Date: Mon, 30 Dec 2013 00:51:25 +0000 (UTC) From: Steve Kargl <kargl@FreeBSD.org> To: src-committers@freebsd.org, svn-src-all@freebsd.org, svn-src-head@freebsd.org Subject: svn commit: r260066 - in head/lib/msun: ld128 ld80 Message-ID: <201312300051.rBU0pPbh053148@svn.freebsd.org>
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Author: kargl Date: Mon Dec 30 00:51:25 2013 New Revision: 260066 URL: http://svnweb.freebsd.org/changeset/base/260066 Log: * ld80/k_expl.h: * ld128/k_expl.h: . Split out a computational kernel,__k_expl(x, &hi, &lo, &k) from expl(x). x must be finite and not tiny or huge. The kernel returns hi and lo values for extra precision and an exponent k for a 2**k scale factor. . Define additional kernels k_hexpl() and hexpl() that include a 1/2 scaling and are used by the hyperbolic functions. * ld80/s_expl.c: * ld128/s_expl.c: . Use the __k_expl() kernel. Obtained from: bde Added: head/lib/msun/ld128/k_expl.h (contents, props changed) head/lib/msun/ld80/k_expl.h (contents, props changed) Modified: head/lib/msun/ld128/s_expl.c head/lib/msun/ld80/s_expl.c Added: head/lib/msun/ld128/k_expl.h ============================================================================== --- /dev/null 00:00:00 1970 (empty, because file is newly added) +++ head/lib/msun/ld128/k_expl.h Mon Dec 30 00:51:25 2013 (r260066) @@ -0,0 +1,328 @@ +/* from: FreeBSD: head/lib/msun/ld128/s_expl.c 251345 2013-06-03 20:09:22Z kargl */ + +/*- + * Copyright (c) 2009-2013 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Optimized by Bruce D. Evans. + */ + +#include <sys/cdefs.h> +__FBSDID("$FreeBSD$"); + +/* + * ld128 version of k_expl.h. See ../ld80/s_expl.c for most comments. + * + * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments + * about the secondary kernels. + */ + +#define INTERVALS 128 +#define LOG2_INTERVALS 7 +#define BIAS (LDBL_MAX_EXP - 1) + +static const double +/* + * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must + * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest + * bits zero so that multiplication of it by n is exact. + */ +INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ +L2 = -1.0253670638894731e-29; /* -0x1.9ff0342542fc3p-97 */ +static const long double +/* 0x1.62e42fefa39ef35793c768000000p-8 */ +L1 = 5.41521234812457272982212595914567508e-3L; + +/* + * XXX values in hex in comments have been lost (or were never present) + * from here. + */ +static const long double +/* + * Domain [-0.002708, 0.002708], range ~[-2.4021e-38, 2.4234e-38]: + * |exp(x) - p(x)| < 2**-124.9 + * (0.002708 is ln2/(2*INTERVALS) rounded up a little). + * + * XXX the coeffs aren't very carefully rounded, and I get 3.6 more bits. + */ +A2 = 0.5, +A3 = 1.66666666666666666666666666651085500e-1L, +A4 = 4.16666666666666666666666666425885320e-2L, +A5 = 8.33333333333333333334522877160175842e-3L, +A6 = 1.38888888888888888889971139751596836e-3L; + +static const double +A7 = 1.9841269841269470e-4, /* 0x1.a01a01a019f91p-13 */ +A8 = 2.4801587301585286e-5, /* 0x1.71de3ec75a967p-19 */ +A9 = 2.7557324277411235e-6, /* 0x1.71de3ec75a967p-19 */ +A10 = 2.7557333722375069e-7; /* 0x1.27e505ab56259p-22 */ + +static const struct { + /* + * hi must be rounded to at most 106 bits so that multiplication + * by r1 in expm1l() is exact, but it is rounded to 88 bits due to + * historical accidents. + * + * XXX it is wasteful to use long double for both hi and lo. ld128 + * exp2l() uses only float for lo (in a very differently organized + * table; ld80 exp2l() is different again. It uses 2 doubles in a + * table organized like this one. 1 double and 1 float would + * suffice). There are different packing/locality/alignment/caching + * problems with these methods. + * + * XXX C's bad %a format makes the bits unreadable. They happen + * to all line up for the hi values 1 before the point and 88 + * in 22 nybbles, but for the low values the nybbles are shifted + * randomly. + */ + long double hi; + long double lo; +} tbl[INTERVALS] = { + 0x1p0L, 0x0p0L, + 0x1.0163da9fb33356d84a66aep0L, 0x3.36dcdfa4003ec04c360be2404078p-92L, + 0x1.02c9a3e778060ee6f7cacap0L, 0x4.f7a29bde93d70a2cabc5cb89ba10p-92L, + 0x1.04315e86e7f84bd738f9a2p0L, 0xd.a47e6ed040bb4bfc05af6455e9b8p-96L, + 0x1.059b0d31585743ae7c548ep0L, 0xb.68ca417fe53e3495f7df4baf84a0p-92L, + 0x1.0706b29ddf6ddc6dc403a8p0L, 0x1.d87b27ed07cb8b092ac75e311753p-88L, + 0x1.0874518759bc808c35f25cp0L, 0x1.9427fa2b041b2d6829d8993a0d01p-88L, + 0x1.09e3ecac6f3834521e060cp0L, 0x5.84d6b74ba2e023da730e7fccb758p-92L, + 0x1.0b5586cf9890f6298b92b6p0L, 0x1.1842a98364291408b3ceb0a2a2bbp-88L, + 0x1.0cc922b7247f7407b705b8p0L, 0x9.3dc5e8aac564e6fe2ef1d431fd98p-92L, + 0x1.0e3ec32d3d1a2020742e4ep0L, 0x1.8af6a552ac4b358b1129e9f966a4p-88L, + 0x1.0fb66affed31af232091dcp0L, 0x1.8a1426514e0b627bda694a400a27p-88L, + 0x1.11301d0125b50a4ebbf1aep0L, 0xd.9318ceac5cc47ab166ee57427178p-92L, + 0x1.12abdc06c31cbfb92bad32p0L, 0x4.d68e2f7270bdf7cedf94eb1cb818p-92L, + 0x1.1429aaea92ddfb34101942p0L, 0x1.b2586d01844b389bea7aedd221d4p-88L, + 0x1.15a98c8a58e512480d573cp0L, 0x1.d5613bf92a2b618ee31b376c2689p-88L, + 0x1.172b83c7d517adcdf7c8c4p0L, 0x1.0eb14a792035509ff7d758693f24p-88L, + 0x1.18af9388c8de9bbbf70b9ap0L, 0x3.c2505c97c0102e5f1211941d2840p-92L, + 0x1.1a35beb6fcb753cb698f68p0L, 0x1.2d1c835a6c30724d5cfae31b84e5p-88L, + 0x1.1bbe084045cd39ab1e72b4p0L, 0x4.27e35f9acb57e473915519a1b448p-92L, + 0x1.1d4873168b9aa7805b8028p0L, 0x9.90f07a98b42206e46166cf051d70p-92L, + 0x1.1ed5022fcd91cb8819ff60p0L, 0x1.121d1e504d36c47474c9b7de6067p-88L, + 0x1.2063b88628cd63b8eeb028p0L, 0x1.50929d0fc487d21c2b84004264dep-88L, + 0x1.21f49917ddc962552fd292p0L, 0x9.4bdb4b61ea62477caa1dce823ba0p-92L, + 0x1.2387a6e75623866c1fadb0p0L, 0x1.c15cb593b0328566902df69e4de2p-88L, + 0x1.251ce4fb2a63f3582ab7dep0L, 0x9.e94811a9c8afdcf796934bc652d0p-92L, + 0x1.26b4565e27cdd257a67328p0L, 0x1.d3b249dce4e9186ddd5ff44e6b08p-92L, + 0x1.284dfe1f5638096cf15cf0p0L, 0x3.ca0967fdaa2e52d7c8106f2e262cp-92L, + 0x1.29e9df51fdee12c25d15f4p0L, 0x1.a24aa3bca890ac08d203fed80a07p-88L, + 0x1.2b87fd0dad98ffddea4652p0L, 0x1.8fcab88442fdc3cb6de4519165edp-88L, + 0x1.2d285a6e4030b40091d536p0L, 0xd.075384589c1cd1b3e4018a6b1348p-92L, + 0x1.2ecafa93e2f5611ca0f45cp0L, 0x1.523833af611bdcda253c554cf278p-88L, + 0x1.306fe0a31b7152de8d5a46p0L, 0x3.05c85edecbc27343629f502f1af2p-92L, + 0x1.32170fc4cd8313539cf1c2p0L, 0x1.008f86dde3220ae17a005b6412bep-88L, + 0x1.33c08b26416ff4c9c8610cp0L, 0x1.96696bf95d1593039539d94d662bp-88L, + 0x1.356c55f929ff0c94623476p0L, 0x3.73af38d6d8d6f9506c9bbc93cbc0p-92L, + 0x1.371a7373aa9caa7145502ep0L, 0x1.4547987e3e12516bf9c699be432fp-88L, + 0x1.38cae6d05d86585a9cb0d8p0L, 0x1.bed0c853bd30a02790931eb2e8f0p-88L, + 0x1.3a7db34e59ff6ea1bc9298p0L, 0x1.e0a1d336163fe2f852ceeb134067p-88L, + 0x1.3c32dc313a8e484001f228p0L, 0xb.58f3775e06ab66353001fae9fca0p-92L, + 0x1.3dea64c12342235b41223ep0L, 0x1.3d773fba2cb82b8244267c54443fp-92L, + 0x1.3fa4504ac801ba0bf701aap0L, 0x4.1832fb8c1c8dbdff2c49909e6c60p-92L, + 0x1.4160a21f72e29f84325b8ep0L, 0x1.3db61fb352f0540e6ba05634413ep-88L, + 0x1.431f5d950a896dc7044394p0L, 0x1.0ccec81e24b0caff7581ef4127f7p-92L, + 0x1.44e086061892d03136f408p0L, 0x1.df019fbd4f3b48709b78591d5cb5p-88L, + 0x1.46a41ed1d005772512f458p0L, 0x1.229d97df404ff21f39c1b594d3a8p-88L, + 0x1.486a2b5c13cd013c1a3b68p0L, 0x1.062f03c3dd75ce8757f780e6ec99p-88L, + 0x1.4a32af0d7d3de672d8bcf4p0L, 0x6.f9586461db1d878b1d148bd3ccb8p-92L, + 0x1.4bfdad5362a271d4397afep0L, 0xc.42e20e0363ba2e159c579f82e4b0p-92L, + 0x1.4dcb299fddd0d63b36ef1ap0L, 0x9.e0cc484b25a5566d0bd5f58ad238p-92L, + 0x1.4f9b2769d2ca6ad33d8b68p0L, 0x1.aa073ee55e028497a329a7333dbap-88L, + 0x1.516daa2cf6641c112f52c8p0L, 0x4.d822190e718226177d7608d20038p-92L, + 0x1.5342b569d4f81df0a83c48p0L, 0x1.d86a63f4e672a3e429805b049465p-88L, + 0x1.551a4ca5d920ec52ec6202p0L, 0x4.34ca672645dc6c124d6619a87574p-92L, + 0x1.56f4736b527da66ecb0046p0L, 0x1.64eb3c00f2f5ab3d801d7cc7272dp-88L, + 0x1.58d12d497c7fd252bc2b72p0L, 0x1.43bcf2ec936a970d9cc266f0072fp-88L, + 0x1.5ab07dd48542958c930150p0L, 0x1.91eb345d88d7c81280e069fbdb63p-88L, + 0x1.5c9268a5946b701c4b1b80p0L, 0x1.6986a203d84e6a4a92f179e71889p-88L, + 0x1.5e76f15ad21486e9be4c20p0L, 0x3.99766a06548a05829e853bdb2b52p-92L, + 0x1.605e1b976dc08b076f592ap0L, 0x4.86e3b34ead1b4769df867b9c89ccp-92L, + 0x1.6247eb03a5584b1f0fa06ep0L, 0x1.d2da42bb1ceaf9f732275b8aef30p-88L, + 0x1.6434634ccc31fc76f8714cp0L, 0x4.ed9a4e41000307103a18cf7a6e08p-92L, + 0x1.66238825522249127d9e28p0L, 0x1.b8f314a337f4dc0a3adf1787ff74p-88L, + 0x1.68155d44ca973081c57226p0L, 0x1.b9f32706bfe4e627d809a85dcc66p-88L, + 0x1.6a09e667f3bcc908b2fb12p0L, 0x1.66ea957d3e3adec17512775099dap-88L, + 0x1.6c012750bdabeed76a9980p0L, 0xf.4f33fdeb8b0ecd831106f57b3d00p-96L, + 0x1.6dfb23c651a2ef220e2cbep0L, 0x1.bbaa834b3f11577ceefbe6c1c411p-92L, + 0x1.6ff7df9519483cf87e1b4ep0L, 0x1.3e213bff9b702d5aa477c12523cep-88L, + 0x1.71f75e8ec5f73dd2370f2ep0L, 0xf.0acd6cb434b562d9e8a20adda648p-92L, + 0x1.73f9a48a58173bd5c9a4e6p0L, 0x8.ab1182ae217f3a7681759553e840p-92L, + 0x1.75feb564267c8bf6e9aa32p0L, 0x1.a48b27071805e61a17b954a2dad8p-88L, + 0x1.780694fde5d3f619ae0280p0L, 0x8.58b2bb2bdcf86cd08e35fb04c0f0p-92L, + 0x1.7a11473eb0186d7d51023ep0L, 0x1.6cda1f5ef42b66977960531e821bp-88L, + 0x1.7c1ed0130c1327c4933444p0L, 0x1.937562b2dc933d44fc828efd4c9cp-88L, + 0x1.7e2f336cf4e62105d02ba0p0L, 0x1.5797e170a1427f8fcdf5f3906108p-88L, + 0x1.80427543e1a11b60de6764p0L, 0x9.a354ea706b8e4d8b718a672bf7c8p-92L, + 0x1.82589994cce128acf88afap0L, 0xb.34a010f6ad65cbbac0f532d39be0p-92L, + 0x1.8471a4623c7acce52f6b96p0L, 0x1.c64095370f51f48817914dd78665p-88L, + 0x1.868d99b4492ec80e41d90ap0L, 0xc.251707484d73f136fb5779656b70p-92L, + 0x1.88ac7d98a669966530bcdep0L, 0x1.2d4e9d61283ef385de170ab20f96p-88L, + 0x1.8ace5422aa0db5ba7c55a0p0L, 0x1.92c9bb3e6ed61f2733304a346d8fp-88L, + 0x1.8cf3216b5448bef2aa1cd0p0L, 0x1.61c55d84a9848f8c453b3ca8c946p-88L, + 0x1.8f1ae991577362b982745cp0L, 0x7.2ed804efc9b4ae1458ae946099d4p-92L, + 0x1.9145b0b91ffc588a61b468p0L, 0x1.f6b70e01c2a90229a4c4309ea719p-88L, + 0x1.93737b0cdc5e4f4501c3f2p0L, 0x5.40a22d2fc4af581b63e8326efe9cp-92L, + 0x1.95a44cbc8520ee9b483694p0L, 0x1.a0fc6f7c7d61b2b3a22a0eab2cadp-88L, + 0x1.97d829fde4e4f8b9e920f8p0L, 0x1.1e8bd7edb9d7144b6f6818084cc7p-88L, + 0x1.9a0f170ca07b9ba3109b8cp0L, 0x4.6737beb19e1eada6825d3c557428p-92L, + 0x1.9c49182a3f0901c7c46b06p0L, 0x1.1f2be58ddade50c217186c90b457p-88L, + 0x1.9e86319e323231824ca78ep0L, 0x6.4c6e010f92c082bbadfaf605cfd4p-92L, + 0x1.a0c667b5de564b29ada8b8p0L, 0xc.ab349aa0422a8da7d4512edac548p-92L, + 0x1.a309bec4a2d3358c171f76p0L, 0x1.0daad547fa22c26d168ea762d854p-88L, + 0x1.a5503b23e255c8b424491cp0L, 0xa.f87bc8050a405381703ef7caff50p-92L, + 0x1.a799e1330b3586f2dfb2b0p0L, 0x1.58f1a98796ce8908ae852236ca94p-88L, + 0x1.a9e6b5579fdbf43eb243bcp0L, 0x1.ff4c4c58b571cf465caf07b4b9f5p-88L, + 0x1.ac36bbfd3f379c0db966a2p0L, 0x1.1265fc73e480712d20f8597a8e7bp-88L, + 0x1.ae89f995ad3ad5e8734d16p0L, 0x1.73205a7fbc3ae675ea440b162d6cp-88L, + 0x1.b0e07298db66590842acdep0L, 0x1.c6f6ca0e5dcae2aafffa7a0554cbp-88L, + 0x1.b33a2b84f15faf6bfd0e7ap0L, 0x1.d947c2575781dbb49b1237c87b6ep-88L, + 0x1.b59728de559398e3881110p0L, 0x1.64873c7171fefc410416be0a6525p-88L, + 0x1.b7f76f2fb5e46eaa7b081ap0L, 0xb.53c5354c8903c356e4b625aacc28p-92L, + 0x1.ba5b030a10649840cb3c6ap0L, 0xf.5b47f297203757e1cc6eadc8bad0p-92L, + 0x1.bcc1e904bc1d2247ba0f44p0L, 0x1.b3d08cd0b20287092bd59be4ad98p-88L, + 0x1.bf2c25bd71e088408d7024p0L, 0x1.18e3449fa073b356766dfb568ff4p-88L, + 0x1.c199bdd85529c2220cb12ap0L, 0x9.1ba6679444964a36661240043970p-96L, + 0x1.c40ab5fffd07a6d14df820p0L, 0xf.1828a5366fd387a7bdd54cdf7300p-92L, + 0x1.c67f12e57d14b4a2137fd2p0L, 0xf.2b301dd9e6b151a6d1f9d5d5f520p-96L, + 0x1.c8f6d9406e7b511acbc488p0L, 0x5.c442ddb55820171f319d9e5076a8p-96L, + 0x1.cb720dcef90691503cbd1ep0L, 0x9.49db761d9559ac0cb6dd3ed599e0p-92L, + 0x1.cdf0b555dc3f9c44f8958ep0L, 0x1.ac51be515f8c58bdfb6f5740a3a4p-88L, + 0x1.d072d4a07897b8d0f22f20p0L, 0x1.a158e18fbbfc625f09f4cca40874p-88L, + 0x1.d2f87080d89f18ade12398p0L, 0x9.ea2025b4c56553f5cdee4c924728p-92L, + 0x1.d5818dcfba48725da05aeap0L, 0x1.66e0dca9f589f559c0876ff23830p-88L, + 0x1.d80e316c98397bb84f9d04p0L, 0x8.805f84bec614de269900ddf98d28p-92L, + 0x1.da9e603db3285708c01a5ap0L, 0x1.6d4c97f6246f0ec614ec95c99392p-88L, + 0x1.dd321f301b4604b695de3cp0L, 0x6.30a393215299e30d4fb73503c348p-96L, + 0x1.dfc97337b9b5eb968cac38p0L, 0x1.ed291b7225a944efd5bb5524b927p-88L, + 0x1.e264614f5a128a12761fa0p0L, 0x1.7ada6467e77f73bf65e04c95e29dp-88L, + 0x1.e502ee78b3ff6273d13014p0L, 0x1.3991e8f49659e1693be17ae1d2f9p-88L, + 0x1.e7a51fbc74c834b548b282p0L, 0x1.23786758a84f4956354634a416cep-88L, + 0x1.ea4afa2a490d9858f73a18p0L, 0xf.5db301f86dea20610ceee13eb7b8p-92L, + 0x1.ecf482d8e67f08db0312fap0L, 0x1.949cef462010bb4bc4ce72a900dfp-88L, + 0x1.efa1bee615a27771fd21a8p0L, 0x1.2dac1f6dd5d229ff68e46f27e3dfp-88L, + 0x1.f252b376bba974e8696fc2p0L, 0x1.6390d4c6ad5476b5162f40e1d9a9p-88L, + 0x1.f50765b6e4540674f84b76p0L, 0x2.862baff99000dfc4352ba29b8908p-92L, + 0x1.f7bfdad9cbe138913b4bfep0L, 0x7.2bd95c5ce7280fa4d2344a3f5618p-92L, + 0x1.fa7c1819e90d82e90a7e74p0L, 0xb.263c1dc060c36f7650b4c0f233a8p-92L, + 0x1.fd3c22b8f71f10975ba4b2p0L, 0x1.2bcf3a5e12d269d8ad7c1a4a8875p-88L +}; + +/* + * Kernel for expl(x). x must be finite and not tiny or huge. + * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). + * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). + */ +static inline void +__k_expl(long double x, long double *hip, long double *lop, int *kp) +{ + long double q, r, r1, t; + double dr, fn, r2; + int n, n2; + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + /* Use a specialized rint() to get fn. Assume round-to-nearest. */ + /* XXX assume no extra precision for the additions, as for trig fns. */ + /* XXX this set of comments is now quadruplicated. */ + /* XXX but see ../src/e_exp.c for a fix using double_t. */ + fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; +#if defined(HAVE_EFFICIENT_IRINT) + n = irint(fn); +#else + n = (int)fn; +#endif + n2 = (unsigned)n % INTERVALS; + /* Depend on the sign bit being propagated: */ + *kp = n >> LOG2_INTERVALS; + r1 = x - fn * L1; + r2 = fn * -L2; + r = r1 + r2; + + /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ + dr = r; + q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + + dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); + t = tbl[n2].lo + tbl[n2].hi; + *hip = tbl[n2].hi; + *lop = tbl[n2].lo + t * (q + r1); +} + +/* + * XXX: the rest of the functions are identical for ld80 and ld128. + * However, we should use scalbnl() for ld128, since long double + * multiplication is very slow on the only supported ld128 arch (sparc64). + */ + +static inline void +k_hexpl(long double x, long double *hip, long double *lop) +{ + float twopkm1; + int k; + + __k_expl(x, hip, lop, &k); + SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23)); + *hip *= twopkm1; + *lop *= twopkm1; +} + +static inline long double +hexpl(long double x) +{ + long double hi, lo, twopkm2; + int k; + + twopkm2 = 1; + __k_expl(x, &hi, &lo, &k); + SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2); + return (lo + hi) * 2 * twopkm2; +} + +#ifdef _COMPLEX_H +/* + * See ../src/k_exp.c for details. + */ +static inline long double complex +__ldexp_cexpl(long double complex z, int expt) +{ + long double exp_x, hi, lo; + long double x, y, scale1, scale2; + int half_expt, k; + + x = creall(z); + y = cimagl(z); + __k_expl(x, &hi, &lo, &k); + + exp_x = (lo + hi) * 0x1p16382; + expt += k - 16382; + + scale1 = 1; + half_expt = expt / 2; + SET_LDBL_EXPSIGN(scale1, BIAS + half_expt); + scale2 = 1; + SET_LDBL_EXPSIGN(scale1, BIAS + expt - half_expt); + + return (cpackl(cos(y) * exp_x * scale1 * scale2, + sinl(y) * exp_x * scale1 * scale2)); +} +#endif /* _COMPLEX_H */ Modified: head/lib/msun/ld128/s_expl.c ============================================================================== --- head/lib/msun/ld128/s_expl.c Mon Dec 30 00:43:24 2013 (r260065) +++ head/lib/msun/ld128/s_expl.c Mon Dec 30 00:51:25 2013 (r260066) @@ -38,16 +38,15 @@ __FBSDID("$FreeBSD$"); #include "fpmath.h" #include "math.h" #include "math_private.h" +#include "k_expl.h" -#define INTERVALS 128 -#define LOG2_INTERVALS 7 -#define BIAS (LDBL_MAX_EXP - 1) +/* XXX Prevent compilers from erroneously constant folding these: */ +static const volatile long double +huge = 0x1p10000L, +tiny = 0x1p-10000L; static const long double -huge = 0x1p10000L, twom10000 = 0x1p-10000L; -/* XXX Prevent gcc from erroneously constant folding this: */ -static volatile const long double tiny = 0x1p-10000L; static const long double /* log(2**16384 - 0.5) rounded towards zero: */ @@ -56,184 +55,16 @@ o_threshold = 11356.5234062941439494919 /* log(2**(-16381-64-1)) rounded towards zero: */ u_threshold = -11433.462743336297878837243843452621503L; -static const double -/* - * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must - * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest - * bits zero so that multiplication of it by n is exact. - */ -INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ -L2 = -1.0253670638894731e-29; /* -0x1.9ff0342542fc3p-97 */ -static const long double -/* 0x1.62e42fefa39ef35793c768000000p-8 */ -L1 = 5.41521234812457272982212595914567508e-3L; - -static const long double -/* - * Domain [-0.002708, 0.002708], range ~[-2.4021e-38, 2.4234e-38]: - * |exp(x) - p(x)| < 2**-124.9 - * (0.002708 is ln2/(2*INTERVALS) rounded up a little). - */ -A2 = 0.5, -A3 = 1.66666666666666666666666666651085500e-1L, -A4 = 4.16666666666666666666666666425885320e-2L, -A5 = 8.33333333333333333334522877160175842e-3L, -A6 = 1.38888888888888888889971139751596836e-3L; - -static const double -A7 = 1.9841269841269471e-4, -A8 = 2.4801587301585284e-5, -A9 = 2.7557324277411234e-6, -A10 = 2.7557333722375072e-7; - -static const struct { - /* - * hi must be rounded to at most 106 bits so that multiplication - * by r1 in expm1l() is exact, but it is rounded to 88 bits due to - * historical accidents. - */ - long double hi; - long double lo; -} tbl[INTERVALS] = { - 0x1p0L, 0x0p0L, - 0x1.0163da9fb33356d84a66aep0L, 0x3.36dcdfa4003ec04c360be2404078p-92L, - 0x1.02c9a3e778060ee6f7cacap0L, 0x4.f7a29bde93d70a2cabc5cb89ba10p-92L, - 0x1.04315e86e7f84bd738f9a2p0L, 0xd.a47e6ed040bb4bfc05af6455e9b8p-96L, - 0x1.059b0d31585743ae7c548ep0L, 0xb.68ca417fe53e3495f7df4baf84a0p-92L, - 0x1.0706b29ddf6ddc6dc403a8p0L, 0x1.d87b27ed07cb8b092ac75e311753p-88L, - 0x1.0874518759bc808c35f25cp0L, 0x1.9427fa2b041b2d6829d8993a0d01p-88L, - 0x1.09e3ecac6f3834521e060cp0L, 0x5.84d6b74ba2e023da730e7fccb758p-92L, - 0x1.0b5586cf9890f6298b92b6p0L, 0x1.1842a98364291408b3ceb0a2a2bbp-88L, - 0x1.0cc922b7247f7407b705b8p0L, 0x9.3dc5e8aac564e6fe2ef1d431fd98p-92L, - 0x1.0e3ec32d3d1a2020742e4ep0L, 0x1.8af6a552ac4b358b1129e9f966a4p-88L, - 0x1.0fb66affed31af232091dcp0L, 0x1.8a1426514e0b627bda694a400a27p-88L, - 0x1.11301d0125b50a4ebbf1aep0L, 0xd.9318ceac5cc47ab166ee57427178p-92L, - 0x1.12abdc06c31cbfb92bad32p0L, 0x4.d68e2f7270bdf7cedf94eb1cb818p-92L, - 0x1.1429aaea92ddfb34101942p0L, 0x1.b2586d01844b389bea7aedd221d4p-88L, - 0x1.15a98c8a58e512480d573cp0L, 0x1.d5613bf92a2b618ee31b376c2689p-88L, - 0x1.172b83c7d517adcdf7c8c4p0L, 0x1.0eb14a792035509ff7d758693f24p-88L, - 0x1.18af9388c8de9bbbf70b9ap0L, 0x3.c2505c97c0102e5f1211941d2840p-92L, - 0x1.1a35beb6fcb753cb698f68p0L, 0x1.2d1c835a6c30724d5cfae31b84e5p-88L, - 0x1.1bbe084045cd39ab1e72b4p0L, 0x4.27e35f9acb57e473915519a1b448p-92L, - 0x1.1d4873168b9aa7805b8028p0L, 0x9.90f07a98b42206e46166cf051d70p-92L, - 0x1.1ed5022fcd91cb8819ff60p0L, 0x1.121d1e504d36c47474c9b7de6067p-88L, - 0x1.2063b88628cd63b8eeb028p0L, 0x1.50929d0fc487d21c2b84004264dep-88L, - 0x1.21f49917ddc962552fd292p0L, 0x9.4bdb4b61ea62477caa1dce823ba0p-92L, - 0x1.2387a6e75623866c1fadb0p0L, 0x1.c15cb593b0328566902df69e4de2p-88L, - 0x1.251ce4fb2a63f3582ab7dep0L, 0x9.e94811a9c8afdcf796934bc652d0p-92L, - 0x1.26b4565e27cdd257a67328p0L, 0x1.d3b249dce4e9186ddd5ff44e6b08p-92L, - 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0x1.9145b0b91ffc588a61b468p0L, 0x1.f6b70e01c2a90229a4c4309ea719p-88L, - 0x1.93737b0cdc5e4f4501c3f2p0L, 0x5.40a22d2fc4af581b63e8326efe9cp-92L, - 0x1.95a44cbc8520ee9b483694p0L, 0x1.a0fc6f7c7d61b2b3a22a0eab2cadp-88L, - 0x1.97d829fde4e4f8b9e920f8p0L, 0x1.1e8bd7edb9d7144b6f6818084cc7p-88L, - 0x1.9a0f170ca07b9ba3109b8cp0L, 0x4.6737beb19e1eada6825d3c557428p-92L, - 0x1.9c49182a3f0901c7c46b06p0L, 0x1.1f2be58ddade50c217186c90b457p-88L, - 0x1.9e86319e323231824ca78ep0L, 0x6.4c6e010f92c082bbadfaf605cfd4p-92L, - 0x1.a0c667b5de564b29ada8b8p0L, 0xc.ab349aa0422a8da7d4512edac548p-92L, - 0x1.a309bec4a2d3358c171f76p0L, 0x1.0daad547fa22c26d168ea762d854p-88L, - 0x1.a5503b23e255c8b424491cp0L, 0xa.f87bc8050a405381703ef7caff50p-92L, - 0x1.a799e1330b3586f2dfb2b0p0L, 0x1.58f1a98796ce8908ae852236ca94p-88L, - 0x1.a9e6b5579fdbf43eb243bcp0L, 0x1.ff4c4c58b571cf465caf07b4b9f5p-88L, - 0x1.ac36bbfd3f379c0db966a2p0L, 0x1.1265fc73e480712d20f8597a8e7bp-88L, - 0x1.ae89f995ad3ad5e8734d16p0L, 0x1.73205a7fbc3ae675ea440b162d6cp-88L, - 0x1.b0e07298db66590842acdep0L, 0x1.c6f6ca0e5dcae2aafffa7a0554cbp-88L, - 0x1.b33a2b84f15faf6bfd0e7ap0L, 0x1.d947c2575781dbb49b1237c87b6ep-88L, - 0x1.b59728de559398e3881110p0L, 0x1.64873c7171fefc410416be0a6525p-88L, - 0x1.b7f76f2fb5e46eaa7b081ap0L, 0xb.53c5354c8903c356e4b625aacc28p-92L, - 0x1.ba5b030a10649840cb3c6ap0L, 0xf.5b47f297203757e1cc6eadc8bad0p-92L, - 0x1.bcc1e904bc1d2247ba0f44p0L, 0x1.b3d08cd0b20287092bd59be4ad98p-88L, - 0x1.bf2c25bd71e088408d7024p0L, 0x1.18e3449fa073b356766dfb568ff4p-88L, - 0x1.c199bdd85529c2220cb12ap0L, 0x9.1ba6679444964a36661240043970p-96L, - 0x1.c40ab5fffd07a6d14df820p0L, 0xf.1828a5366fd387a7bdd54cdf7300p-92L, - 0x1.c67f12e57d14b4a2137fd2p0L, 0xf.2b301dd9e6b151a6d1f9d5d5f520p-96L, - 0x1.c8f6d9406e7b511acbc488p0L, 0x5.c442ddb55820171f319d9e5076a8p-96L, - 0x1.cb720dcef90691503cbd1ep0L, 0x9.49db761d9559ac0cb6dd3ed599e0p-92L, - 0x1.cdf0b555dc3f9c44f8958ep0L, 0x1.ac51be515f8c58bdfb6f5740a3a4p-88L, - 0x1.d072d4a07897b8d0f22f20p0L, 0x1.a158e18fbbfc625f09f4cca40874p-88L, - 0x1.d2f87080d89f18ade12398p0L, 0x9.ea2025b4c56553f5cdee4c924728p-92L, - 0x1.d5818dcfba48725da05aeap0L, 0x1.66e0dca9f589f559c0876ff23830p-88L, - 0x1.d80e316c98397bb84f9d04p0L, 0x8.805f84bec614de269900ddf98d28p-92L, - 0x1.da9e603db3285708c01a5ap0L, 0x1.6d4c97f6246f0ec614ec95c99392p-88L, - 0x1.dd321f301b4604b695de3cp0L, 0x6.30a393215299e30d4fb73503c348p-96L, - 0x1.dfc97337b9b5eb968cac38p0L, 0x1.ed291b7225a944efd5bb5524b927p-88L, - 0x1.e264614f5a128a12761fa0p0L, 0x1.7ada6467e77f73bf65e04c95e29dp-88L, - 0x1.e502ee78b3ff6273d13014p0L, 0x1.3991e8f49659e1693be17ae1d2f9p-88L, - 0x1.e7a51fbc74c834b548b282p0L, 0x1.23786758a84f4956354634a416cep-88L, - 0x1.ea4afa2a490d9858f73a18p0L, 0xf.5db301f86dea20610ceee13eb7b8p-92L, - 0x1.ecf482d8e67f08db0312fap0L, 0x1.949cef462010bb4bc4ce72a900dfp-88L, - 0x1.efa1bee615a27771fd21a8p0L, 0x1.2dac1f6dd5d229ff68e46f27e3dfp-88L, - 0x1.f252b376bba974e8696fc2p0L, 0x1.6390d4c6ad5476b5162f40e1d9a9p-88L, - 0x1.f50765b6e4540674f84b76p0L, 0x2.862baff99000dfc4352ba29b8908p-92L, - 0x1.f7bfdad9cbe138913b4bfep0L, 0x7.2bd95c5ce7280fa4d2344a3f5618p-92L, - 0x1.fa7c1819e90d82e90a7e74p0L, 0xb.263c1dc060c36f7650b4c0f233a8p-92L, - 0x1.fd3c22b8f71f10975ba4b2p0L, 0x1.2bcf3a5e12d269d8ad7c1a4a8875p-88L -}; - long double expl(long double x) { - union IEEEl2bits u, v; - long double q, r, r1, t, twopk, twopkp10000; - double dr, fn, r2; - int k, n, n2; + union IEEEl2bits u; + long double hi, lo, t, twopk; + int k; uint16_t hx, ix; + DOPRINT_START(&x); + /* Filter out exceptional cases. */ u.e = x; hx = u.xbits.expsign; @@ -241,60 +72,33 @@ expl(long double x) if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ if (ix == BIAS + LDBL_MAX_EXP) { if (hx & 0x8000) /* x is -Inf or -NaN */ - return (-1 / x); - return (x + x); /* x is +Inf or +NaN */ + RETURNP(-1 / x); + RETURNP(x + x); /* x is +Inf or +NaN */ } if (x > o_threshold) - return (huge * huge); + RETURNP(huge * huge); if (x < u_threshold) - return (tiny * tiny); + RETURNP(tiny * tiny); } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ - return (1 + x); /* 1 with inexact iff x != 0 */ + RETURN2P(1, x); /* 1 with inexact iff x != 0 */ } ENTERI(); - /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ - /* Use a specialized rint() to get fn. Assume round-to-nearest. */ - /* XXX assume no extra precision for the additions, as for trig fns. */ - /* XXX this set of comments is now quadruplicated. */ - fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; -#if defined(HAVE_EFFICIENT_IRINT) - n = irint(fn); -#else - n = (int)fn; -#endif - n2 = (unsigned)n % INTERVALS; - k = n >> LOG2_INTERVALS; - r1 = x - fn * L1; - r2 = fn * -L2; - r = r1 + r2; - - /* Prepare scale factors. */ - /* XXX sparc64 multiplication is so slow that scalbnl() is faster. */ - v.e = 1; - if (k >= LDBL_MIN_EXP) { - v.xbits.expsign = BIAS + k; - twopk = v.e; - } else { - v.xbits.expsign = BIAS + k + 10000; - twopkp10000 = v.e; - } - - /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ - dr = r; - q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + - dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); - t = tbl[n2].lo + tbl[n2].hi; - t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + twopk = 1; + __k_expl(x, &hi, &lo, &k); + t = SUM2P(hi, lo); /* Scale by 2**k. */ + /* XXX sparc64 multiplication is so slow that scalbnl() is faster. */ if (k >= LDBL_MIN_EXP) { if (k == LDBL_MAX_EXP) RETURNI(t * 2 * 0x1p16383L); + SET_LDBL_EXPSIGN(twopk, BIAS + k); RETURNI(t * twopk); } else { - RETURNI(t * twopkp10000 * twom10000); + SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); + RETURNI(t * twopk * twom10000); } } @@ -312,6 +116,12 @@ expl(long double x) * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear * in both subintervals, so set T3 = 2**-5, which places the condition * into the [T1, T3] interval. + * + * XXX we now do this more to (partially) balance the number of terms + * in the C and D polys than to avoid checking the condition in both + * intervals. + * + * XXX these micro-optimizations are excessive. */ static const double T1 = -0.1659, /* ~-30.625/128 * log(2) */ @@ -321,6 +131,12 @@ T3 = 0.03125; /* * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]: * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03 +/* + * XXX none of the long double C or D coeffs except C10 is correctly printed. + * If you re-print their values in %.35Le format, the result is always + * different. For example, the last 2 digits in C3 should be 59, not 67. + * 67 is apparently from rounding an extra-precision value to 36 decimal + * places. */ static const long double C3 = 1.66666666666666666666666666666666667e-1L, @@ -335,6 +151,13 @@ C11 = 2.5052108385441720361903196015125 C12 = 2.08767569878679576457272282566520649e-9L, C13 = 1.60590438367252471783548748824255707e-10L; +/* + * XXX this has 1 more coeff than needed. + * XXX can start the double coeffs but not the double mults at C10. + * With my coeffs (C10-C17 double; s = best_s): + * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 + */ static const double C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */ C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */ @@ -359,6 +182,13 @@ D11 = 2.5052108385508457004648045093526 D12 = 2.08767569819738524488686318024854942e-9L, D13 = 1.60590442297008495301927448122499313e-10L; +/* + * XXX this has 1 more coeff than needed. + * XXX can start the double coeffs but not the double mults at D11. + * With my coeffs (D11-D16 double): + * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 + */ static const double D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */ D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */ @@ -375,6 +205,8 @@ expm1l(long double x) int k, n, n2; uint16_t hx, ix; + DOPRINT_START(&x); + /* Filter out exceptional cases. */ u.e = x; hx = u.xbits.expsign; @@ -382,11 +214,11 @@ expm1l(long double x) if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ if (ix == BIAS + LDBL_MAX_EXP) { if (hx & 0x8000) /* x is -Inf or -NaN */ - return (-1 / x - 1); - return (x + x); /* x is +Inf or +NaN */ + RETURNP(-1 / x - 1); + RETURNP(x + x); /* x is +Inf or +NaN */ } if (x > o_threshold) - return (huge * huge); + RETURNP(huge * huge); /* * expm1l() never underflows, but it must avoid * unrepresentable large negative exponents. We used a @@ -395,7 +227,7 @@ expm1l(long double x) * in the same way as large ones here. */ if (hx & 0x8000) /* x <= -128 */ - return (tiny - 1); /* good for x < -114ln2 - eps */ + RETURN2P(tiny, -1); /* good for x < -114ln2 - eps */ } ENTERI(); @@ -407,7 +239,7 @@ expm1l(long double x) if (x < T3) { if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ /* x (rounded) with inexact if x != 0: */ - RETURNI(x == 0 ? x : + RETURNPI(x == 0 ? x : (0x1p200 * x + fabsl(x)) * 0x1p-200); } q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + @@ -428,9 +260,9 @@ expm1l(long double x) hx2_hi = x_hi * x_hi / 2; hx2_lo = x_lo * (x + x_hi) / 2; if (ix >= BIAS - 7) - RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi)); + RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); else - RETURNI(hx2_lo + q + hx2_hi + x); + RETURN2PI(x, hx2_lo + q + hx2_hi); } /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ @@ -463,21 +295,21 @@ expm1l(long double x) t = tbl[n2].lo + tbl[n2].hi; if (k == 0) { - t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + - (tbl[n2].hi - 1); + t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); RETURNI(t); } if (k == -1) { - t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + - (tbl[n2].hi - 2); + t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); RETURNI(t / 2); } if (k < -7) { - t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk - 1); } if (k > 2 * LDBL_MANT_DIG - 1) { - t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); if (k == LDBL_MAX_EXP) RETURNI(t * 2 * 0x1p16383L - 1); RETURNI(t * twopk - 1); @@ -487,8 +319,8 @@ expm1l(long double x) twomk = v.e; if (k > LDBL_MANT_DIG - 1) - t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi; + t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); else - t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk); + t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk); } Added: head/lib/msun/ld80/k_expl.h ============================================================================== --- /dev/null 00:00:00 1970 (empty, because file is newly added) +++ head/lib/msun/ld80/k_expl.h Mon Dec 30 00:51:25 2013 (r260066) @@ -0,0 +1,305 @@ +/* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */ + +/*- + * Copyright (c) 2009-2013 Steven G. Kargl + * All rights reserved. + * + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions + * are met: + * 1. Redistributions of source code must retain the above copyright + * notice unmodified, this list of conditions, and the following + * disclaimer. + * 2. Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * + * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR + * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES + * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. + * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, + * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT + * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF + * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * Optimized by Bruce D. Evans. + */ + +#include <sys/cdefs.h> +__FBSDID("$FreeBSD$"); + +/* + * See s_expl.c for more comments about __k_expl(). + * + * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments + * about the secondary kernels. + */ + +#define INTERVALS 128 +#define LOG2_INTERVALS 7 +#define BIAS (LDBL_MAX_EXP - 1) + +static const double +/* + * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must + * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest + * bits zero so that multiplication of it by n is exact. + */ +INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ +L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ +L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ +/* + * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: + * |exp(x) - p(x)| < 2**-77.2 + * (0.002708 is ln2/(2*INTERVALS) rounded up a little). + */ +A2 = 0.5, +A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ +A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ +A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ +A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ + +/* + * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where + * the first 53 bits of the significand are stored in hi and the next 53 + * bits are in lo. Tang's paper states that the trailing 6 bits of hi must + * be zero for his algorithm in both single and double precision, because + * the table is re-used in the implementation of expm1() where a floating + * point addition involving hi must be exact. Here hi is double, so + * converting it to long double gives 11 trailing zero bits. + */ +static const struct { + double hi; + double lo; +} tbl[INTERVALS] = { + 0x1p+0, 0x0p+0, + /* + * XXX hi is rounded down, and the formatting is not quite normal. + * But I rather like both. The 0x1.*p format is good for 4N+1 + * mantissa bits. Rounding down makes the lo terms positive, + * so that the columnar formatting can be simpler. + */ + 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54, + 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53, + 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53, + 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55, + 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53, + 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57, + 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54, + 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54, + 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54, + 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59, + 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53, + 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53, + 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53, + 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53, + 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55, + 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53, + 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53, + 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55, + 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53, + 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54, + 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53, + 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55, + 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55, + 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54, + 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55, + 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55, + 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53, + 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55, + 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53, + 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54, + 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56, + 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55, + 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55, + 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54, + 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53, + 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53, + 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53, + 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53, + 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53, + 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55, + 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0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53, + 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53, + 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53, + 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53, + 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53, + 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56, + 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56, + 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53, + 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54, + 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53, + 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54, + 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54, + 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53, + 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54, + 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53, + 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53, + 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53, + 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53, + 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53, + 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55, + 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53, + 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55, + 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54, + 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54, + 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56, + 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56, + 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53, + 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53, + 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53, + 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55, + 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53, + 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54, + 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54, + 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53, + 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53, + 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55, + 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54, + 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53, + 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53, + 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54, + 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54, + 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54, + 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53, + 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55, + 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 +}; + +/* + * Kernel for expl(x). x must be finite and not tiny or huge. + * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). + * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). + */ +static inline void +__k_expl(long double x, long double *hip, long double *lop, int *kp) +{ + long double fn, q, r, r1, r2, t, z; + int n, n2; + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + /* Use a specialized rint() to get fn. Assume round-to-nearest. */ + fn = x * INV_L + 0x1.8p63 - 0x1.8p63; + r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ +#if defined(HAVE_EFFICIENT_IRINTL) + n = irintl(fn); +#elif defined(HAVE_EFFICIENT_IRINT) + n = irint(fn); *** DIFF OUTPUT TRUNCATED AT 1000 LINES ***
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