dccp0, 0x1.41188f42c3e32p0, 618 0x1.439d443f5f159p0, 0x1.462707b2bac21p0, 0x1.48b5e3c3e8186p0, 619 0x1.4b49e2ae5ac67p0, 0x1.4de30ec211e60p0, 0x1.50817263c13cdp0, 620 0x1.5325180cfacf7p0, 0x1.55ce0a4c58c7cp0, 0x1.587c53c5a7af0p0, 621 0x1.5b2fff3210fd9p0, 0x1.5de9176045ff5p0, 0x1.60a7a734ab0e8p0, 622 0x1.636bb9a983258p0, 0x1.663559cf1bc7cp0, 0x1.690492cbf9433p0, 623 0x1.6bd96fdd034a2p0, 0x1.6eb3fc55b1e76p0, 0x1.719443a03acb9p0, 624 0x1.747a513dbef6ap0, 0x1.776630c678bc1p0, 0x1.7a57ede9ea23ep0, 625 0x1.7d4f946f0ba8dp0, 0x1.804d30347b546p0, 0x1.8350cd30ac390p0, 626 0x1.865a7772164c5p0, 0x1.896a3b1f66a0ep0, 0x1.8c802477b0010p0, 627 0x1.8f9c3fd29beafp0, 0x1.92be99a09bf00p0, 0x1.95e73e6b1b75ep0, 628 0x1.99163ad4b1dccp0, 0x1.9c4b9b995509bp0, 0x1.9f876d8e8c566p0, 629 0x1.a2c9bda3a3e78p0, 0x1.a61298e1e069cp0, 0x1.a9620c6cb3374p0, 630 0x1.acb82581eee54p0, 0x1.b014f179fc3b8p0, 0x1.b3787dc80f95fp0, 631 0x1.b6e2d7fa5eb18p0, 0x1.ba540dba56e56p0, 0x1.bdcc2cccd3c85p0, 632 0x1.c14b431256446p0, 0x1.c4d15e873c193p0, 0x1.c85e8d43f7cd0p0, 633 0x1.cbf2dd7d490f2p0, 0x1.cf8e5d84758a9p0, 0x1.d3311bc7822b4p0, 634 0x1.d6db26d16cd67p0, 0x1.da8c8d4a66969p0, 0x1.de455df80e3c0p0, 635 0x1.e205a7bdab73ep0, 0x1.e5cd799c6a54ep0, 0x1.e99ce2b397649p0, 636 0x1.ed73f240dc142p0, 0x1.f152b7a07bb76p0, 0x1.f539424d90f5ep0, 637 0x1.f927a1e24bb76p0, 0x1.fd1de6182f8c9p0, 0x1.008e0f64294abp1, 638 0x1.02912df5ce72ap1, 0x1.049856cd84339p1, 0x1.06a39207f0a09p1, 639 0x1.08b2e7d2035cfp1, 0x1.0ac6606916501p1, 0x1.0cde041b0e9aep1, 640 0x1.0ef9db467dcf8p1, 0x1.1119ee5ac36b6p1, 0x1.133e45d82e952p1, 641 0x1.1566ea50201d7p1, 0x1.1793e4652cc50p1, 0x1.19c53ccb3fc6bp1, 642 0x1.1bfafc47bda73p1, 0x1.1e352bb1a74adp1, 0x1.2073d3f1bd518p1, 643 0x1.22b6fe02a3b9cp1, 0x1.24feb2f105cb8p1, 0x1.274afbdbba4a6p1, 644 0x1.299be1f3e7f1cp1, 0x1.2bf16e7d2a38cp1, 0x1.2e4baacdb6614p1, 645 0x1.30aaa04e80d05p1, 0x1.330e587b62b28p1, 0x1.3576dce33feadp1, 646 0x1.37e437282d4eep1, 0x1.3a5670ff972edp1, 0x1.3ccd9432682b4p1, 647 0x1.3f49aa9d30590p1, 0x1.41cabe304cb34p1, 0x1.4450d8f00edd4p1, 648 0x1.46dc04f4e5338p1, 0x1.496c4c6b832dap1, 0x1.4c01b9950a111p1, 649 0x1.4e9c56c731f5dp1, 0x1.513c2e6c731d7p1, 0x1.53e14b042f9cap1, 650 0x1.568bb722dd593p1, 0x1.593b7d72305bbp1, 651 }; 652 653 // Lookup table for 2^(k * 2^-6) with k = 0..63. 654 // Generated by Sollya with: 655 // > display=hexadecimal; 656 // > prec = 500; 657 // > for i from 0 to 63 do { 658 // a = 2^(i * 2^-6); 659 // b = round(a, D, RN); 660 // c = round(a - b, D, RN); 661 // d = round(a - b - c, D, RN); 662 // print("{", d, ",", c, ",", b, "},"); 663 // }; 664 const fputil::TripleDouble EXP2_MID1[64] = { 665 {0, 0, 0x1p0}, 666 {-0x1.9085b0a3d74d5p-110, -0x1.19083535b085dp-56, 0x1.02c9a3e778061p0}, 667 {0x1.05ff94f8d257ep-110, 0x1.d73e2a475b465p-55, 0x1.059b0d3158574p0}, 668 {0x1.15820d96b414fp-111, 0x1.186be4bb284ffp-57, 0x1.0874518759bc8p0}, 669 {-0x1.67c9bd6ebf74cp-108, 0x1.8a62e4adc610bp-54, 0x1.0b5586cf9890fp0}, 670 {-0x1.5aa76994e9ddbp-113, 0x1.03a1727c57b53p-59, 0x1.0e3ec32d3d1a2p0}, 671 {0x1.9d58b988f562dp-109, -0x1.6c51039449b3ap-54, 0x1.11301d0125b51p0}, 672 {-0x1.2fe7bb4c76416p-108, -0x1.32fbf9af1369ep-54, 0x1.1429aaea92dep0}, 673 {0x1.4f2406aa13ffp-109, -0x1.19041b9d78a76p-55, 0x1.172b83c7d517bp0}, 674 {0x1.ad36183926ae8p-111, 0x1.e5b4c7b4968e4p-55, 0x1.1a35beb6fcb75p0}, 675 {0x1.ea62d0881b918p-110, 0x1.e016e00a2643cp-54, 0x1.1d4873168b9aap0}, 676 {-0x1.781dbc16f1ea4p-111, 0x1.dc775814a8495p-55, 0x1.2063b88628cd6p0}, 677 {-0x1.4d89f9af532ep-109, 0x1.9b07eb6c70573p-54, 0x1.2387a6e756238p0}, 678 {0x1.277393a461b77p-110, 0x1.2bd339940e9d9p-55, 0x1.26b4565e27cddp0}, 679 {0x1.de5448560469p-111, 0x1.612e8afad1255p-55, 0x1.29e9df51fdee1p0}, 680 {-0x1.ee9d8f8cb9307p-110, 0x1.0024754db41d5p-54, 0x1.2d285a6e4030bp0}, 681 {0x1.7b7b2f09cd0d9p-110, 0x1.6f46ad23182e4p-55, 0x1.306fe0a31b715p0}, 682 {-0x1.406a2ea6cfc6bp-108, 0x1.32721843659a6p-54, 0x1.33c08b26416ffp0}, 683 {0x1.87e3e12516bfap-108, -0x1.63aeabf42eae2p-54, 0x1.371a7373aa9cbp0}, 684 {0x1.9b0b1ff17c296p-111, -0x1.5e436d661f5e3p-56, 0x1.3a7db34e59ff7p0}, 685 {-0x1.808ba68fa8fb7p-109, 0x1.ada0911f09ebcp-55, 0x1.3dea64c123422p0}, 686 {-0x1.32b43eafc6518p-114, -0x1.ef3691c309278p-58, 0x1.4160a21f72e2ap0}, 687 {-0x1.0ac312de3d922p-114, 0x1.89b7a04ef80dp-59, 0x1.44e086061892dp0}, 688 {0x1.e1eebae743acp-111, 0x1.3c1a3b69062fp-56, 0x1.486a2b5c13cdp0}, 689 {0x1.c06c7745c2b39p-113, 0x1.d4397afec42e2p-56, 0x1.4bfdad5362a27p0}, 690 {-0x1.1aa1fd7b685cdp-112, -0x1.4b309d25957e3p-54, 0x1.4f9b2769d2ca7p0}, 691 {0x1.fa733951f214cp-111, -0x1.07abe1db13cadp-55, 0x1.5342b569d4f82p0}, 692 {-0x1.ff86852a613ffp-111, 0x1.9bb2c011d93adp-54, 0x1.56f4736b527dap0}, 693 {-0x1.744ee506fdafep-109, 0x1.6324c054647adp-54, 0x1.5ab07dd485429p0}, 694 {-0x1.95f9ab75fa7d6p-108, 0x1.ba6f93080e65ep-54, 0x1.5e76f15ad2148p0}, 695 {0x1.5d8e757cfb991p-111, -0x1.383c17e40b497p-54, 0x1.6247eb03a5585p0}, 696 {0x1.4a337f4dc0a3bp-108, -0x1.bb60987591c34p-54, 0x1.6623882552225p0}, 697 {0x1.57d3e3adec175p-108, -0x1.bdd3413b26456p-54, 0x1.6a09e667f3bcdp0}, 698 {0x1.a59f88abbe778p-115, -0x1.bbe3a683c88abp-57, 0x1.6dfb23c651a2fp0}, 699 {-0x1.269796953a4c3p-109, -0x1.16e4786887a99p-55, 0x1.71f75e8ec5f74p0}, 700 {-0x1.8f8e7fa19e5e8p-108, -0x1.0245957316dd3p-54, 0x1.75feb564267c9p0}, 701 {-0x1.4217a932d10d4p-113, -0x1.41577ee04992fp-55, 0x1.7a11473eb0187p0}, 702 {0x1.70a1427f8fcdfp-112, 0x1.05d02ba15797ep-56, 0x1.7e2f336cf4e62p0}, 703 {0x1.0f6ad65cbbac1p-112, -0x1.d4c1dd41532d8p-54, 0x1.82589994cce13p0}, 704 {-0x1.f16f65181d921p-109, -0x1.fc6f89bd4f6bap-54, 0x1.868d99b4492edp0}, 705 {-0x1.30644a7836333p-110, 0x1.6e9f156864b27p-54, 0x1.8ace5422aa0dbp0}, 706 {0x1.3bf26d2b85163p-114, 0x1.5cc13a2e3976cp-55, 0x1.8f1ae99157736p0}, 707 {0x1.697e257ac0db2p-111, -0x1.75fc781b57ebcp-57, 0x1.93737b0cdc5e5p0}, 708 {0x1.7edb9d7144b6fp-108, -0x1.d185b7c1b85d1p-54, 0x1.97d829fde4e5p0}, 709 {0x1.6376b7943085cp-110, 0x1.c7c46b071f2bep-56, 0x1.9c49182a3f09p0}, 710 {0x1.354084551b4fbp-109, -0x1.359495d1cd533p-54, 0x1.a0c667b5de565p0}, 711 {-0x1.bfd7adfd63f48p-111, -0x1.d2f6edb8d41e1p-54, 0x1.a5503b23e255dp0}, 712 {0x1.8b16ae39e8cb9p-109, 0x1.0fac90ef7fd31p-54, 0x1.a9e6b5579fdbfp0}, 713 {0x1.a7fbc3ae675eap-108, 0x1.7a1cd345dcc81p-54, 0x1.ae89f995ad3adp0}, 714 {0x1.2babc0edda4d9p-111, -0x1.2805e3084d708p-57, 0x1.b33a2b84f15fbp0}, 715 {0x1.aa64481e1ab72p-111, -0x1.5584f7e54ac3bp-56, 0x1.b7f76f2fb5e47p0}, 716 {0x1.9a164050e1258p-109, 0x1.23dd07a2d9e84p-55, 0x1.bcc1e904bc1d2p0}, 717 {0x1.99e51125928dap-110, 0x1.11065895048ddp-55, 0x1.c199bdd85529cp0}, 718 {-0x1.fc44c329d5cb2p-109, 0x1.2884dff483cadp-54, 0x1.c67f12e57d14bp0}, 719 {0x1.d8765566b032ep-110, 0x1.503cbd1e949dbp-56, 0x1.cb720dcef9069p0}, 720 {-0x1.e7044039da0f6p-108, -0x1.cbc3743797a9cp-54, 0x1.d072d4a07897cp0}, 721 {-0x1.ab053b05531fcp-111, 0x1.2ed02d75b3707p-55, 0x1.d5818dcfba487p0}, 722 {0x1.7f6246f0ec615p-108, 0x1.c2300696db532p-54, 0x1.da9e603db3285p0}, 723 {0x1.b7225a944efd6p-108, -0x1.1a5cd4f184b5cp-54, 0x1.dfc97337b9b5fp0}, 724 {0x1.1e92cb3c2d278p-109, 0x1.39e8980a9cc8fp-55, 0x1.e502ee78b3ff6p0}, 725 {-0x1.fc0f242bbf3dep-109, -0x1.e9c23179c2893p-54, 0x1.ea4afa2a490dap0}, 726 {0x1.f6dd5d229ff69p-108, 0x1.dc7f486a4b6bp-54, 0x1.efa1bee615a27p0}, 727 {-0x1.4019bffc80ef3p-110, 0x1.9d3e12dd8a18bp-54, 0x1.f50765b6e454p0}, 728 {0x1.dc060c36f7651p-112, 0x1.74853f3a5931ep-55, 0x1.fa7c1819e90d8p0}, 729 }; 730 731 // Lookup table for 2^(k * 2^-12) with k = 0..63. 732 // Generated by Sollya with: 733 // > display=hexadecimal; 734 // > prec = 500; 735 // > for i from 0 to 63 do { 736 // a = 2^(i * 2^-12); 737 // b = round(a, D, RN); 738 // c = round(a - b, D, RN); 739 // d = round(a - b - c, D, RN); 740 // print("{", d, ",", c, ",", b, "},"); 741 // }; 742 const fputil::TripleDouble EXP2_MID2[64] = { 743 {0, 0, 0x1p0}, 744 {0x1.39726694630e3p-108, 0x1.ae8e38c59c72ap-54, 0x1.000b175effdc7p0}, 745 {0x1.e5e06ddd31156p-112, -0x1.7b5d0d58ea8f4p-58, 0x1.00162f3904052p0}, 746 {0x1.5a0768b51f609p-111, 0x1.4115cb6b16a8ep-54, 0x1.0021478e11ce6p0}, 747 {0x1.d008403605217p-111, -0x1.d7c96f201bb2fp-55, 0x1.002c605e2e8cfp0}, 748 {0x1.89bc16f765708p-109, 0x1.84711d4c35e9fp-54, 0x1.003779a95f959p0}, 749 {-0x1.4535b7f8c1e2dp-109, -0x1.0484245243777p-55, 0x1.0042936faa3d8p0}, 750 {-0x1.8ba92f6b25456p-108, -0x1.4b237da2025f9p-54, 0x1.004dadb113dap0}, 751 {-0x1.30c72e81f4294p-113, -0x1.5e00e62d6b30dp-56, 0x1.0058c86da1c0ap0}, 752 {-0x1.34a5384e6f0b9p-110, 0x1.a1d6cedbb9481p-54, 0x1.0063e3a559473p0}, 753 {0x1.f8d0580865d2ep-108, -0x1.4acf197a00142p-54, 0x1.006eff583fc3dp0}, 754 {-0x1.002bcb3ae9a99p-111, -0x1.eaf2ea42391a5p-57, 0x1.007a1b865a8cap0}, 755 {0x1.c3c5aedee9851p-111, 0x1.da93f90835f75p-56, 0x1.0085382faef83p0}, 756 {0x1.7217851d1ec6ep-109, -0x1.6a79084ab093cp-55, 0x1.00905554425d4p0}, 757 {-0x1.80cbca335a7c3p-110, 0x1.86364f8fbe8f8p-54, 0x1.009b72f41a12bp0}, 758 {-0x1.706bd4eb22595p-110, -0x1.82e8e14e3110ep-55, 0x1.00a6910f3b6fdp0}, 759 {-0x1.b55dd523f3c08p-111, -0x1.4f6b2a7609f71p-55, 0x1.00b1afa5abcbfp0}, 760 {0x1.90a1e207cced1p-110, -0x1.e1a258ea8f71bp-56, 0x1.00bcceb7707ecp0}, 761 {0x1.78d0472db37c5p-110, 0x1.4362ca5bc26f1p-56, 0x1.00c7ee448ee02p0}, 762 {-0x1.bcd4db3cb52fep-109, 0x1.095a56c919d02p-54, 0x1.00d30e4d0c483p0}, 763 {-0x1.cf1b131575ec2p-112, -0x1.406ac4e81a645p-57, 0x1.00de2ed0ee0f5p0}, 764 {-0x1.6aaa1fa7ff913p-112, 0x1.b5a6902767e09p-54, 0x1.00e94fd0398ep0}, 765 {0x1.68f236dff3218p-110, -0x1.91b2060859321p-54, 0x1.00f4714af41d3p0}, 766 {-0x1.e8bb58067e60ap-109, 0x1.427068ab22306p-55, 0x1.00ff93412315cp0}, 767 {0x1.d4cd5e1d71fdfp-108, 0x1.c1d0660524e08p-54, 0x1.010ab5b2cbd11p0}, 768 {0x1.e4ecf350ebe88p-108, -0x1.e7bdfb3204be8p-54, 0x1.0115d89ff3a8bp0}, 769 {0x1.6a2aa2c89c4f8p-109, 0x1.843aa8b9cbbc6p-55, 0x1.0120fc089ff63p0}, 770 {0x1.1ca368a20ed05p-110, -0x1.34104ee7edae9p-56, 0x1.012c1fecd613bp0}, 771 {0x1.edb1095d925cfp-114, -0x1.2b6aeb6176892p-56, 0x1.0137444c9b5b5p0}, 772 {-0x1.488c78eded75fp-111, 0x1.a8cd33b8a1bb3p-56, 0x1.01426927f5278p0}, 773 {-0x1.7480f5ea1b3c9p-113, 0x1.2edc08e5da99ap-56, 0x1.014d8e7ee8d2fp0}, 774 {-0x1.ae45989a04dd5p-111, 0x1.57ba2dc7e0c73p-55, 0x1.0158b4517bb88p0}, 775 {0x1.bf48007d80987p-109, 0x1.b61299ab8cdb7p-54, 0x1.0163da9fb3335p0}, 776 {0x1.1aa91a059292cp-109, -0x1.90565902c5f44p-54, 0x1.016f0169949edp0}, 777 {0x1.b6663292855f5p-110, 0x1.70fc41c5c2d53p-55, 0x1.017a28af25567p0}, 778 {0x1.e7fbca6793d94p-108, 0x1.4b9a6e145d76cp-54, 0x1.018550706ab62p0}, 779 {-0x1.5b9f5c7de3b93p-110, -0x1.008eff5142bf9p-56, 0x1.019078ad6a19fp0}, 780 {0x1.4638bf2f6acabp-110, -0x1.77669f033c7dep-54, 0x1.019ba16628de2p0}, 781 {-0x1.ab237b9a069c5p-109, -0x1.09bb78eeead0ap-54, 0x1.01a6ca9aac5f3p0}, 782 {0x1.3ab358be97cefp-108, 0x1.371231477ece5p-54, 0x1.01b1f44af9f9ep0}, 783 {-0x1.4027b2294bb64p-110, 0x1.5e7626621eb5bp-56, 0x1.01bd1e77170b4p0}, 784 {0x1.656394426c99p-111, -0x1.bc72b100828a5p-54, 0x1.01c8491f08f08p0}, 785 {0x1.bf9785189bdd8p-111, -0x1.ce39cbbab8bbep-57, 0x1.01d37442d507p0}, 786 {0x1.7c12f86114fe3p-109, 0x1.16996709da2e2p-55, 0x1.01de9fe280ac8p0}, 787 {-0x1.653d5d24b5d28p-109, -0x1.c11f5239bf535p-55, 0x1.01e9cbfe113efp0}, 788 {0x1.04a0cdc1d86d7p-109, 0x1.e1d4eb5edc6b3p-55, 0x1.01f4f8958c1c6p0}, 789 {0x1.c678c46149782p-109, -0x1.afb99946ee3fp-54, 0x1.020025a8f6a35p0}, 790 {0x1.48524e1e9df7p-108, -0x1.8f06d8a148a32p-54, 0x1.020b533856324p0}, 791 {0x1.9953ea727ff0bp-109, -0x1.2bf310fc54eb6p-55, 0x1.02168143b0281p0}, 792 {-0x1.ccfbbec22d28ep-108, -0x1.c95a035eb4175p-54, 0x1.0221afcb09e3ep0}, 793 {0x1.9e2bb6e181de1p-108, -0x1.491793e46834dp-54, 0x1.022cdece68c4fp0}, 794 {0x1.f17609ae29308p-110, -0x1.3e8d0d9c49091p-56, 0x1.02380e4dd22adp0}, 795 {-0x1.c7dc2c476bfb8p-110, -0x1.314aa16278aa3p-54, 0x1.02433e494b755p0}, 796 {-0x1.fab994971d4a3p-109, 0x1.48daf888e9651p-55, 0x1.024e6ec0da046p0}, 797 {0x1.848b62cbdd0afp-109, 0x1.56dc8046821f4p-55, 0x1.02599fb483385p0}, 798 {-0x1.bf603ba715d0cp-109, 0x1.45b42356b9d47p-54, 0x1.0264d1244c719p0}, 799 {0x1.89434e751e1aap-110, -0x1.082ef51b61d7ep-56, 0x1.027003103b10ep0}, 800 {-0x1.03b54fd64e8acp-110, 0x1.2106ed0920a34p-56, 0x1.027b357854772p0}, 801 {0x1.7785ea0acc486p-109, -0x1.fd4cf26ea5d0fp-54, 0x1.0286685c9e059p0}, 802 {-0x1.ce447fdb35ff9p-109, -0x1.09f8775e78084p-54, 0x1.02919bbd1d1d8p0}, 803 {0x1.5b884aab5642ap-112, 0x1.64cbba902ca27p-58, 0x1.029ccf99d720ap0}, 804 {-0x1.cfb3e46d7c1cp-108, 0x1.4383ef231d207p-54, 0x1.02a803f2d170dp0}, 805 {-0x1.0d40cee4b81afp-112, 0x1.4a47a505b3a47p-54, 0x1.02b338c811703p0}, 806 {0x1.6ae7d36d7c1f7p-109, 0x1.e47120223467fp-54, 0x1.02be6e199c811p0}, 807 }; 808 809 } // namespace LIBC_NAMESPACE_DECL 810