/* Correctly-rounded error function for binary32 value. Copyright (c) 2022-2024 Alexei Sibidanov. This file is part of the CORE-MATH project project (file src/binary32/erf/erff.c revision bc385c2). Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */ #include #include #include #include "math_config.h" float __erff (float x) { /* for 7 <= i < 63, C[i-7] is a degree-7 polynomial approximation of erf(i/16+1/32+x) for -1/32 <= x <= 1/32 */ static const double C[56][8] = { { 0x1.f86faa9428f9cp-2, 0x1.cfc41e36c7dfap-1, -0x1.b2c7dc53508b9p-2, -0x1.5a9de93fa556ep-3, 0x1.731793dbb01b5p-3, 0x1.133e06426cf18p-6, -0x1.a12a6289cafd8p-5, 0x1.717d6f1d6f557p-9 }, { 0x1.1855a5fd3dd50p-1, 0x1.b3aafcc27502fp-1, -0x1.cee5ac8e92bb2p-2, -0x1.fa02983ca2d79p-4, 0x1.77cd746cb1922p-3, -0x1.fa6f277886487p-10, -0x1.8de75458db416p-5, 0x1.00899c98551c9p-7 }, { 0x1.32a54cb8db67ap-1, 0x1.96164fafd8de5p-1, -0x1.e23a7ea0c9ad3p-2, -0x1.3f5ee15671cf4p-4, 0x1.70e468a3d72d9p-3, -0x1.3da68037cfc99p-6, -0x1.69ed9ba1f9839p-5, 0x1.8cab9244a4ff4p-7 }, { 0x1.4b13713ad3513p-1, 0x1.7791b886e7405p-1, -0x1.ecef423109bf5p-2, -0x1.15c3c5cec6847p-5, 0x1.5f688fc931ba6p-3, -0x1.1da63ed190037p-5, -0x1.38427ca63cca4p-5, 0x1.fa00e52525e17p-7 }, { 0x1.61955607dd15dp-1, 0x1.58a445da7c74ep-1, -0x1.ef6c246a0f66cp-2, 0x1.e83e0d9d61330p-8, 0x1.44cc65535bc9fp-3, -0x1.87d3c4860435dp-5, -0x1.f90b10501169bp-6, 0x1.22295856d427ap-6 }, { 0x1.762870f720c6fp-1, 0x1.39ccc1b136d5cp-1, -0x1.ea4feea4e4744p-2, 0x1.715e5952ebfbap-5, 0x1.22cdbd83c75c4p-3, -0x1.da50aa1d925b6p-5, -0x1.754dc0a29b4ddp-6, 0x1.350b6bef9392cp-6 }, { 0x1.88d1cd474a2e0p-1, 0x1.1b7e98fe26219p-1, -0x1.de65a22ce1419p-2, 0x1.40686a3f16400p-4, 0x1.f6b0cbb216b2bp-4, -0x1.09c7c903edd57p-4, -0x1.da7529fde641p-7, 0x1.362a7a0588eabp-6 }, { 0x1.999d4192a5717p-1, 0x1.fc3ee5d1524b3p-2, -0x1.cc990045b55c8p-2, 0x1.b37338e68b37dp-4, 0x1.a0d120c872ea7p-4, -0x1.19bb2b07ecff6p-4, -0x1.a110f5f593aafp-8, 0x1.272c15a57720ep-6 }, { 0x1.a89c850b7d54dp-1, 0x1.c40b0729ed54ap-2, -0x1.b5eaaef0a2346p-2, 0x1.0847c7dacbae1p-3, 0x1.47de0ba6d18fbp-4, -0x1.1d9de77a4b648p-4, 0x1.30ffbe56f0726p-10, 0x1.0a9cb99feea01p-6 }, { 0x1.b5e62fce16096p-1, 0x1.8eed36b886d95p-2, -0x1.9b64a06e50705p-2, 0x1.2bb6e2c744df5p-3, 0x1.dee3261ca61bcp-5, -0x1.16996004f7da5p-4, 0x1.fdff37bae983ep-8, 0x1.c750083e65f9ap-7 }, { 0x1.c194b1d49a184p-1, 0x1.5d4fd33729015p-2, -0x1.7e0f4f045addbp-2, 0x1.444bc66c31a1bp-3, 0x1.356dbf16ec8f1p-5, -0x1.0643de0906cd8p-4, 0x1.b281af7bd3a2cp-7, 0x1.6b97eaa2c6abdp-7 }, { 0x1.cbc54b476248ep-1, 0x1.2f7cc3fe6f423p-2, -0x1.5ee8429e36de8p-2, 0x1.52a8395f96177p-3, 0x1.313761ba257dcp-6, -0x1.dcf844d5fed8fp-5, 0x1.1e1420f475fa9p-6, 0x1.091c7dc1e18b2p-7 }, { 0x1.d4970f9ce00d9p-1, 0x1.059f59af7a905p-2, -0x1.3eda354de36c3p-2, 0x1.57b85ad439779p-3, 0x1.8e913b9778136p-10, -0x1.a2893bd3435f4p-5, 0x1.4d3a90e37164ap-6, 0x1.4ce7f6e19a902p-8 }, { 0x1.dc29fb60715b0p-1, 0x1.bf8e1b1ca2277p-3, -0x1.1eb7095e5d6d2p-2, 0x1.549ea6f7a64f4p-3, -0x1.b10f12f3877a3p-7, -0x1.61420c8f7156ap-5, 0x1.674f1f92a8812p-6, 0x1.25543ffd74d52p-9 }, { 0x1.e29e22a89d767p-1, 0x1.7bd5c7df3fe99p-3, -0x1.fe674494077bfp-3, 0x1.4a9feacf86578p-3, -0x1.a008269076644p-6, -0x1.1cf0e8fb4f1cbp-5, 0x1.6e0d2ef105fb3p-6, -0x1.367205fbd7876p-12 }, { 0x1.e812fc64db36ap-1, 0x1.3fda6bc016991p-3, -0x1.c1cb278627920p-3, 0x1.3b10512314f1ep-3, -0x1.1e6457bb1b9a9p-5, -0x1.b1f6474e2388cp-6, 0x1.640a5345f7ec7p-6, -0x1.3dae5a997fdbp-9 }, { 0x1.eca6ccd709544p-1, 0x1.0b3f52ce8c380p-3, -0x1.8885019f63c6dp-3, 0x1.274275fc91a05p-3, -0x1.57f73699a8372p-5, -0x1.3076a305fc7cep-6, 0x1.4c6ae04843a41p-6, -0x1.0be5fcf5ecc91p-8 }, { 0x1.f0762fde45ee7p-1, 0x1.bb1c972f23e4ap-4, -0x1.5341e3c01b58dp-3, 0x1.107929f6f0b60p-3, -0x1.7e1b34f976c02p-5, -0x1.73b62589c234ap-7, 0x1.2a97ee1876486p-6, -0x1.595f40a3150fep-8 }, { 0x1.f39bc242e43e6p-1, 0x1.6c7e64e7281c5p-4, -0x1.2274b86835fd3p-3, 0x1.efb890e5c770dp-4, -0x1.92c7db16847e0p-5, -0x1.45477db5e2dd4p-8, 0x1.01fc6165fc866p-6, -0x1.8845509030c2cp-8 }, { 0x1.f62fe80272419p-1, 0x1.297db960e4f5dp-4, -0x1.ecb83b087c04fp-4, 0x1.bce18363ca3d1p-4, -0x1.985aaf776482cp-5, 0x1.cd953efdae886p-12, 0x1.ab9a0b89b54ffp-7, -0x1.9b5e576ccc31cp-8 }, { 0x1.f848acb544e95p-1, 0x1.e1d4cf1e24501p-5, -0x1.9e12e1fde5552p-4, 0x1.8a27806df3d1bp-4, -0x1.91674e5eb3319p-5, 0x1.3bc75595b2db8p-8, 0x1.51bc537ac61afp-7, -0x1.96b23b19ea04dp-8 }, { 0x1.f9f9ba8d3c733p-1, 0x1.83298d7172108p-5, -0x1.58d101f905a75p-4, 0x1.58f1456f8639bp-4, -0x1.808d1850b8231p-5, 0x1.0c1bd99c348a7p-7, 0x1.f61e9d7bc48cap-8, -0x1.7f07c13441774p-8 }, { 0x1.fb54641aebbc9p-1, 0x1.34ac36ad8dafap-5, -0x1.1c8ec267f9405p-4, 0x1.2a52c5d841848p-4, -0x1.68541c02b3b6bp-5, 0x1.5afe400196379p-7, 0x1.565b2d6eda3d6p-8, -0x1.596aaff29e739p-8 }, { 0x1.fc67bcf2d7b8fp-1, 0x1.e85c449e377efp-6, -0x1.d177f166c07c6p-5, 0x1.fe23b7584b504p-5, -0x1.4b12109613313p-5, 0x1.8d9905c0acf7dp-7, 0x1.9265032a669dap-9, -0x1.2ac4a6dbcbf3ep-8 }, { 0x1.fd40bd6d7a785p-1, 0x1.7f5188610ddc7p-6, -0x1.7954423f7c998p-5, 0x1.af5baae33887fp-5, -0x1.2ad77c7cbc474p-5, 0x1.a7b8c47ec2a51p-7, 0x1.46646ee094bccp-10, -0x1.ef19d8db8673p-9 }, { 0x1.fdea6e062d0c9p-1, 0x1.2a875b5ffab58p-6, -0x1.2f3178cd6dcd5p-5, 0x1.68d1c45b94182p-5, -0x1.09648ed3aeaefp-5, 0x1.ad8b150d38164p-7, -0x1.e9a6023d9429fp-13, -0x1.8722d19ee2e8ep-9 }, { 0x1.fe6e1742f7cf5p-1, 0x1.cd5ec93c1243ap-7, -0x1.e2ff3aaacb386p-6, 0x1.2aa4e5823cc89p-5, -0x1.d049842dbe399p-6, 0x1.a34edb21ab302p-7, -0x1.676e5996c7f9bp-10, -0x1.23b01a35140bfp-9 }, { 0x1.fed37386190fbp-1, 0x1.61beae53b72c2p-7, -0x1.7d6193f22c3c1p-6, 0x1.e947279e3bb7dp-6, -0x1.906031b97ca97p-6, 0x1.8d14d62561755p-7, -0x1.1f245e7178882p-9, -0x1.9257d4eb47685p-10 }, { 0x1.ff20e0a7ba8c2p-1, 0x1.0d1d69569b839p-7, -0x1.2a8ca0dc02752p-6, 0x1.8cc071b709751p-6, -0x1.54a149f1b070cp-6, 0x1.6e9137b13412cp-7, -0x1.6577ed3d8e83bp-9, -0x1.e9c1a5178a289p-11 }, { 0x1.ff5b8fb26f5f6p-1, 0x1.9646f35a7663cp-8, -0x1.cf68ed9311b0bp-7, 0x1.3e8735b5a694fp-6, -0x1.1e1612d026fdfp-6, 0x1.4afd8e6ca636dp-7, -0x1.8c375170ccb22p-9, -0x1.c799443c4fd3bp-12 }, { 0x1.ff87b1913e853p-1, 0x1.30499b5039596p-8, -0x1.64964201ec8bap-7, 0x1.fa73d7eafba98p-7, -0x1.daa3022141fbbp-7, 0x1.2509444c063b7p-7, -0x1.99482a2f8a0a1p-9, -0x1.403d1f76c9454p-15 }, { 0x1.ffa89fe5b3625p-1, 0x1.c4412bf4b8f35p-9, -0x1.100f347126cf0p-7, 0x1.8ebda07671d40p-7, -0x1.850c6a31c98c1p-7, 0x1.fdac860c67d21p-8, -0x1.927d03d2ba12cp-9, 0x1.0ff620b4190fep-12 }, { 0x1.ffc10194fcb64p-1, 0x1.4d78bba8ca621p-9, -0x1.9ba107a443e02p-8, 0x1.36f273fbc04ccp-7, -0x1.3b38716ac7e6fp-7, 0x1.b3fe0181914acp-8, -0x1.7d3fe7de98c5cp-9, 0x1.ea31f8e5317f7p-12 }, { 0x1.ffd2eae369a07p-1, 0x1.e7f232d9e266cp-10, -0x1.34c7442dd48d9p-8, 0x1.e066bed070a0bp-8, -0x1.f914f3c42fc0dp-8, 0x1.6f4664ed2260fp-8, -0x1.5e59910761d24p-9, 0x1.39cbb6e84c126p-11 }, { 0x1.ffdff92db56e5p-1, 0x1.6235fbd7a4373p-10, -0x1.cb5e029b9e56ap-9, 0x1.6fa4c7ef274dap-8, -0x1.903a089a835f3p-8, 0x1.30f12e0ca1901p-8, -0x1.39d21b6957f99p-9, 0x1.5d3f8495a703cp-11 }, { 0x1.ffe96a78a04a9p-1, 0x1.fe41cd9bb4f2cp-11, -0x1.52d7b28966c0cp-9, 0x1.16c192d86a1a7p-8, -0x1.39bfce951100cp-8, 0x1.f376a7869f9e3p-9, -0x1.12e6cef999c4fp-9, 0x1.66acd4d667b5p-11 }, { 0x1.fff0312b010b5p-1, 0x1.6caa0d3583018p-11, -0x1.efb729f4cf75bp-10, 0x1.a2da7cebe12acp-9, -0x1.e6c27a24bc759p-9, 0x1.93b1f4d8ea65p-9, -0x1.d82050aa94a08p-10, 0x1.5cd7dc75d6cbap-11 }, { 0x1.fff50456dab8cp-1, 0x1.0295ef6591865p-11, -0x1.679880e95a4dap-10, 0x1.37d38e3a5c8ebp-9, -0x1.75b3708aebb8fp-9, 0x1.4231c4b4b0296p-9, -0x1.8e26476489318p-10, 0x1.45c3b570dd924p-11 }, { 0x1.fff86cfd3e657p-1, 0x1.6be02102b353dp-12, -0x1.02b157780d6aep-10, 0x1.cc1d886861133p-10, -0x1.1bff6f12ec9abp-9, 0x1.fc0f77bd9c736p-10, -0x1.4a3320bd0959dp-10, 0x1.267f8b4f95d2p-11 }, { 0x1.fffad0b901755p-1, 0x1.fc0d55470cf5ep-13, -0x1.7121aff5e820ep-11, 0x1.506d6992f7de5p-10, -0x1.ab595d3ecd0d6p-10, 0x1.8bdd79daaf754p-10, -0x1.0d9b090f997c1p-10, 0x1.031ab9fd1c7dap-11 }, { 0x1.fffc7a37857d2p-1, 0x1.5feada379d8a5p-13, -0x1.05304df58f3aap-11, 0x1.e79c081b8600fp-11, -0x1.3e5dbe33232e0p-10, 0x1.30eb208200729p-10, -0x1.b1d493b147945p-11, 0x1.bd587bbc071bep-12 }, { 0x1.fffd9fdeabccep-1, 0x1.e3bcf436a1a49p-14, -0x1.6e953111ef0a1p-12, 0x1.5e3edf6768654p-11, -0x1.d5be67c0547a4p-11, 0x1.d07d9ffa1d435p-11, -0x1.58328f5f358cap-11, 0x1.76d42d95c42c4p-12 }, { 0x1.fffe68f4fa777p-1, 0x1.49e17724f4cddp-14, -0x1.fe48c44e229c1p-13, 0x1.f2bd95d76f188p-12, -0x1.57388cb12d011p-11, 0x1.5decc25c5c079p-11, -0x1.0d7499d1b0d2dp-11, 0x1.359332c94ecdcp-12 }, { 0x1.fffef1960d85dp-1, 0x1.be6abbb10a4cdp-15, -0x1.6040381a8c313p-13, 0x1.5fff1dde9ee9dp-12, -0x1.f0c933efa9971p-12, 0x1.04cbf4a5cd760p-11, -0x1.a07f150af6dadp-12, 0x1.f68dd183426bap-13 }, { 0x1.ffff4db27f146p-1, 0x1.2bb5cc22e5cd8p-15, -0x1.e25894899f526p-14, 0x1.ec8a8e5a72757p-13, -0x1.64256ae0a3cf9p-12, 0x1.80a836c18c46cp-12, -0x1.3dea401af6775p-12, 0x1.915ddff3fe0d1p-13 }, { 0x1.ffff8b500e77cp-1, 0x1.8f4ccca7fc769p-16, -0x1.478cffe305946p-14, 0x1.559f04adde504p-13, -0x1.f9e1577d6961dp-13, 0x1.18bda53c14716p-12, -0x1.df8634c35541cp-13, 0x1.3bb5c6b616337p-13 }, { 0x1.ffffb43555b5fp-1, 0x1.07ebd2a2d26c8p-16, -0x1.b93e442a37f2bp-15, 0x1.d5cf15159ce28p-14, -0x1.63f5e1469c006p-13, 0x1.95a03acebac18p-13, -0x1.656e5e2a1f8e2p-13, 0x1.e98c437189bdep-14 }, { 0x1.ffffcf23ff5fcp-1, 0x1.5a2adfa0b492cp-17, -0x1.26c88270759f0p-15, 0x1.40473572b99a8p-14, -0x1.f057cbde578a5p-14, 0x1.22178d1c3c948p-13, -0x1.0765b61a0d859p-13, 0x1.765b3ea03ddbep-14 }, { 0x1.ffffe0bd3e852p-1, 0x1.c282cd3957a72p-18, -0x1.86ad6dfa44faap-16, 0x1.b0f313f03a029p-15, -0x1.56e44abecd255p-14, 0x1.9ad1ecfe34a89p-14, -0x1.7fe4033478618p-14, 0x1.1a8184e049fbfp-14 }, { 0x1.ffffec2641a9ep-1, 0x1.22df29821407ep-18, -0x1.00c902a6cfd98p-16, 0x1.22234eb88671fp-15, -0x1.d57a181c9e6e1p-15, 0x1.200c283b54a90p-14, -0x1.14b4c3295a7d0p-14, 0x1.a4f966f713bdep-15 }, { 0x1.fffff37d63a36p-1, 0x1.74adc8f405eecp-19, -0x1.4ed4228e44858p-17, 0x1.81918baea92bap-16, -0x1.3e81b17a0009cp-15, 0x1.9004a36116436p-15, -0x1.8aa1ba400e076p-15, 0x1.35cd4e2340a9ep-15 }, { 0x1.fffff82cdcf1bp-1, 0x1.d9c73698fa87dp-20, -0x1.b11017ec67115p-18, 0x1.fc0dfadf653f8p-17, -0x1.ac4e03cd2dfc2p-16, 0x1.131806b5abbc5p-15, -0x1.1672ef66fcaafp-15, 0x1.c2882c7debed7p-16 }, { 0x1.fffffb248c39dp-1, 0x1.2acee2f5ec66ap-20, -0x1.15cc570408a36p-18, 0x1.4be757bbb75a3p-17, -0x1.1d6aa5f8d2940p-16, 0x1.76c5937d5105ep-16, -0x1.84dffc3ca9302p-16, 0x1.43c8315f2c30ap-16 }, { 0x1.fffffd01f36afp-1, 0x1.75fa8dbc840bap-21, -0x1.6186da0133f5ap-19, 0x1.ae023231e1af5p-18, -0x1.790812f7ca394p-17, 0x1.f9c25656d0ef2p-17, -0x1.0cc66682e304cp-16, 0x1.cc170a75d6f9cp-17 }, { 0x1.fffffe2ba0ea5p-1, 0x1.d06ad6ecde88ep-22, -0x1.be46aa8edc9a1p-20, 0x1.143860c7840b8p-18, -0x1.edaba78fb1260p-18, 0x1.52138a96ecee2p-17, -0x1.6fca538c4e2eep-17, 0x1.434040640bcefp-17 }, { 0x1.fffffee3cc32cp-1, 0x1.1e1e857adb8ddp-22, -0x1.1769ce5f2a6e8p-20, 0x1.5fe5d479b0543p-19, -0x1.405d865c94c2ap-18, 0x1.bfc94feb96afcp-18, -0x1.f245d5f3e8358p-18, 0x1.c142456acf443p-18 }, }; float ax = fabsf (x); uint32_t ux = asuint (ax); double s = x; double z = ax; /* 0x407ad444 corresponds to x = 0x1.f5a888p+1 = 3.91921..., which is the largest float such that erf(x) does not round to 1 (to nearest). */ if (__glibc_unlikely (ux > 0x407ad444u)) { float os = copysignf (1.0f, x); if (ux > (0xffu << 23)) return x + x; /* nan */ if (ux == (0xffu << 23)) return os; /* +-inf */ return os - 0x1p-25f * os; } double v = floor (16.0 * z); uint32_t i = 16.0f * ax; /* 0x3ee00000 corresponds to x = 0.4375, for smaller x we have i < 7. */ if (__glibc_unlikely (ux < 0x3ee00000u)) { static const double c[] = { 0x1.20dd750429b6dp+0, -0x1.812746b0375fbp-2, 0x1.ce2f219fd6f45p-4, -0x1.b82ce2cbf0838p-6, 0x1.565bb655adb85p-8, -0x1.c025bfc879c94p-11, 0x1.f81718f61309cp-14, -0x1.cc67bd88f5867p-17 }; double z2 = s * s, z4 = z2 * z2, z8 = z4 * z4; double c0 = c[0] + z2 * c[1]; double c2 = c[2] + z2 * c[3]; double c4 = c[4] + z2 * c[5]; double c6 = c[6] + z2 * c[7]; c0 += z4 * c2; c4 += z4 * c6; c0 += z8 * c4; return s * c0; } z = (z - 0.03125) - 0.0625 * v; const double *c = C[i - 7]; double z2 = z * z, z4 = z2 * z2; double c0 = c[0] + z * c[1]; double c2 = c[2] + z * c[3]; double c4 = c[4] + z * c[5]; double c6 = c[6] + z * c[7]; c0 += z2 * c2; c4 += z2 * c6; c0 += z4 * c4; return copysign (c0, s); } libm_alias_float (__erf, erf)