diff options
Diffstat (limited to 'lib/msun')
29 files changed, 2140 insertions, 723 deletions
diff --git a/lib/msun/Makefile b/lib/msun/Makefile index 4bca55b..654d0f5 100644 --- a/lib/msun/Makefile +++ b/lib/msun/Makefile @@ -72,7 +72,7 @@ COMMON_SRCS= b_exp.c b_log.c b_tgamma.c \ s_lround.c s_lroundf.c s_lroundl.c s_modff.c \ s_nan.c s_nearbyint.c s_nextafter.c s_nextafterf.c \ s_nexttowardf.c s_remquo.c s_remquof.c \ - s_rint.c s_rintf.c s_round.c s_roundf.c s_roundl.c \ + s_rint.c s_rintf.c s_round.c s_roundf.c \ s_scalbln.c s_scalbn.c s_scalbnf.c s_signbit.c \ s_signgam.c s_significand.c s_significandf.c s_sin.c s_sinf.c \ s_tan.c s_tanf.c s_tanh.c s_tanhf.c s_tgammaf.c s_trunc.c s_truncf.c \ @@ -97,13 +97,14 @@ COMMON_SRCS+= s_copysignl.c s_fabsl.c s_llrintl.c s_lrintl.c s_modfl.c .if ${LDBL_PREC} != 53 # If long double != double use these; otherwise, we alias the double versions. COMMON_SRCS+= e_acoshl.c e_acosl.c e_asinl.c e_atan2l.c e_atanhl.c \ - e_fmodl.c e_hypotl.c e_remainderl.c e_sqrtl.c \ + e_coshl.c e_fmodl.c e_hypotl.c \ + e_remainderl.c e_sinhl.c e_sqrtl.c \ invtrig.c k_cosl.c k_sinl.c k_tanl.c \ s_asinhl.c s_atanl.c s_cbrtl.c s_ceill.c s_cosl.c s_cprojl.c \ - s_csqrtl.c s_exp2l.c s_expl.c s_floorl.c s_fmal.c \ + s_csqrtl.c s_erfl.c s_exp2l.c s_expl.c s_floorl.c s_fmal.c \ s_frexpl.c s_logbl.c s_logl.c s_nanl.c s_nextafterl.c \ - s_nexttoward.c s_remquol.c s_rintl.c s_scalbnl.c \ - s_sinl.c s_tanl.c s_truncl.c w_cabsl.c + s_nexttoward.c s_remquol.c s_rintl.c s_roundl.c s_scalbnl.c \ + s_sinl.c s_tanhl.c s_tanl.c s_truncl.c w_cabsl.c .endif # C99 complex functions @@ -161,9 +162,9 @@ MLINKS+=cimag.3 cimagf.3 cimag.3 cimagl.3 \ cimag.3 creal.3 cimag.3 crealf.3 cimag.3 creall.3 MLINKS+=copysign.3 copysignf.3 copysign.3 copysignl.3 MLINKS+=cos.3 cosf.3 cos.3 cosl.3 -MLINKS+=cosh.3 coshf.3 +MLINKS+=cosh.3 coshf.3 cosh.3 coshl.3 MLINKS+=csqrt.3 csqrtf.3 csqrt.3 csqrtl.3 -MLINKS+=erf.3 erfc.3 erf.3 erff.3 erf.3 erfcf.3 +MLINKS+=erf.3 erfc.3 erf.3 erff.3 erf.3 erfcf.3 erf.3 erfl.3 erf.3 erfcl.3 MLINKS+=exp.3 expm1.3 exp.3 expm1f.3 exp.3 expm1l.3 exp.3 pow.3 exp.3 powf.3 \ exp.3 exp2.3 exp.3 exp2f.3 exp.3 exp2l.3 exp.3 expf.3 exp.3 expl.3 MLINKS+=fabs.3 fabsf.3 fabs.3 fabsl.3 @@ -209,11 +210,11 @@ MLINKS+=round.3 roundf.3 round.3 roundl.3 MLINKS+=scalbn.3 scalbln.3 scalbn.3 scalblnf.3 scalbn.3 scalblnl.3 MLINKS+=scalbn.3 scalbnf.3 scalbn.3 scalbnl.3 MLINKS+=sin.3 sinf.3 sin.3 sinl.3 -MLINKS+=sinh.3 sinhf.3 +MLINKS+=sinh.3 sinhf.3 sinh.3 sinhl.3 MLINKS+=sqrt.3 cbrt.3 sqrt.3 cbrtf.3 sqrt.3 cbrtl.3 sqrt.3 sqrtf.3 \ sqrt.3 sqrtl.3 MLINKS+=tan.3 tanf.3 tan.3 tanl.3 -MLINKS+=tanh.3 tanhf.3 +MLINKS+=tanh.3 tanhf.3 tanh.3 tanhl.3 MLINKS+=trunc.3 truncf.3 trunc.3 truncl.3 .include <bsd.lib.mk> diff --git a/lib/msun/Symbol.map b/lib/msun/Symbol.map index 037659d..e53ca07 100644 --- a/lib/msun/Symbol.map +++ b/lib/msun/Symbol.map @@ -260,23 +260,23 @@ FBSD_1.3 { ccosf; ccosh; ccoshf; + coshl; ctan; ctanf; ctanh; ctanhf; + erfcl; + erfl; expl; expm1l; log10l; log1pl; log2l; logl; + sinhl; + tanhl; /* Implemented as weak aliases for imprecise versions */ - coshl; - erfcl; - erfl; lgammal; powl; - sinhl; - tanhl; tgammal; }; diff --git a/lib/msun/ld128/k_expl.h b/lib/msun/ld128/k_expl.h new file mode 100644 index 0000000..a5668fd --- /dev/null +++ b/lib/msun/ld128/k_expl.h @@ -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 */ diff --git a/lib/msun/ld128/s_erfl.c b/lib/msun/ld128/s_erfl.c new file mode 100644 index 0000000..e29c969 --- /dev/null +++ b/lib/msun/ld128/s_erfl.c @@ -0,0 +1,329 @@ +/* @(#)s_erf.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include <sys/cdefs.h> +__FBSDID("$FreeBSD$"); + +/* + * See s_erf.c for complete comments. + * + * Converted to long double by Steven G. Kargl. + */ +#include <float.h> + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* XXX Prevent compilers from erroneously constant folding these: */ +static const volatile long double tiny = 0x1p-10000L; + +static const double +half= 0.5, +one = 1, +two = 2; +/* + * In the domain [0, 2**-40], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-120. + */ +static const long double +efx = 1.28379167095512573896158903121545167e-01L, /* 0xecbff6a7, 0x481dd788, 0xb64d21a8, 0xeb06fc3f */ +efx8 = 1.02703333676410059116927122497236133e+00L, /* 0xecbff6a7, 0x481dd788, 0xb64d21a8, 0xeb06ff3f */ +/* + * Domain [0, 0.84375], range ~[-1.919e-38, 1.919e-38]: + * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-125.29 + */ +pp0 = 1.28379167095512573896158903121545167e-01L, /* 0x3ffc06eb, 0xa8214db6, 0x88d71d48, 0xa7f6bfec */ +pp1 = -3.14931554396568573802046931159683404e-01L, /* 0xbffd427d, 0x6ada7263, 0x547eb096, 0x95f37463 */ +pp2 = -5.27514920282183487103576956956725309e-02L, /* 0xbffab023, 0xe5a271e3, 0xb0e79b01, 0x2f7ac962 */ +pp3 = -1.13202828509005281355609495523452713e-02L, /* 0xbff872f1, 0x6a5023a1, 0xe08b3884, 0x326af20f */ +pp4 = -9.18626155872522453865998391206048506e-04L, /* 0xbff4e19f, 0xea5fb024, 0x43247a37, 0xe430b06c */ +pp5 = -7.87518862406176274922506447157284230e-05L, /* 0xbff14a4f, 0x31a85fe0, 0x7fff2204, 0x09c49b37 */ +pp6 = -3.42357944472240436548115331090560881e-06L, /* 0xbfeccb81, 0x4b43c336, 0xcd2eb6c2, 0x903f2d87 */ +pp7 = -1.37317432573890412634717890726745428e-07L, /* 0xbfe826e3, 0x0e915eb6, 0x42aee414, 0xf7e36805 */ +pp8 = -2.71115170113861755855049008732113726e-09L, /* 0xbfe2749e, 0x2b94fd00, 0xecb4d166, 0x0efb91f8 */ +pp9 = -3.37925756196555959454018189718117864e-11L, /* 0xbfdc293e, 0x1d9060cb, 0xd043204a, 0x314cd7f0 */ +qq1 = 4.76672625471551170489978555182449450e-01L, /* 0x3ffde81c, 0xde6531f0, 0x76803bee, 0x526e29e9 */ +qq2 = 1.06713144672281502058807525850732240e-01L, /* 0x3ffbb518, 0xd7a6bb74, 0xcd9bdd33, 0x7601eee5 */ +qq3 = 1.47747613127513761102189201923147490e-02L, /* 0x3ff8e423, 0xae527e18, 0xf12cb447, 0x723b4749 */ +qq4 = 1.39939377672028671891148770908874816e-03L, /* 0x3ff56ed7, 0xba055d84, 0xc21b45c4, 0x388d1812 */ +qq5 = 9.44302939359455241271983309378738276e-05L, /* 0x3ff18c11, 0xc18c99a4, 0x86d0fe09, 0x46387b4c */ +qq6 = 4.56199342312522842161301671745365650e-06L, /* 0x3fed3226, 0x73421d05, 0x08875300, 0x32fa1432 */ +qq7 = 1.53019260483764773845294600092361197e-07L, /* 0x3fe8489b, 0x3a63f627, 0x2b9ad2ce, 0x26516e57 */ +qq8 = 3.25542691121324805094777901250005508e-09L, /* 0x3fe2bf6c, 0x26d93a29, 0x9142be7c, 0x9f1dd043 */ +qq9 = 3.37405581964478060434410167262684979e-11L; /* 0x3fdc28c8, 0xfb8fa1be, 0x10e57eec, 0xaa19e49f */ + +static const long double +erx = 8.42700792949714894142232424201210961e-01L, /* 0x3ffeaf76, 0x7a741088, 0xb0000000, 0x00000000 */ +/* + * Domain [0.84375, 1.25], range ~[-2.521e-36, 2.523e-36]: + * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-120.15 + */ +pa0 = -2.48010117891186017024438233323795897e-17L, /* 0xbfc7c97f, 0x77812279, 0x6c877f22, 0xef4bfb2e */ +pa1 = 4.15107497420594680894327969504526489e-01L, /* 0x3ffda911, 0xf096fbc2, 0x55662005, 0x2337fa64 */ +pa2 = -3.94180628087084846724448515851892609e-02L, /* 0xbffa42e9, 0xab54528c, 0xad529da1, 0x6efc2af3 */ +pa3 = 4.48897599625192107295954790681677462e-02L, /* 0x3ffa6fbc, 0xa65edba1, 0x0e4cbcea, 0x73ef9a31 */ +pa4 = 8.02069252143016600110972019232995528e-02L, /* 0x3ffb4887, 0x0e8b548e, 0x3230b417, 0x11b553b3 */ +pa5 = -1.02729816533435279443621120242391295e-02L, /* 0xbff850a0, 0x041de3ee, 0xd5bca6c9, 0x4ef5f9f2 */ +pa6 = 5.70777694530755634864821094419982095e-03L, /* 0x3ff77610, 0x9b501e10, 0x4c978382, 0x742df68f */ +pa7 = 1.22635150233075521018231779267077071e-03L, /* 0x3ff5417b, 0x0e623682, 0x60327da0, 0x96b9219e */ +pa8 = 5.36100234820204569428412542856666503e-04L, /* 0x3ff41912, 0x27ceb4c1, 0x1d3298ec, 0x84ced627 */ +pa9 = -1.97753571846365167177187858667583165e-04L, /* 0xbff29eb8, 0x23f5bcf3, 0x15c83c46, 0xe4fda98b */ +pa10 = 6.19333039900846970674794789568415105e-05L, /* 0x3ff103c4, 0x60f88e46, 0xc0c9fb02, 0x13cc7fc1 */ +pa11 = -5.40531400436645861492290270311751349e-06L, /* 0xbfed6abe, 0x9665f8a8, 0xdd0ad3ba, 0xe5dc0ee3 */ +qa1 = 9.05041313265490487793231810291907851e-01L, /* 0x3ffecf61, 0x93340222, 0xe9930620, 0xc4e61168 */ +qa2 = 6.79848064708886864767240880834868092e-01L, /* 0x3ffe5c15, 0x0ba858dc, 0xf7900ae9, 0xfea1e09a */ +qa3 = 4.04720609926471677581066689316516445e-01L, /* 0x3ffd9e6f, 0x145e9b00, 0x6d8c1749, 0xd2928623 */ +qa4 = 1.69183273898369996364661075664302225e-01L, /* 0x3ffc5a7c, 0xc2a363c1, 0xd6c19097, 0xef9b4063 */ +qa5 = 7.44476185988067992342479750486764248e-02L, /* 0x3ffb30ef, 0xfc7259ef, 0x1bcbb089, 0x686dd62d */ +qa6 = 2.02981172725892407200420389604788573e-02L, /* 0x3ff94c90, 0x7976cb0e, 0x21e1d36b, 0x0f09ca2b */ +qa7 = 6.94281866271607668268269403102277234e-03L, /* 0x3ff7c701, 0x2b193250, 0xc5d46ecc, 0x374843d8 */ +qa8 = 1.12952275469171559611651594706820034e-03L, /* 0x3ff52818, 0xfd2a7c06, 0xd13e38fd, 0xda4b34f5 */ +qa9 = 3.13736683241992737197226578597710179e-04L, /* 0x3ff348fa, 0x0cb48d18, 0x051f849b, 0x135ccf74 */ +qa10 = 1.17037675204033225470121134087771410e-05L, /* 0x3fee88b6, 0x98f47704, 0xa5d8f8f2, 0xc6422e11 */ +qa11 = 4.61312518293853991439362806880973592e-06L, /* 0x3fed3594, 0xe31db94f, 0x3592b693, 0xed4386b4 */ +qa12 = -1.02158572037456893687737553657431771e-06L; /* 0xbfeb123a, 0xd60d9b1e, 0x1f6fdeb9, 0x7dc8410a */ +/* + * Domain [1.25,2.85715], range ~[-2.922e-37,2.922e-37]: + * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-121.36 + */ +static const long double +ra0 = -9.86494292470069009555706994426014461e-03L, /* 0xbff84341, 0x239e8709, 0xe941b06a, 0xcb4b6ec5 */ +ra1 = -1.13580436992565640457579040117568870e+00L, /* 0xbfff22c4, 0x133f7c0d, 0x72d5e231, 0x2eb1ee3f */ +ra2 = -4.89744330295291950661185707066921755e+01L, /* 0xc00487cb, 0xa38b4fc2, 0xc136695b, 0xc1df8047 */ +ra3 = -1.10766149300215937173768072715352140e+03L, /* 0xc00914ea, 0x55e6beb3, 0xabc50e07, 0xb6e5664d */ +ra4 = -1.49991031232170934967642795601952100e+04L, /* 0xc00cd4b8, 0xd33243e6, 0xffbf6545, 0x3c57ef6e */ +ra5 = -1.29805749738318462882524181556996692e+05L, /* 0xc00ffb0d, 0xbfeed9b6, 0x5b2a3ff4, 0xe245bd3c */ +ra6 = -7.42828497044940065828871976644647850e+05L, /* 0xc0126ab5, 0x8fe7caca, 0x473352d9, 0xcd4e0c90 */ +ra7 = -2.85637299581890734287995171242421106e+06L, /* 0xc0145cad, 0xa7f76fe7, 0x3e358051, 0x1799f927 */ +ra8 = -7.40674797129824999383748865571026084e+06L, /* 0xc015c412, 0x6fe29c02, 0x298ad158, 0x7d24e45c */ +ra9 = -1.28653420911930973914078724204151759e+07L, /* 0xc016889e, 0x7c2eb0dc, 0x95d5863b, 0x0aa34dc3 */ +ra10 = -1.47198163599330179552932489109452638e+07L, /* 0xc016c136, 0x90b84923, 0xf9bcb497, 0x19bbd0f5 */ +ra11 = -1.07812992258382800318665248311522624e+07L, /* 0xc0164904, 0xe673a113, 0x35d7f079, 0xe13701f3 */ +ra12 = -4.83545565681708642630419905537756076e+06L, /* 0xc0152721, 0xfea094a8, 0x869eb39d, 0x413d6f13 */ +ra13 = -1.23956521201673964822976917356685286e+06L, /* 0xc0132ea0, 0xd3646baa, 0x2fe62b0d, 0xbae5ce85 */ +ra14 = -1.62289333553652417591275333240371812e+05L, /* 0xc0103cf8, 0xaab1e2d6, 0x4c25e014, 0x248d76ab */ +ra15 = -8.82890392601176969729168894389833110e+03L, /* 0xc00c13e7, 0x3b3d8f94, 0x6fbda6f6, 0xe7049a82 */ +ra16 = -1.22591866337261720023681535568334619e+02L, /* 0xc005ea5e, 0x12358891, 0xcfa712c5, 0x77f050d4 */ +sa1 = 6.44508918884710829371852723353794047e+01L, /* 0x400501cd, 0xb69a6c0f, 0x5716de14, 0x47161af6 */ +sa2 = 1.76118475473171481523704824327358534e+03L, /* 0x4009b84b, 0xd305829f, 0xc4c771b0, 0xbf1f7f9b */ +sa3 = 2.69448346969488374857087646131950188e+04L, /* 0x400da503, 0x56bacc05, 0x4fdba68d, 0x2cca27e6 */ +sa4 = 2.56826633369941456778326497384543763e+05L, /* 0x4010f59d, 0x51124428, 0x69c41de6, 0xbd0d5753 */ +sa5 = 1.60647413092257206847700054645905859e+06L, /* 0x40138834, 0xa2184244, 0x557a1bed, 0x68c9d556 */ +sa6 = 6.76963075165099718574753447122393797e+06L, /* 0x40159d2f, 0x7b01b0cc, 0x8bac9e95, 0x5d35d56e */ +sa7 = 1.94295690905361884290986932493647741e+07L, /* 0x40172878, 0xc1172d61, 0x3068501e, 0x2f3c71da */ +sa8 = 3.79774781017759149060839255547073541e+07L, /* 0x401821be, 0xc30d06fe, 0x410563d7, 0x032111fd */ +sa9 = 5.00659831846029484248302236457727397e+07L, /* 0x40187df9, 0x1f97a111, 0xc51d6ac2, 0x4b389793 */ +sa10 = 4.36486287620506484276130525941972541e+07L, /* 0x40184d03, 0x3a618ae0, 0x2a723357, 0xfa45c60a */ +sa11 = 2.43779678791333894255510508253951934e+07L, /* 0x401773fa, 0x6fe10ee2, 0xc467850d, 0xc6b7ff30 */ +sa12 = 8.30732360384443202039372372212966542e+06L, /* 0x4015fb09, 0xee6a5631, 0xdd98de7e, 0x8b00461a */ +sa13 = 1.60160846942050515734192397495105693e+06L, /* 0x40138704, 0x8782bf13, 0x5b8fb315, 0xa898abe5 */ +sa14 = 1.54255505242533291014555153757001825e+05L, /* 0x40102d47, 0xc0abc98e, 0x843c9490, 0xb4352440 */ +sa15 = 5.87949220002375547561467275493888824e+03L, /* 0x400b6f77, 0xe00d21d1, 0xec4d41e8, 0x2f8e1673 */ +sa16 = 4.97272976346793193860385983372237710e+01L; /* 0x40048dd1, 0x816c1b3f, 0x24f540a6, 0x4cfe03cc */ +/* + * Domain [2.85715,9], range ~[-7.886e-37,7.918e-37]: + * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-120 + */ +static const long double +rb0 = -9.86494292470008707171371994479162369e-3L, /* 0xbff84341, 0x239e86f4, 0x2f57e561, 0xf4469360 */ +rb1 = -1.57047326624110727986326503729442830L, /* 0xbfff920a, 0x8935bf73, 0x8803b894, 0x4656482d */ +rb2 = -1.03228196364885474342132255440317065e2L, /* 0xc0059ce9, 0xac4ed0ff, 0x2cff0ff7, 0x5e70d1ab */ +rb3 = -3.74000570653418227179358710865224376e3L, /* 0xc00ad380, 0x2ebf7835, 0xf6b07ed2, 0x861242f7 */ +rb4 = -8.35435477739098044190860390632813956e4L, /* 0xc00f4657, 0x8c3ae934, 0x3647d7b3, 0x80e76fb7 */ +rb5 = -1.21398672055223642118716640216747152e6L, /* 0xc0132862, 0x2b8761c8, 0x27d18c0f, 0x137c9463 */ +rb6 = -1.17669175877248796101665344873273970e7L, /* 0xc0166719, 0x0b2cea46, 0x81f14174, 0x11602ea5 */ +rb7 = -7.66108006086998253606773064264599615e7L, /* 0xc019243f, 0x3c26f4f0, 0x1cc05241, 0x3b953728 */ +rb8 = -3.32547117558141845968704725353130804e8L, /* 0xc01b3d24, 0x42d8ee26, 0x24ef6f3b, 0x604a8c65 */ +rb9 = -9.41561252426350696802167711221739746e8L, /* 0xc01cc0f8, 0xad23692a, 0x8ddb2310, 0xe9937145 */ +rb10 = -1.67157110805390944549427329626281063e9L, /* 0xc01d8e88, 0x9a903734, 0x09a55fa3, 0xd205c903 */ +rb11 = -1.74339631004410841337645931421427373e9L, /* 0xc01d9fa8, 0x77582d2a, 0xc183b8ab, 0x7e00cb05 */ +rb12 = -9.57655233596934915727573141357471703e8L, /* 0xc01cc8a5, 0x460cc685, 0xd0271fa0, 0x6a70e3da */ +rb13 = -2.26320062731339353035254704082495066e8L, /* 0xc01aafab, 0xd7d76721, 0xc9720e11, 0x6a8bd489 */ +rb14 = -1.42777302996263256686002973851837039e7L, /* 0xc016b3b8, 0xc499689f, 0x2b88d965, 0xc32414f9 */ +sb1 = 1.08512869705594540211033733976348506e2L, /* 0x4005b20d, 0x2db7528d, 0x00d20dcb, 0x858f6191 */ +sb2 = 5.02757713761390460534494530537572834e3L, /* 0x400b3a39, 0x3bf4a690, 0x3025d28d, 0xfd40a891 */ +sb3 = 1.31019107205412870059331647078328430e5L, /* 0x400fffcb, 0x1b71d05e, 0x3b28361d, 0x2a3c3690 */ +sb4 = 2.13021555152296846166736757455018030e6L, /* 0x40140409, 0x3c6984df, 0xc4491d7c, 0xb04aa08d */ +sb5 = 2.26649105281820861953868568619768286e7L, /* 0x401759d6, 0xce8736f0, 0xf28ad037, 0x2a901e0c */ +sb6 = 1.61071939490875921812318684143076081e8L, /* 0x401a3338, 0x686fb541, 0x6bd27d06, 0x4f95c9ac */ +sb7 = 7.66895673844301852676056750497991966e8L, /* 0x401c6daf, 0x31cec121, 0x54699126, 0x4bd9bf9e */ +sb8 = 2.41884450436101936436023058196042526e9L, /* 0x401e2059, 0x46b0b8d7, 0x87b64cbf, 0x78bc296d */ +sb9 = 4.92403055884071695093305291535107666e9L, /* 0x401f257e, 0xbe5ed739, 0x39e17346, 0xcadd2e55 */ +sb10 = 6.18627786365587486459633615573786416e9L, /* 0x401f70bb, 0x1be7a7e7, 0x6a45b5ae, 0x607c70f0 */ +sb11 = 4.45898013426501378097430226324743199e9L, /* 0x401f09c6, 0xa32643d7, 0xf1724620, 0x9ea46c32 */ +sb12 = 1.63006115763329848117160344854224975e9L, /* 0x401d84a3, 0x0996887f, 0x65a4f43b, 0x978c1d74 */ +sb13 = 2.39216717012421697446304015847567721e8L, /* 0x401ac845, 0x09a065c2, 0x30095da7, 0x9d72d6ae */ +sb14 = 7.84837329009278694937250358810225609e6L; /* 0x4015df06, 0xd5290e15, 0x63031fac, 0x4d9c894c */ +/* + * Domain [9,108], range ~[-5.324e-38,5.340e-38]: + * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-124 + */ +static const long double +rc0 = -9.86494292470008707171367567652935673e-3L, /* 0xbff84341, 0x239e86f4, 0x2f57e55b, 0x1aa10fd3 */ +rc1 = -1.26229447747315096406518846411562266L, /* 0xbfff4325, 0xbb1aab28, 0xda395cd9, 0xfb861c15 */ +rc2 = -6.13742634438922591780742637728666162e1L, /* 0xc004eafe, 0x7dd51cd8, 0x3c7c5928, 0x751e50cf */ +rc3 = -1.50455835478908280402912854338421517e3L, /* 0xc0097823, 0xbc15b9ab, 0x3d60745c, 0x523e80a5 */ +rc4 = -2.04415631865861549920184039902945685e4L, /* 0xc00d3f66, 0x40b3fc04, 0x5388f2ec, 0xb009e1f0 */ +rc5 = -1.57625662981714582753490610560037638e5L, /* 0xc01033dc, 0xd4dc95b6, 0xfd4da93b, 0xf355b4a9 */ +rc6 = -6.73473451616752528402917538033283794e5L, /* 0xc01248d8, 0x2e73a4f9, 0xcded49c5, 0xfa3bfeb7 */ +rc7 = -1.47433165421387483167186683764364857e6L, /* 0xc01367f1, 0xba77a8f7, 0xcfdd0dbb, 0x25d554b3 */ +rc8 = -1.38811981807868828563794929997744139e6L, /* 0xc01352e5, 0x7d16d9ad, 0xbbdcbf38, 0x38fbc5ea */ +rc9 = -3.59659700530831825640766479698155060e5L, /* 0xc0115f3a, 0xecd57f45, 0x21f8ad6c, 0x910a5958 */ +sc1 = 7.72730753022908298637508998072635696e1L, /* 0x40053517, 0xa10d52bc, 0xdabb55b6, 0xbd0328cd */ +sc2 = 2.36825757341694050500333261769082182e3L, /* 0x400a2808, 0x3e0a9b42, 0x82977842, 0x9c5de29e */ +sc3 = 3.72210540173034735352888847134073099e4L, /* 0x400e22ca, 0x1ba827ef, 0xac8390d7, 0x1fc39a41 */ +sc4 = 3.24136032646418336712461033591393412e5L, /* 0x40113c8a, 0x0216e100, 0xc59d1e44, 0xf0e68d9d */ +sc5 = 1.57836135851134393802505823370009175e6L, /* 0x40138157, 0x95bc7664, 0x17575961, 0xdbe58eeb */ +sc6 = 4.12881981392063738026679089714182355e6L, /* 0x4014f801, 0x9e82e8d2, 0xb8b3a70e, 0xfd84185d */ +sc7 = 5.24438427289213488410596395361544142e6L, /* 0x40154017, 0x81177109, 0x2aa6c3b0, 0x1f106625 */ +sc8 = 2.59909544563616121735963429710382149e6L, /* 0x40143d45, 0xbb90a9b1, 0x12bf9390, 0xa827a700 */ +sc9 = 2.80930665169282501639651995082335693e5L; /* 0x40111258, 0xaa92222e, 0xa97e3216, 0xa237fa6c */ + +long double +erfl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx, llx; + int32_t i; + uint16_t hx; + + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfl(nan)=nan */ + i = (hx>>15)<<1; + return (1-i)+one/x; /* erfl(+-inf)=+-1 */ + } + + ax = fabsl(x); + if(ax < 0.84375) { + if(ax < 0x1p-40L) { + if(ax < 0x1p-16373L) + return (8*x+efx8*x)/8; /* avoid spurious underflow */ + return x + efx*x; + } + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*(pp5+z*(pp6+z*(pp7+ + z*(pp8+z*pp9)))))))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*(qq6+z*(qq7+ + z*(qq8+z*qq9)))))))); + y = r/s; + return x + x*y; + } + if(ax < 1.25) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*(pa7+ + s*(pa8+s*(pa9+s*(pa10+s*pa11)))))))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*(qa7+ + s*(qa8+s*(qa9+s*(qa10+s*(qa11+s*qa12))))))))))); + if(x>=0) return (erx + P/Q); else return (-erx - P/Q); + } + if (ax >= 9) { /* inf>|x|>= 9 */ + if(x>=0) return (one-tiny); else return (tiny-one); + } + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*(ra9+s*(ra10+s*(ra11+s*(ra12+s*(ra13+s*(ra14+ + s*(ra15+s*ra16))))))))))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*(sa9+s*(sa10+s*(sa11+s*(sa12+s*(sa13+s*(sa14+ + s*(sa15+s*sa16))))))))))))))); + } else { /* |x| >= 2.85715 */ + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*(rb7+ + s*(rb8+s*(rb9+s*(rb10+s*(rb11+s*(rb12+s*(rb13+ + s*rb14))))))))))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*(sb7+ + s*(sb8+s*(sb9+s*(sb10+s*(sb11+s*(sb12+s*(sb13+ + s*sb14))))))))))))); + } + z = (float)ax; + r = expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>=0) return (one-r/ax); else return (r/ax-one); +} + +long double +erfcl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx, llx; + uint16_t hx; + + EXTRACT_LDBL128_WORDS(hx, lx, llx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfcl(nan)=nan */ + /* erfcl(+-inf)=0,2 */ + return ((hx>>15)<<1)+one/x; + } + + ax = fabsl(x); + if(ax < 0.84375L) { + if(ax < 0x1p-34L) + return one-x; + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*(pp5+z*(pp6+z*(pp7+ + z*(pp8+z*pp9)))))))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*(qq6+z*(qq7+ + z*(qq8+z*qq9)))))))); + y = r/s; + if(ax < 0.25L) { /* x<1/4 */ + return one-(x+x*y); + } else { + r = x*y; + r += (x-half); + return half - r; + } + } + if(ax < 1.25L) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*(pa7+ + s*(pa8+s*(pa9+s*(pa10+s*pa11)))))))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*(qa7+ + s*(qa8+s*(qa9+s*(qa10+s*(qa11+s*qa12))))))))))); + if(x>=0) { + z = one-erx; return z - P/Q; + } else { + z = erx+P/Q; return one+z; + } + } + + if(ax < 108) { /* |x| < 108 */ + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*(ra9+s*(ra10+s*(ra11+s*(ra12+s*(ra13+s*(ra14+ + s*(ra15+s*ra16))))))))))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*(sa9+s*(sa10+s*(sa11+s*(sa12+s*(sa13+s*(sa14+ + s*(sa15+s*sa16))))))))))))))); + } else if(ax < 9) { + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*(rb7+ + s*(rb8+s*(rb9+s*(rb10+s*(rb11+s*(rb12+s*(rb13+ + s*rb14))))))))))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*(sb7+ + s*(sb8+s*(sb9+s*(sb10+s*(sb11+s*(sb12+s*(sb13+ + s*sb14))))))))))))); + } else { + if(x < -9) return two-tiny; /* x < -9 */ + R=rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+s*(rc6+s*(rc7+ + s*(rc8+s*rc9)))))))); + S=one+s*(sc1+s*(sc2+s*(sc3+s*(sc4+s*(sc5+s*(sc6+s*(sc7+ + s*(sc8+s*sc9)))))))); + } + z = (float)ax; + r = expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>0) return r/ax; else return two-r/ax; + } else { + if(x>0) return tiny*tiny; else return two-tiny; + } +} diff --git a/lib/msun/ld128/s_expl.c b/lib/msun/ld128/s_expl.c index 176c932..1357e70 100644 --- a/lib/msun/ld128/s_expl.c +++ b/lib/msun/ld128/s_expl.c @@ -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 = 11356.523406294143949491931077970763428L, /* 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, - 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, - 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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 = 2.50521083854417203619031960151253944e-8L, 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 = 2.50521083855084570046480450935267433e-8L, 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); } diff --git a/lib/msun/ld80/k_expl.h b/lib/msun/ld80/k_expl.h new file mode 100644 index 0000000..ebfb9a8 --- /dev/null +++ b/lib/msun/ld80/k_expl.h @@ -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, + 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53, + 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53, + 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53, + 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59, + 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54, + 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56, + 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54, + 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56, + 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54, + 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53, + 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53, + 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53, + 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53, + 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54, + 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55, + 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54, + 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60, + 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54, + 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53, + 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53, + 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53, + 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53, + 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57, + 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53, + 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53, + 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53, + 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53, + 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53, + 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53, + 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53, + 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54, + 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53, + 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54, + 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56, + 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53, + 0x1.82589994cce12p+0, 0x1.159f115f56694p-53, + 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53, + 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53, + 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54, + 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54, + 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53, + 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55, + 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); +#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; + + /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ + z = r * r; +#if 0 + q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; +#else + q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6)); +#endif + t = (long double)tbl[n2].lo + tbl[n2].hi; + *hip = tbl[n2].hi; + *lop = tbl[n2].lo + t * (q + r1); +} + +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 */ diff --git a/lib/msun/ld80/s_erfl.c b/lib/msun/ld80/s_erfl.c new file mode 100644 index 0000000..1ae2f90 --- /dev/null +++ b/lib/msun/ld80/s_erfl.c @@ -0,0 +1,337 @@ +/* @(#)s_erf.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include <sys/cdefs.h> +__FBSDID("$FreeBSD$"); + +/* + * See s_erf.c for complete comments. + * + * Converted to long double by Steven G. Kargl. + */ +#include <float.h> +#ifdef __i386__ +#include <ieeefp.h> +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" + +/* XXX Prevent compilers from erroneously constant folding: */ +static const volatile long double tiny = 0x1p-10000L; + +static const double +half= 0.5, +one = 1, +two = 2; +/* + * In the domain [0, 2**-34], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-102. + */ +static const union IEEEl2bits +efxu = LD80C(0x8375d410a6db446c, -3, 1.28379167095512573902e-1L), +efx8u = LD80C(0x8375d410a6db446c, 0, 1.02703333676410059122e+0L), +/* + * Domain [0, 0.84375], range ~[-1.423e-22, 1.423e-22]: + * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-72.573 + */ +pp0u = LD80C(0x8375d410a6db446c, -3, 1.28379167095512573902e-1L), +pp1u = LD80C(0xa46c7d09ec3d0cec, -2, -3.21140201054840180596e-1L), +pp2u = LD80C(0x9b31e66325576f86, -5, -3.78893851760347812082e-2L), +pp3u = LD80C(0x804ac72c9a0b97dd, -7, -7.83032847030604679616e-3L), +pp4u = LD80C(0x9f42bcbc3d5a601d, -12, -3.03765663857082048459e-4L), +pp5u = LD80C(0x9ec4ad6193470693, -16, -1.89266527398167917502e-5L), +qq1u = LD80C(0xdb4b8eb713188d6b, -2, 4.28310832832310510579e-1L), +qq2u = LD80C(0xa5750835b2459bd1, -4, 8.07896272074540216658e-2L), +qq3u = LD80C(0x8b85d6bd6a90b51c, -7, 8.51579638189385354266e-3L), +qq4u = LD80C(0x87332f82cff4ff96, -11, 5.15746855583604912827e-4L), +qq5u = LD80C(0x83466cb6bf9dca00, -16, 1.56492109706256700009e-5L), +qq6u = LD80C(0xf5bf98c2f996bf63, -24, 1.14435527803073879724e-7L); +#define efx (efxu.e) +#define efx8 (efx8u.e) +#define pp0 (pp0u.e) +#define pp1 (pp1u.e) +#define pp2 (pp2u.e) +#define pp3 (pp3u.e) +#define pp4 (pp4u.e) +#define pp5 (pp5u.e) +#define qq1 (qq1u.e) +#define qq2 (qq2u.e) +#define qq3 (qq3u.e) +#define qq4 (qq4u.e) +#define qq5 (qq5u.e) +#define qq6 (qq6u.e) +static const union IEEEl2bits +erxu = LD80C(0xd7bb3d0000000000, -1, 8.42700779438018798828e-1L), +/* + * Domain [0.84375, 1.25], range ~[-8.132e-22, 8.113e-22]: + * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-71.762 + */ +pa0u = LD80C(0xe8211158da02c692, -27, 1.35116960705131296711e-8L), +pa1u = LD80C(0xd488f89f36988618, -2, 4.15107507167065612570e-1L), +pa2u = LD80C(0xece74f8c63fa3942, -4, -1.15675565215949226989e-1L), +pa3u = LD80C(0xc8d31e020727c006, -4, 9.80589241379624665791e-2L), +pa4u = LD80C(0x985d5d5fafb0551f, -5, 3.71984145558422368847e-2L), +pa5u = LD80C(0xa5b6c4854d2f5452, -8, -5.05718799340957673661e-3L), +pa6u = LD80C(0x85c8d58fe3993a47, -8, 4.08277919612202243721e-3L), +pa7u = LD80C(0xddbfbc23677b35cf, -13, 2.11476292145347530794e-4L), +qa1u = LD80C(0xb8a977896f5eff3f, -1, 7.21335860303380361298e-1L), +qa2u = LD80C(0x9fcd662c3d4eac86, -1, 6.24227891731886593333e-1L), +qa3u = LD80C(0x9d0b618eac67ba07, -2, 3.06727455774491855801e-1L), +qa4u = LD80C(0x881a4293f6d6c92d, -3, 1.32912674218195890535e-1L), +qa5u = LD80C(0xbab144f07dea45bf, -5, 4.55792134233613027584e-2L), +qa6u = LD80C(0xa6c34ba438bdc900, -7, 1.01783980070527682680e-2L), +qa7u = LD80C(0x8fa866dc20717a91, -9, 2.19204436518951438183e-3L); +#define erx (erxu.e) +#define pa0 (pa0u.e) +#define pa1 (pa1u.e) +#define pa2 (pa2u.e) +#define pa3 (pa3u.e) +#define pa4 (pa4u.e) +#define pa5 (pa5u.e) +#define pa6 (pa6u.e) +#define pa7 (pa7u.e) +#define qa1 (qa1u.e) +#define qa2 (qa2u.e) +#define qa3 (qa3u.e) +#define qa4 (qa4u.e) +#define qa5 (qa5u.e) +#define qa6 (qa6u.e) +#define qa7 (qa7u.e) +static const union IEEEl2bits +/* + * Domain [1.25,2.85715], range ~[-2.334e-22,2.334e-22]: + * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-71.860 + */ +ra0u = LD80C(0xa1a091e0fb4f335a, -7, -9.86494298915814308249e-3L), +ra1u = LD80C(0xc2b0d045ae37df6b, -1, -7.60510460864878271275e-1L), +ra2u = LD80C(0xf2cec3ee7da636c5, 3, -1.51754798236892278250e+1L), +ra3u = LD80C(0x813cc205395adc7d, 7, -1.29237335516455333420e+2L), +ra4u = LD80C(0x8737c8b7b4062c2f, 9, -5.40871625829510494776e+2L), +ra5u = LD80C(0x8ffe5383c08d4943, 10, -1.15194769466026108551e+3L), +ra6u = LD80C(0x983573e64d5015a9, 10, -1.21767039790249025544e+3L), +ra7u = LD80C(0x92a794e763a6d4db, 9, -5.86618463370624636688e+2L), +ra8u = LD80C(0xd5ad1fae77c3d9a3, 6, -1.06838132335777049840e+2L), +ra9u = LD80C(0x934c1a247807bb9c, 2, -4.60303980944467334806e+0L), +sa1u = LD80C(0xd342f90012bb1189, 4, 2.64077014928547064865e+1L), +sa2u = LD80C(0x839be13d9d5da883, 8, 2.63217811300123973067e+2L), +sa3u = LD80C(0x9f8cba6d1ae1b24b, 10, 1.27639775710344617587e+3L), +sa4u = LD80C(0xcaa83f403713e33e, 11, 3.24251544209971162003e+3L), +sa5u = LD80C(0x8796aff2f3c47968, 12, 4.33883591261332837874e+3L), +sa6u = LD80C(0xb6ef97f9c753157b, 11, 2.92697460344182158454e+3L), +sa7u = LD80C(0xe02aee5f83773d1c, 9, 8.96670799139389559818e+2L), +sa8u = LD80C(0xc82b83855b88e07e, 6, 1.00084987800048510018e+2L), +sa9u = LD80C(0x92f030aefadf28ad, 1, 2.29591004455459083843e+0L); +#define ra0 (ra0u.e) +#define ra1 (ra1u.e) +#define ra2 (ra2u.e) +#define ra3 (ra3u.e) +#define ra4 (ra4u.e) +#define ra5 (ra5u.e) +#define ra6 (ra6u.e) +#define ra7 (ra7u.e) +#define ra8 (ra8u.e) +#define ra9 (ra9u.e) +#define sa1 (sa1u.e) +#define sa2 (sa2u.e) +#define sa3 (sa3u.e) +#define sa4 (sa4u.e) +#define sa5 (sa5u.e) +#define sa6 (sa6u.e) +#define sa7 (sa7u.e) +#define sa8 (sa8u.e) +#define sa9 (sa9u.e) +/* + * Domain [2.85715,7], range ~[-8.323e-22,8.390e-22]: + * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-70.326 + */ +static const union IEEEl2bits +rb0u = LD80C(0xa1a091cf43abcd26, -7, -9.86494292470284646962e-3L), +rb1u = LD80C(0xd19d2df1cbb8da0a, -1, -8.18804618389296662837e-1L), +rb2u = LD80C(0x9a4dd1383e5daf5b, 4, -1.92879967111618594779e+1L), +rb3u = LD80C(0xbff0ae9fc0751de6, 7, -1.91940164551245394969e+2L), +rb4u = LD80C(0xdde08465310b472b, 9, -8.87508080766577324539e+2L), +rb5u = LD80C(0xe796e1d38c8c70a9, 10, -1.85271506669474503781e+3L), +rb6u = LD80C(0xbaf655a76e0ab3b5, 10, -1.49569795581333675349e+3L), +rb7u = LD80C(0x95d21e3e75503c21, 8, -2.99641547972948019157e+2L), +sb1u = LD80C(0x814487ed823c8cbd, 5, 3.23169247732868256569e+1L), +sb2u = LD80C(0xbe4bfbb1301304be, 8, 3.80593618534539961773e+2L), +sb3u = LD80C(0x809c4ade46b927c7, 11, 2.05776827838541292848e+3L), +sb4u = LD80C(0xa55284359f3395a8, 12, 5.29031455540062116327e+3L), +sb5u = LD80C(0xbcfa72da9b820874, 12, 6.04730608102312640462e+3L), +sb6u = LD80C(0x9d09a35988934631, 11, 2.51260238030767176221e+3L), +sb7u = LD80C(0xd675bbe542c159fa, 7, 2.14459898308561015684e+2L); +#define rb0 (rb0u.e) +#define rb1 (rb1u.e) +#define rb2 (rb2u.e) +#define rb3 (rb3u.e) +#define rb4 (rb4u.e) +#define rb5 (rb5u.e) +#define rb6 (rb6u.e) +#define rb7 (rb7u.e) +#define sb1 (sb1u.e) +#define sb2 (sb2u.e) +#define sb3 (sb3u.e) +#define sb4 (sb4u.e) +#define sb5 (sb5u.e) +#define sb6 (sb6u.e) +#define sb7 (sb7u.e) +/* + * Domain [7,108], range ~[-4.422e-22,4.422e-22]: + * |log(x*erfc(x)) + x**2 + 0.5625 - rc(x)/sc(x)| < 2**-70.938 + */ +static const union IEEEl2bits +/* err = -4.422092275318925082e-22 -70.937689 */ +rc0u = LD80C(0xa1a091cf437a17ad, -7, -9.86494292470008707260e-3L), +rc1u = LD80C(0xbe79c5a978122b00, -1, -7.44045595049165939261e-1L), +rc2u = LD80C(0xdb26f9bbe31a2794, 3, -1.36970155085888424425e+1L), +rc3u = LD80C(0xb5f69a38f5747ac8, 6, -9.09816453742625888546e+1L), +rc4u = LD80C(0xd79676d970d0a21a, 7, -2.15587750997584074147e+2L), +rc5u = LD80C(0xfe528153c45ec97c, 6, -1.27161142938347796666e+2L), +sc1u = LD80C(0xc5e8cd46d5604a96, 4, 2.47386727842204312937e+1L), +sc2u = LD80C(0xc5f0f5a5484520eb, 7, 1.97941248254913378865e+2L), +sc3u = LD80C(0x964e3c7b34db9170, 9, 6.01222441484087787522e+2L), +sc4u = LD80C(0x99be1b89faa0596a, 9, 6.14970430845978077827e+2L), +sc5u = LD80C(0xf80dfcbf37ffc5ea, 6, 1.24027318931184605891e+2L); +#define rc0 (rc0u.e) +#define rc1 (rc1u.e) +#define rc2 (rc2u.e) +#define rc3 (rc3u.e) +#define rc4 (rc4u.e) +#define rc5 (rc5u.e) +#define sc1 (sc1u.e) +#define sc2 (sc2u.e) +#define sc3 (sc3u.e) +#define sc4 (sc4u.e) +#define sc5 (sc5u.e) + +long double +erfl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx; + int32_t i; + uint16_t hx; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfl(nan)=nan */ + i = (hx>>15)<<1; + return (1-i)+one/x; /* erfl(+-inf)=+-1 */ + } + + ENTERI(); + + ax = fabsl(x); + if(ax < 0.84375) { + if(ax < 0x1p-34L) { + if(ax < 0x1p-16373L) + RETURNI((8*x+efx8*x)/8); /* avoid spurious underflow */ + RETURNI(x + efx*x); + } + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*pp5)))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*qq6))))); + y = r/s; + RETURNI(x + x*y); + } + if(ax < 1.25) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*pa7)))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*qa7)))))); + if(x>=0) RETURNI(erx + P/Q); else RETURNI(-erx - P/Q); + } + if(ax >= 7) { /* inf>|x|>= 7 */ + if(x>=0) RETURNI(one-tiny); else RETURNI(tiny-one); + } + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*ra9)))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*sa9)))))))); + } else { /* |x| >= 2.85715 */ + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*rb7)))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); + } + z=(float)ax; + r=expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>=0) RETURNI(one-r/ax); else RETURNI(r/ax-one); +} + +long double +erfcl(long double x) +{ + long double ax,R,S,P,Q,s,y,z,r; + uint64_t lx; + uint16_t hx; + + EXTRACT_LDBL80_WORDS(hx, lx, x); + + if((hx & 0x7fff) == 0x7fff) { /* erfcl(nan)=nan */ + /* erfcl(+-inf)=0,2 */ + return ((hx>>15)<<1)+one/x; + } + + ENTERI(); + + ax = fabsl(x); + if(ax < 0.84375L) { + if(ax < 0x1p-34L) + RETURNI(one-x); + z = x*x; + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*(pp4+z*pp5)))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*(qq5+z*qq6))))); + y = r/s; + if(ax < 0.25L) { /* x<1/4 */ + RETURNI(one-(x+x*y)); + } else { + r = x*y; + r += (x-half); + RETURNI(half - r); + } + } + if(ax < 1.25L) { + s = ax-one; + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*(pa6+s*pa7)))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*(qa6+s*qa7)))))); + if(x>=0) { + z = one-erx; RETURNI(z - P/Q); + } else { + z = (erx+P/Q); RETURNI(one+z); + } + } + + if(ax < 108) { /* |x| < 108 */ + s = one/(ax*ax); + if(ax < 2.85715) { /* |x| < 2.85715 */ + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*(ra7+ + s*(ra8+s*ra9)))))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*(sa8+s*sa9)))))))); + } else if(ax < 7) { /* | |x| < 7 */ + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*(rb6+s*rb7)))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); + } else { + if(x < -7) RETURNI(two-tiny);/* x < -7 */ + R=rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*rc5)))); + S=one+s*(sc1+s*(sc2+s*(sc3+s*(sc4+s*sc5)))); + } + z = (float)ax; + r = expl(-z*z-0.5625)*expl((z-ax)*(z+ax)+R/S); + if(x>0) RETURNI(r/ax); else RETURNI(two-r/ax); + } else { + if(x>0) RETURNI(tiny*tiny); else RETURNI(two-tiny); + } +} diff --git a/lib/msun/ld80/s_expl.c b/lib/msun/ld80/s_expl.c index ec748d3..3147d35 100644 --- a/lib/msun/ld80/s_expl.c +++ b/lib/msun/ld80/s_expl.c @@ -48,16 +48,15 @@ __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 union IEEEl2bits /* log(2**16384 - 0.5) rounded towards zero: */ @@ -68,178 +67,16 @@ o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); #define u_threshold (u_thresholdu.e) -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, - 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, - 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53, - 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53, - 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53, - 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59, - 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54, - 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56, - 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54, - 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56, - 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54, - 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53, - 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53, - 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53, - 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53, - 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54, - 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55, - 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54, - 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60, - 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54, - 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53, - 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53, - 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53, - 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53, - 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57, - 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53, - 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53, - 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53, - 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53, - 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53, - 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53, - 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53, - 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54, - 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53, - 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54, - 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56, - 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53, - 0x1.82589994cce12p+0, 0x1.159f115f56694p-53, - 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53, - 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53, - 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54, - 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54, - 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53, - 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55, - 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 -}; - long double expl(long double x) { - union IEEEl2bits u, v; - long double fn, q, r, r1, r2, t, twopk, twopkp10000; - long double z; - 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; @@ -247,59 +84,32 @@ 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, -NaN or unsupported */ - return (-1 / x); - return (x + x); /* x is +Inf, +NaN or unsupported */ + RETURNP(-1 / x); + RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ } if (x > o_threshold) - return (huge * huge); + RETURNP(huge * huge); if (x < u_threshold) - return (tiny * tiny); - } else if (ix < BIAS - 65) { /* |x| < 0x1p-65 (includes pseudos) */ - return (1 + x); /* 1 with inexact iff x != 0 */ + RETURNP(tiny * tiny); + } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ + 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. */ - 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); -#else - n = (int)fn; -#endif - n2 = (unsigned)n % INTERVALS; - /* Depend on the sign bit being propagated: */ - k = n >> LOG2_INTERVALS; - r1 = x - fn * L1; - r2 = fn * -L2; - - /* Prepare scale factors. */ - 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). */ - z = r * r; - q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; - t = (long double)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. */ 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); } } @@ -326,8 +136,11 @@ T1 = -0.1659, /* ~-30.625/128 * log(2) */ T2 = 0.1659; /* ~30.625/128 * log(2) */ /* - * Domain [-0.1659, 0.1659], range ~[-1.2027e-22, 3.4417e-22]: - * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.2 + * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 + * + * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, + * but unlike for ld128 we can't drop any terms. */ static const union IEEEl2bits B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), @@ -353,6 +166,8 @@ 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; @@ -360,11 +175,11 @@ expm1l(long double x) if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ if (ix == BIAS + LDBL_MAX_EXP) { if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ - return (-1 / x - 1); - return (x + x); /* x is +Inf, +NaN or unsupported */ + RETURNP(-1 / x - 1); + RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ } if (x > o_threshold) - return (huge * huge); + RETURNP(huge * huge); /* * expm1l() never underflows, but it must avoid * unrepresentable large negative exponents. We used a @@ -373,15 +188,15 @@ expm1l(long double x) * in the same way as large ones here. */ if (hx & 0x8000) /* x <= -64 */ - return (tiny - 1); /* good for x < -65ln2 - eps */ + RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */ } ENTERI(); if (T1 < x && x < T2) { - if (ix < BIAS - 64) { /* |x| < 0x1p-64 (includes pseudos) */ + if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ /* x (rounded) with inexact if x != 0: */ - RETURNI(x == 0 ? x : + RETURNPI(x == 0 ? x : (0x1p100 * x + fabsl(x)) * 0x1p-100); } @@ -402,9 +217,9 @@ 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). */ @@ -438,21 +253,21 @@ expm1l(long double x) t = (long double)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); @@ -462,8 +277,8 @@ 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); } diff --git a/lib/msun/man/cosh.3 b/lib/msun/man/cosh.3 index 96abc5a..334b564 100644 --- a/lib/msun/man/cosh.3 +++ b/lib/msun/man/cosh.3 @@ -28,12 +28,13 @@ .\" from: @(#)cosh.3 5.1 (Berkeley) 5/2/91 .\" $FreeBSD$ .\" -.Dd January 14, 2005 +.Dd August 17, 2013 .Dt COSH 3 .Os .Sh NAME .Nm cosh , -.Nm coshf +.Nm coshf , +.Nm coshl .Nd hyperbolic cosine functions .Sh LIBRARY .Lb libm @@ -43,11 +44,14 @@ .Fn cosh "double x" .Ft float .Fn coshf "float x" +.Ft long double +.Fn coshl "long double x" .Sh DESCRIPTION The -.Fn cosh -and the -.Fn coshf +.Fn cosh , +.Fn coshf , +and +.Fn coshl functions compute the hyperbolic cosine of .Fa x . .Sh SEE ALSO diff --git a/lib/msun/man/erf.3 b/lib/msun/man/erf.3 index 46886b0..a9a3e0a 100644 --- a/lib/msun/man/erf.3 +++ b/lib/msun/man/erf.3 @@ -28,14 +28,16 @@ .\" from: @(#)erf.3 6.4 (Berkeley) 4/20/91 .\" $FreeBSD$ .\" -.Dd April 20, 1991 +.Dd July 13, 2014 .Dt ERF 3 .Os .Sh NAME .Nm erf , .Nm erff , +.Nm erfl , .Nm erfc , -.Nm erfcf +.Nm erfcf , +.Nm erfcl .Nd error function operators .Sh LIBRARY .Lb libm @@ -45,18 +47,23 @@ .Fn erf "double x" .Ft float .Fn erff "float x" +.Ft "long double" +.Fn erfl "long double x" .Ft double .Fn erfc "double x" .Ft float .Fn erfcf "float x" +.Ft "long double" +.Fn erfcl "long double x" .Sh DESCRIPTION These functions calculate the error function of .Fa x . .Pp The -.Fn erf -and the -.Fn erff +.Fn erf , +.Fn erff , +and +.Fn erfl functions calculate the error function of x; where .Bd -ragged -offset indent .if n \{\ @@ -69,9 +76,10 @@ erf\|(x) := .Ed .Pp The -.Fn erfc -and the -.Fn erfcf +.Fn erfc , +.Fn erfcf , +and +.Fn erfcl functions calculate the complementary error function of .Fa x ; that is @@ -79,9 +87,6 @@ that is subtracts the result of the error function .Fn erf x from 1.0. -This is useful, since for large -.Fa x -places disappear. .Sh SEE ALSO .Xr math 3 .Sh HISTORY diff --git a/lib/msun/man/sinh.3 b/lib/msun/man/sinh.3 index 02944cc..b34cc38 100644 --- a/lib/msun/man/sinh.3 +++ b/lib/msun/man/sinh.3 @@ -27,12 +27,14 @@ .\" .\" from: @(#)sinh.3 6.6 (Berkeley) 4/19/91 .\" $FreeBSD$ -.Dd January 14, 2005 +.\" +.Dd August 17, 2013 .Dt SINH 3 .Os .Sh NAME .Nm sinh , -.Nm sinhf +.Nm sinhf , +.Nm sinhl .Nd hyperbolic sine function .Sh LIBRARY .Lb libm @@ -42,11 +44,14 @@ .Fn sinh "double x" .Ft float .Fn sinhf "float x" +.Ft long double +.Fn sinhl "long double x" .Sh DESCRIPTION The -.Fn sinh -and the -.Fn sinhf +.Fn sinh , +.Fn sinhf , +and +.Fn sinhl functions compute the hyperbolic sine of .Fa x . .Sh SEE ALSO diff --git a/lib/msun/man/tanh.3 b/lib/msun/man/tanh.3 index 6fb185c..ea2468f 100644 --- a/lib/msun/man/tanh.3 +++ b/lib/msun/man/tanh.3 @@ -28,12 +28,13 @@ .\" from: @(#)tanh.3 5.1 (Berkeley) 5/2/91 .\" $FreeBSD$ .\" -.Dd May 2, 1991 +.Dd August 17, 2013 .Dt TANH 3 .Os .Sh NAME .Nm tanh , -.Nm tanhf +.Nm tanhf , +.Nm tanhl .Nd hyperbolic tangent functions .Sh LIBRARY .Lb libm @@ -43,20 +44,24 @@ .Fn tanh "double x" .Ft float .Fn tanhf "float x" +.Ft long double +.Fn tanhl "long double x" .Sh DESCRIPTION The -.Fn tanh -and the -.Fn tanhf +.Fn tanh , +.Fn tanhf , +and +.Fn tanhl functions compute the hyperbolic tangent of .Fa x . For a discussion of error due to roundoff, see .Xr math 3 . .Sh RETURN VALUES The -.Fn tanh +.Fn tanh , +.Fn tanhf , and the -.Fn tanhf +.Fn tanhl functions return the hyperbolic tangent value. .Sh SEE ALSO .Xr acos 3 , diff --git a/lib/msun/src/e_cosh.c b/lib/msun/src/e_cosh.c index a363695..246b5fb 100644 --- a/lib/msun/src/e_cosh.c +++ b/lib/msun/src/e_cosh.c @@ -35,6 +35,8 @@ __FBSDID("$FreeBSD$"); * only cosh(0)=1 is exact for finite x. */ +#include <float.h> + #include "math.h" #include "math_private.h" @@ -77,3 +79,7 @@ __ieee754_cosh(double x) /* |x| > overflowthresold, cosh(x) overflow */ return huge*huge; } + +#if (LDBL_MANT_DIG == 53) +__weak_reference(cosh, coshl); +#endif diff --git a/lib/msun/src/e_coshl.c b/lib/msun/src/e_coshl.c new file mode 100644 index 0000000..0a21277 --- /dev/null +++ b/lib/msun/src/e_coshl.c @@ -0,0 +1,130 @@ +/* from: FreeBSD: head/lib/msun/src/e_coshl.c XXX */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include <sys/cdefs.h> +__FBSDID("$FreeBSD$"); + +/* + * See e_cosh.c for complete comments. + * + * Converted to long double by Bruce D. Evans. + */ + +#include <float.h> +#ifdef __i386__ +#include <ieeefp.h> +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" +#include "k_expl.h" + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const volatile long double huge = 0x1p10000L, tiny = 0x1p-10000L; +#if LDBL_MANT_DIG == 64 +/* + * Domain [-1, 1], range ~[-1.8211e-21, 1.8211e-21]: + * |cosh(x) - c(x)| < 2**-68.8 + */ +static const union IEEEl2bits +C4u = LD80C(0xaaaaaaaaaaaaac78, -5, 4.16666666666666682297e-2L); +#define C4 C4u.e +static const double +C2 = 0.5, +C6 = 1.3888888888888616e-3, /* 0x16c16c16c16b99.0p-62 */ +C8 = 2.4801587301767953e-5, /* 0x1a01a01a027061.0p-68 */ +C10 = 2.7557319163300398e-7, /* 0x127e4fb6c9b55f.0p-74 */ +C12 = 2.0876768371393075e-9, /* 0x11eed99406a3f4.0p-81 */ +C14 = 1.1469537039374480e-11, /* 0x1938c67cd18c48.0p-89 */ +C16 = 4.8473490896852041e-14; /* 0x1b49c429701e45.0p-97 */ +#elif LDBL_MANT_DIG == 113 +/* + * Domain [-1, 1], range ~[-2.3194e-37, 2.3194e-37]: + * |cosh(x) - c(x)| < 2**-121.69 + */ +static const long double +C4 = 4.16666666666666666666666666666666225e-2L, /* 0x1555555555555555555555555554e.0p-117L */ +C6 = 1.38888888888888888888888888889434831e-3L, /* 0x16c16c16c16c16c16c16c16c1dd7a.0p-122L */ +C8 = 2.48015873015873015873015871870962089e-5L, /* 0x1a01a01a01a01a01a01a017af2756.0p-128L */ +C10 = 2.75573192239858906525574318600800201e-7L, /* 0x127e4fb7789f5c72ef01c8a040640.0p-134L */ +C12 = 2.08767569878680989791444691755468269e-9L, /* 0x11eed8eff8d897b543d0679607399.0p-141L */ +C14= 1.14707455977297247387801189650495351e-11L, /* 0x193974a8c07c9d24ae169a7fa9b54.0p-149L */ +C16 = 4.77947733238737883626416876486279985e-14L; /* 0x1ae7f3e733b814d4e1b90f5727fe4.0p-157L */ +static const double +C2 = 0.5, +C18 = 1.5619206968597871e-16, /* 0x16827863b9900b.0p-105 */ +C20 = 4.1103176218528049e-19, /* 0x1e542ba3d3c269.0p-114 */ +C22 = 8.8967926401641701e-22, /* 0x10ce399542a014.0p-122 */ +C24 = 1.6116681626523904e-24, /* 0x1f2c981d1f0cb7.0p-132 */ +C26 = 2.5022374732804632e-27; /* 0x18c7ecf8b2c4a0.0p-141 */ +#else +#error "Unsupported long double format" +#endif /* LDBL_MANT_DIG == 64 */ + +/* log(2**16385 - 0.5) rounded up: */ +static const float +o_threshold = 1.13572168e4; /* 0xb174de.0p-10 */ + +long double +coshl(long double x) +{ + long double hi,lo,x2,x4; + double dx2; + uint16_t ix; + + GET_LDBL_EXPSIGN(ix,x); + ix &= 0x7fff; + + /* x is INF or NaN */ + if(ix>=0x7fff) return x*x; + + ENTERI(); + + /* |x| < 1, return 1 or c(x) */ + if(ix<0x3fff) { + if (ix<BIAS-(LDBL_MANT_DIG+1)/2) /* |x| < TINY */ + RETURNI(1+tiny); /* cosh(tiny) = 1(+) with inexact */ + x2 = x*x; +#if LDBL_MANT_DIG == 64 + x4 = x2*x2; + RETURNI(((C16*x2 + C14)*x4 + (C12*x2 + C10))*(x4*x4*x2) + + ((C8*x2 + C6)*x2 + C4)*x4 + C2*x2 + 1); +#elif LDBL_MANT_DIG == 113 + dx2 = x2; + RETURNI((((((((((((C26*dx2 + C24)*dx2 + C22)*dx2 + + C20)*x2 + C18)*x2 + + C16)*x2 + C14)*x2 + C12)*x2 + C10)*x2 + C8)*x2 + C6)*x2 + + C4)*(x2*x2) + C2*x2 + 1); +#endif + } + + /* |x| in [1, 64), return accurate exp(|x|)/2+1/exp(|x|)/2 */ + if (ix < 0x4005) { + k_hexpl(fabsl(x), &hi, &lo); + RETURNI(lo + 0.25/(hi + lo) + hi); + } + + /* |x| in [64, o_threshold], return correctly-overflowing exp(|x|)/2 */ + if (fabsl(x) <= o_threshold) + RETURNI(hexpl(fabsl(x))); + + /* |x| > o_threshold, cosh(x) overflow */ + RETURNI(huge*huge); +} diff --git a/lib/msun/src/e_lgamma_r.c b/lib/msun/src/e_lgamma_r.c index 1cff592..7a95ea4 100644 --- a/lib/msun/src/e_lgamma_r.c +++ b/lib/msun/src/e_lgamma_r.c @@ -86,8 +86,10 @@ __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" +static const volatile double vzero = 0; + static const double -two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ +zero= 0.00000000000000000000e+00, half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ @@ -154,39 +156,35 @@ w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ -static const double zero= 0.00000000000000000000e+00; - - static double sin_pi(double x) +/* + * Compute sin(pi*x) without actually doing the pi*x multiplication. + * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is + * the precision of x. + */ +static double +sin_pi(double x) { + volatile double vz; double y,z; - int n,ix; + int n; + + y = -x; - GET_HIGH_WORD(ix,x); - ix &= 0x7fffffff; + vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */ + z = vz-0x1p52; /* rint(y) for the above range */ + if (z == y) + return zero; - if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0); - y = -x; /* x is assume negative */ + vz = y+0x1p50; + GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */ + z = vz-0x1p50; /* y rounded to a multiple of 0.25 */ + if (z > y) { + z -= 0.25; /* adjust to round down */ + n--; + } + n &= 7; /* octant of y mod 2 */ + y = y - z + n * 0.25; /* y mod 2 */ - /* - * argument reduction, make sure inexact flag not raised if input - * is an integer - */ - z = floor(y); - if(z!=y) { /* inexact anyway */ - y *= 0.5; - y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */ - n = (int) (y*4.0); - } else { - if(ix>=0x43400000) { - y = zero; n = 0; /* y must be even */ - } else { - if(ix<0x43300000) z = y+two52; /* exact */ - GET_LOW_WORD(n,z); - n &= 1; - y = n; - n<<= 2; - } - } switch (n) { case 0: y = __kernel_sin(pi*y,zero,0); break; case 1: @@ -206,7 +204,7 @@ __ieee754_lgamma_r(double x, int *signgamp) { double t,y,z,nadj,p,p1,p2,p3,q,r,w; int32_t hx; - int i,lx,ix; + int i,ix,lx; EXTRACT_WORDS(hx,lx,x); @@ -214,7 +212,7 @@ __ieee754_lgamma_r(double x, int *signgamp) *signgamp = 1; ix = hx&0x7fffffff; if(ix>=0x7ff00000) return x*x; - if((ix|lx)==0) return one/zero; + if((ix|lx)==0) return one/vzero; if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ if(hx<0) { *signgamp = -1; @@ -223,9 +221,9 @@ __ieee754_lgamma_r(double x, int *signgamp) } if(hx<0) { if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ - return one/zero; + return one/vzero; t = sin_pi(x); - if(t==zero) return one/zero; /* -integer */ + if(t==zero) return one/vzero; /* -integer */ nadj = __ieee754_log(pi/fabs(t*x)); if(t<zero) *signgamp = -1; x = -x; diff --git a/lib/msun/src/e_lgammaf_r.c b/lib/msun/src/e_lgammaf_r.c index e2d90ef..9a7ab39 100644 --- a/lib/msun/src/e_lgammaf_r.c +++ b/lib/msun/src/e_lgammaf_r.c @@ -19,8 +19,10 @@ __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" +static const volatile float vzero = 0; + static const float -two23= 8.3886080000e+06, /* 0x4b000000 */ +zero= 0.0000000000e+00, half= 5.0000000000e-01, /* 0x3f000000 */ one = 1.0000000000e+00, /* 0x3f800000 */ pi = 3.1415927410e+00, /* 0x40490fdb */ @@ -87,39 +89,30 @@ w4 = -5.9518753551e-04, /* 0xba1c065c */ w5 = 8.3633989561e-04, /* 0x3a5b3dd2 */ w6 = -1.6309292987e-03; /* 0xbad5c4e8 */ -static const float zero= 0.0000000000e+00; - - static float sin_pif(float x) +static float +sin_pif(float x) { + volatile float vz; float y,z; - int n,ix; + int n; - GET_FLOAT_WORD(ix,x); - ix &= 0x7fffffff; + y = -x; - if(ix<0x3e800000) return __kernel_sindf(pi*x); - y = -x; /* x is assume negative */ + vz = y+0x1p23F; /* depend on 0 <= y < 0x1p23 */ + z = vz-0x1p23F; /* rintf(y) for the above range */ + if (z == y) + return zero; + + vz = y+0x1p21F; + GET_FLOAT_WORD(n,vz); /* bits for rounded y (units 0.25) */ + z = vz-0x1p21F; /* y rounded to a multiple of 0.25 */ + if (z > y) { + z -= 0.25F; /* adjust to round down */ + n--; + } + n &= 7; /* octant of y mod 2 */ + y = y - z + n * 0.25F; /* y mod 2 */ - /* - * argument reduction, make sure inexact flag not raised if input - * is an integer - */ - z = floorf(y); - if(z!=y) { /* inexact anyway */ - y *= (float)0.5; - y = (float)2.0*(y - floorf(y)); /* y = |x| mod 2.0 */ - n = (int) (y*(float)4.0); - } else { - if(ix>=0x4b800000) { - y = zero; n = 0; /* y must be even */ - } else { - if(ix<0x4b000000) z = y+two23; /* exact */ - GET_FLOAT_WORD(n,z); - n &= 1; - y = n; - n<<= 2; - } - } switch (n) { case 0: y = __kernel_sindf(pi*y); break; case 1: @@ -147,7 +140,7 @@ __ieee754_lgammaf_r(float x, int *signgamp) *signgamp = 1; ix = hx&0x7fffffff; if(ix>=0x7f800000) return x*x; - if(ix==0) return one/zero; + if(ix==0) return one/vzero; if(ix<0x35000000) { /* |x|<2**-21, return -log(|x|) */ if(hx<0) { *signgamp = -1; @@ -156,9 +149,9 @@ __ieee754_lgammaf_r(float x, int *signgamp) } if(hx<0) { if(ix>=0x4b000000) /* |x|>=2**23, must be -integer */ - return one/zero; + return one/vzero; t = sin_pif(x); - if(t==zero) return one/zero; /* -integer */ + if(t==zero) return one/vzero; /* -integer */ nadj = __ieee754_logf(pi/fabsf(t*x)); if(t<zero) *signgamp = -1; x = -x; diff --git a/lib/msun/src/e_pow.c b/lib/msun/src/e_pow.c index 7607a4a..d54af9d 100644 --- a/lib/msun/src/e_pow.c +++ b/lib/msun/src/e_pow.c @@ -19,20 +19,20 @@ __FBSDID("$FreeBSD$"); * 1. Compute and return log2(x) in two pieces: * log2(x) = w1 + w2, * where w1 has 53-24 = 29 bit trailing zeros. - * 2. Perform y*log2(x) = n+y' by simulating muti-precision + * 2. Perform y*log2(x) = n+y' by simulating multi-precision * arithmetic, where |y'|<=0.5. * 3. Return x**y = 2**n*exp(y'*log2) * * Special cases: * 1. (anything) ** 0 is 1 * 2. (anything) ** 1 is itself - * 3. (anything) ** NAN is NAN + * 3. (anything) ** NAN is NAN except 1 ** NAN = 1 * 4. NAN ** (anything except 0) is NAN * 5. +-(|x| > 1) ** +INF is +INF * 6. +-(|x| > 1) ** -INF is +0 * 7. +-(|x| < 1) ** +INF is +0 * 8. +-(|x| < 1) ** -INF is +INF - * 9. +-1 ** +-INF is NAN + * 9. +-1 ** +-INF is 1 * 10. +0 ** (+anything except 0, NAN) is +0 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 * 12. +0 ** (-anything except 0, NAN) is +INF @@ -141,7 +141,7 @@ __ieee754_pow(double x, double y) if(ly==0) { if (iy==0x7ff00000) { /* y is +-inf */ if(((ix-0x3ff00000)|lx)==0) - return one; /* (-1)**+-inf is NaN */ + return one; /* (-1)**+-inf is 1 */ else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ return (hy>=0)? y: zero; else /* (|x|<1)**-,+inf = inf,0 */ diff --git a/lib/msun/src/e_sinh.c b/lib/msun/src/e_sinh.c index 17442d0..6c01f4a 100644 --- a/lib/msun/src/e_sinh.c +++ b/lib/msun/src/e_sinh.c @@ -32,6 +32,8 @@ __FBSDID("$FreeBSD$"); * only sinh(0)=0 is exact for finite x. */ +#include <float.h> + #include "math.h" #include "math_private.h" @@ -71,3 +73,7 @@ __ieee754_sinh(double x) /* |x| > overflowthresold, sinh(x) overflow */ return x*shuge; } + +#if (LDBL_MANT_DIG == 53) +__weak_reference(sinh, sinhl); +#endif diff --git a/lib/msun/src/e_sinhl.c b/lib/msun/src/e_sinhl.c new file mode 100644 index 0000000..ce7e333 --- /dev/null +++ b/lib/msun/src/e_sinhl.c @@ -0,0 +1,131 @@ +/* from: FreeBSD: head/lib/msun/src/e_sinhl.c XXX */ + +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include <sys/cdefs.h> +__FBSDID("$FreeBSD$"); + +/* + * See e_sinh.c for complete comments. + * + * Converted to long double by Bruce D. Evans. + */ + +#include <float.h> +#ifdef __i386__ +#include <ieeefp.h> +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" +#include "k_expl.h" + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const long double shuge = 0x1p16383L; +#if LDBL_MANT_DIG == 64 +/* + * Domain [-1, 1], range ~[-6.6749e-22, 6.6749e-22]: + * |sinh(x)/x - s(x)| < 2**-70.3 + */ +static const union IEEEl2bits +S3u = LD80C(0xaaaaaaaaaaaaaaaa, -3, 1.66666666666666666658e-1L); +#define S3 S3u.e +static const double +S5 = 8.3333333333333332e-3, /* 0x11111111111111.0p-59 */ +S7 = 1.9841269841270074e-4, /* 0x1a01a01a01a070.0p-65 */ +S9 = 2.7557319223873889e-6, /* 0x171de3a5565fe6.0p-71 */ +S11 = 2.5052108406704084e-8, /* 0x1ae6456857530f.0p-78 */ +S13 = 1.6059042748655297e-10, /* 0x161245fa910697.0p-85 */ +S15 = 7.6470006914396920e-13, /* 0x1ae7ce4eff2792.0p-93 */ +S17 = 2.8346142308424267e-15; /* 0x19882ce789ffc6.0p-101 */ +#elif LDBL_MANT_DIG == 113 +/* + * Domain [-1, 1], range ~[-2.9673e-36, 2.9673e-36]: + * |sinh(x)/x - s(x)| < 2**-118.0 + */ +static const long double +S3 = 1.66666666666666666666666666666666033e-1L, /* 0x1555555555555555555555555553b.0p-115L */ +S5 = 8.33333333333333333333333333337643193e-3L, /* 0x111111111111111111111111180f5.0p-119L */ +S7 = 1.98412698412698412698412697391263199e-4L, /* 0x1a01a01a01a01a01a01a0176aad11.0p-125L */ +S9 = 2.75573192239858906525574406205464218e-6L, /* 0x171de3a556c7338faac243aaa9592.0p-131L */ +S11 = 2.50521083854417187749675637460977997e-8L, /* 0x1ae64567f544e38fe59b3380d7413.0p-138L */ +S13 = 1.60590438368216146368737762431552702e-10L, /* 0x16124613a86d098059c7620850fc2.0p-145L */ +S15 = 7.64716373181980539786802470969096440e-13L, /* 0x1ae7f3e733b814193af09ce723043.0p-153L */ +S17 = 2.81145725434775409870584280722701574e-15L; /* 0x1952c77030c36898c3fd0b6dfc562.0p-161L */ +static const double +S19= 8.2206352435411005e-18, /* 0x12f49b4662b86d.0p-109 */ +S21= 1.9572943931418891e-20, /* 0x171b8f2fab9628.0p-118 */ +S23 = 3.8679983530666939e-23, /* 0x17617002b73afc.0p-127 */ +S25 = 6.5067867911512749e-26; /* 0x1423352626048a.0p-136 */ +#else +#error "Unsupported long double format" +#endif /* LDBL_MANT_DIG == 64 */ + +/* log(2**16385 - 0.5) rounded up: */ +static const float +o_threshold = 1.13572168e4; /* 0xb174de.0p-10 */ + +long double +sinhl(long double x) +{ + long double hi,lo,x2,x4; + double dx2,s; + int16_t ix,jx; + + GET_LDBL_EXPSIGN(jx,x); + ix = jx&0x7fff; + + /* x is INF or NaN */ + if(ix>=0x7fff) return x+x; + + ENTERI(); + + s = 1; + if (jx<0) s = -1; + + /* |x| < 64, return x, s(x), or accurate s*(exp(|x|)/2-1/exp(|x|)/2) */ + if (ix<0x4005) { /* |x|<64 */ + if (ix<BIAS-(LDBL_MANT_DIG+1)/2) /* |x|<TINY */ + if(shuge+x>1) RETURNI(x); /* sinh(tiny) = tiny with inexact */ + if (ix<0x3fff) { /* |x|<1 */ + x2 = x*x; +#if LDBL_MANT_DIG == 64 + x4 = x2*x2; + RETURNI(((S17*x2 + S15)*x4 + (S13*x2 + S11))*(x2*x*x4*x4) + + ((S9*x2 + S7)*x2 + S5)*(x2*x*x2) + S3*(x2*x) + x); +#elif LDBL_MANT_DIG == 113 + dx2 = x2; + RETURNI(((((((((((S25*dx2 + S23)*dx2 + + S21)*x2 + S19)*x2 + + S17)*x2 + S15)*x2 + S13)*x2 + S11)*x2 + S9)*x2 + S7)*x2 + + S5)* (x2*x*x2) + + S3*(x2*x) + x); +#endif + } + k_hexpl(fabsl(x), &hi, &lo); + RETURNI(s*(lo - 0.25/(hi + lo) + hi)); + } + + /* |x| in [64, o_threshold], return correctly-overflowing s*exp(|x|)/2 */ + if (fabsl(x) <= o_threshold) + RETURNI(s*hexpl(fabsl(x))); + + /* |x| > o_threshold, sinh(x) overflow */ + return x*shuge; +} diff --git a/lib/msun/src/imprecise.c b/lib/msun/src/imprecise.c index a7503bf..92fb2d0 100644 --- a/lib/msun/src/imprecise.c +++ b/lib/msun/src/imprecise.c @@ -60,10 +60,5 @@ DECLARE_WEAK(powl); long double imprecise_ ## f ## l(long double v) { return f(v); }\ DECLARE_WEAK(f ## l) -DECLARE_IMPRECISE(cosh); -DECLARE_IMPRECISE(erfc); -DECLARE_IMPRECISE(erf); DECLARE_IMPRECISE(lgamma); -DECLARE_IMPRECISE(sinh); -DECLARE_IMPRECISE(tanh); DECLARE_IMPRECISE(tgamma); diff --git a/lib/msun/src/math.h b/lib/msun/src/math.h index 1bd931c..3ab76f8 100644 --- a/lib/msun/src/math.h +++ b/lib/msun/src/math.h @@ -451,7 +451,10 @@ long double atanl(long double); long double cbrtl(long double); long double ceill(long double); long double copysignl(long double, long double) __pure2; +long double coshl(long double); long double cosl(long double); +long double erfcl(long double); +long double erfl(long double); long double exp2l(long double); long double expl(long double); long double expm1l(long double); @@ -466,6 +469,7 @@ long double frexpl(long double value, int *); /* fundamentally !__pure2 */ long double hypotl(long double, long double); int ilogbl(long double) __pure2; long double ldexpl(long double, int); +long double lgammal(long double); long long llrintl(long double); long long llroundl(long double); long double log10l(long double); @@ -482,45 +486,22 @@ long double nextafterl(long double, long double); double nexttoward(double, long double); float nexttowardf(float, long double); long double nexttowardl(long double, long double); +long double powl(long double, long double); long double remainderl(long double, long double); long double remquol(long double, long double, int *); long double rintl(long double); long double roundl(long double); long double scalblnl(long double, long); long double scalbnl(long double, int); +long double sinhl(long double); long double sinl(long double); long double sqrtl(long double); +long double tanhl(long double); long double tanl(long double); +long double tgammal(long double); long double truncl(long double); #endif /* __ISO_C_VISIBLE >= 1999 */ __END_DECLS #endif /* !_MATH_H_ */ - -/* separate header for cmath */ -#ifndef _MATH_EXTRA_H_ -#if __ISO_C_VISIBLE >= 1999 -#if _DECLARE_C99_LDBL_MATH - -#define _MATH_EXTRA_H_ - -/* - * extra long double versions of math functions for C99 and cmath - */ -__BEGIN_DECLS - -long double coshl(long double); -long double erfcl(long double); -long double erfl(long double); -long double lgammal(long double); -long double powl(long double, long double); -long double sinhl(long double); -long double tanhl(long double); -long double tgammal(long double); - -__END_DECLS - -#endif /* !_DECLARE_C99_LDBL_MATH */ -#endif /* __ISO_C_VISIBLE >= 1999 */ -#endif /* !_MATH_EXTRA_H_ */ diff --git a/lib/msun/src/s_erf.c b/lib/msun/src/s_erf.c index 854767b..e1d63bc 100644 --- a/lib/msun/src/s_erf.c +++ b/lib/msun/src/s_erf.c @@ -111,18 +111,25 @@ __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" +/* XXX Prevent compilers from erroneously constant folding: */ +static const volatile double tiny= 1e-300; + static const double -tiny = 1e-300, -half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ -one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ -two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ - /* c = (float)0.84506291151 */ +half= 0.5, +one = 1, +two = 2, +/* c = (float)0.84506291151 */ erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ /* - * Coefficients for approximation to erf on [0,0.84375] + * In the domain [0, 2**-28], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-84. */ efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ +/* + * Coefficients for approximation to erf on [0,0.84375] + */ pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ @@ -134,7 +141,7 @@ qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ /* - * Coefficients for approximation to erf in [0.84375,1.25] + * Coefficients for approximation to erf in [0.84375,1.25] */ pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ @@ -150,7 +157,7 @@ qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ /* - * Coefficients for approximation to erfc in [1.25,1/0.35] + * Coefficients for approximation to erfc in [1.25,1/0.35] */ ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ @@ -169,7 +176,7 @@ sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ /* - * Coefficients for approximation to erfc in [1/.35,28] + * Coefficients for approximation to erfc in [1/.35,28] */ rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ @@ -222,15 +229,12 @@ erf(double x) x = fabs(x); s = one/(x*x); if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ - R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( - ra5+s*(ra6+s*ra7)))))); - S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( - sa5+s*(sa6+s*(sa7+s*sa8))))))); + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*sa8))))))); } else { /* |x| >= 1/0.35 */ - R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( - rb5+s*rb6))))); - S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( - sb5+s*(sb6+s*sb7)))))); + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); } z = x; SET_LOW_WORD(z,0); @@ -238,6 +242,10 @@ erf(double x) if(hx>=0) return one-r/x; else return r/x-one; } +#if (LDBL_MANT_DIG == 53) +__weak_reference(erf, erfl); +#endif + double erfc(double x) { @@ -279,23 +287,23 @@ erfc(double x) x = fabs(x); s = one/(x*x); if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ - R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( - ra5+s*(ra6+s*ra7)))))); - S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( - sa5+s*(sa6+s*(sa7+s*sa8))))))); + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ + s*sa8))))))); } else { /* |x| >= 1/.35 ~ 2.857143 */ if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ - R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( - rb5+s*rb6))))); - S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( - sb5+s*(sb6+s*sb7)))))); + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); } z = x; SET_LOW_WORD(z,0); - r = __ieee754_exp(-z*z-0.5625)* - __ieee754_exp((z-x)*(z+x)+R/S); + r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); if(hx>0) return r/x; else return two-r/x; } else { if(hx>0) return tiny*tiny; else return two-tiny; } } + +#if (LDBL_MANT_DIG == 53) +__weak_reference(erfc, erfcl); +#endif diff --git a/lib/msun/src/s_erff.c b/lib/msun/src/s_erff.c index b97ca1d..d6cfbd2 100644 --- a/lib/msun/src/s_erff.c +++ b/lib/msun/src/s_erff.c @@ -19,64 +19,63 @@ __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" +/* XXX Prevent compilers from erroneously constant folding: */ +static const volatile float tiny = 1e-30; + static const float -tiny = 1e-30, -half= 5.0000000000e-01, /* 0x3F000000 */ -one = 1.0000000000e+00, /* 0x3F800000 */ -two = 2.0000000000e+00, /* 0x40000000 */ +half= 0.5, +one = 1, +two = 2, +erx = 8.42697144e-01, /* 0x3f57bb00 */ /* - * Coefficients for approximation to erf on [0,0.84375] + * In the domain [0, 2**-14], only the first term in the power series + * expansion of erf(x) is used. The magnitude of the first neglected + * terms is less than 2**-42. */ -efx = 1.2837916613e-01, /* 0x3e0375d4 */ -efx8= 1.0270333290e+00, /* 0x3f8375d4 */ +efx = 1.28379166e-01, /* 0x3e0375d4 */ +efx8= 1.02703333e+00, /* 0x3f8375d4 */ /* - * Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]: - * |(erf(x) - x)/x - p(x)/q(x)| < 2**-31. + * Domain [0, 0.84375], range ~[-5.4419e-10, 5.5179e-10]: + * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-31 */ -pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */ -pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */ -pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */ -qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */ -qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */ -qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */ +pp0 = 1.28379166e-01, /* 0x3e0375d4 */ +pp1 = -3.36030394e-01, /* 0xbeac0c2d */ +pp2 = -1.86261395e-03, /* 0xbaf422f4 */ +qq1 = 3.12324315e-01, /* 0x3e9fe8f9 */ +qq2 = 2.16070414e-02, /* 0x3cb10140 */ +qq3 = -1.98859372e-03, /* 0xbb025311 */ /* - * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]: - * |(erf(x) - erx) - p(x)/q(x)| < 2**-36. + * Domain [0.84375, 1.25], range ~[-1.023e-9, 1.023e-9]: + * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-31 */ -erx = 8.42697144e-01F, /* 0x1.af7600p-1. erf(1) rounded to 16 bits. */ -pa0 = 3.64939137e-06F, /* 0x1.e9d022p-19 */ -pa1 = 4.15109694e-01F, /* 0x1.a91284p-2 */ -pa2 = -1.65179938e-01F, /* -0x1.5249dcp-3 */ -pa3 = 1.10914491e-01F, /* 0x1.c64e46p-4 */ -qa1 = 6.02074385e-01F, /* 0x1.344318p-1 */ -qa2 = 5.35934687e-01F, /* 0x1.126608p-1 */ -qa3 = 1.68576106e-01F, /* 0x1.593e6ep-3 */ -qa4 = 5.62181212e-02F, /* 0x1.cc89f2p-5 */ +pa0 = 3.65041046e-06, /* 0x3674f993 */ +pa1 = 4.15109307e-01, /* 0x3ed48935 */ +pa2 = -2.09395722e-01, /* 0xbe566bd5 */ +pa3 = 8.67677554e-02, /* 0x3db1b34b */ +qa1 = 4.95560974e-01, /* 0x3efdba2b */ +qa2 = 3.71248513e-01, /* 0x3ebe1449 */ +qa3 = 3.92478965e-02, /* 0x3d20c267 */ /* - * Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]: - * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30 + * Domain [1.25,1/0.35], range ~[-4.821e-9, 4.927e-9]: + * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-28 */ -ra0 = -9.87132732e-03F, /* -0x1.4376b2p-7 */ -ra1 = -5.53605914e-01F, /* -0x1.1b723cp-1 */ -ra2 = -2.17589188e+00F, /* -0x1.1683a0p+1 */ -ra3 = -1.43268085e+00F, /* -0x1.6ec42cp+0 */ -sa1 = 5.45995426e+00F, /* 0x1.5d6fe4p+2 */ -sa2 = 6.69798088e+00F, /* 0x1.acabb8p+2 */ -sa3 = 1.43113089e+00F, /* 0x1.6e5e98p+0 */ -sa4 = -5.77397496e-02F, /* -0x1.d90108p-5 */ +ra0 = -9.88156721e-03, /* 0xbc21e64c */ +ra1 = -5.43658376e-01, /* 0xbf0b2d32 */ +ra2 = -1.66828310e+00, /* 0xbfd58a4d */ +ra3 = -6.91554189e-01, /* 0xbf3109b2 */ +sa1 = 4.48581553e+00, /* 0x408f8bcd */ +sa2 = 4.10799170e+00, /* 0x408374ab */ +sa3 = 5.53855181e-01, /* 0x3f0dc974 */ /* - * Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]: - * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42 + * Domain [2.85715, 11], range ~[-1.484e-9, 1.505e-9]: + * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-30 */ -rb0 = -9.86494310e-03F, /* -0x1.434124p-7 */ -rb1 = -6.25171244e-01F, /* -0x1.401672p-1 */ -rb2 = -6.16498327e+00F, /* -0x1.8a8f16p+2 */ -rb3 = -1.66696873e+01F, /* -0x1.0ab70ap+4 */ -rb4 = -9.53764343e+00F, /* -0x1.313460p+3 */ -sb1 = 1.26884899e+01F, /* 0x1.96081cp+3 */ -sb2 = 4.51839523e+01F, /* 0x1.6978bcp+5 */ -sb3 = 4.72810211e+01F, /* 0x1.7a3f88p+5 */ -sb4 = 8.93033314e+00F; /* 0x1.1dc54ap+3 */ +rb0 = -9.86496918e-03, /* 0xbc21a0ae */ +rb1 = -5.48049808e-01, /* 0xbf0c4cfe */ +rb2 = -1.84115684e+00, /* 0xbfebab07 */ +sb1 = 4.87132740e+00, /* 0x409be1ea */ +sb2 = 3.04982710e+00, /* 0x4043305e */ +sb3 = -7.61900663e-01; /* 0xbf430bec */ float erff(float x) @@ -85,9 +84,9 @@ erff(float x) float R,S,P,Q,s,y,z,r; GET_FLOAT_WORD(hx,x); ix = hx&0x7fffffff; - if(ix>=0x7f800000) { /* erf(nan)=nan */ + if(ix>=0x7f800000) { /* erff(nan)=nan */ i = ((u_int32_t)hx>>31)<<1; - return (float)(1-i)+one/x; /* erf(+-inf)=+-1 */ + return (float)(1-i)+one/x; /* erff(+-inf)=+-1 */ } if(ix < 0x3f580000) { /* |x|<0.84375 */ @@ -105,7 +104,7 @@ erff(float x) if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ s = fabsf(x)-one; P = pa0+s*(pa1+s*(pa2+s*pa3)); - Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4))); + Q = one+s*(qa1+s*(qa2+s*qa3)); if(hx>=0) return erx + P/Q; else return -erx - P/Q; } if (ix >= 0x40800000) { /* inf>|x|>=4 */ @@ -113,12 +112,12 @@ erff(float x) } x = fabsf(x); s = one/(x*x); - if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */ + if(ix< 0x4036db8c) { /* |x| < 2.85715 ~ 1/0.35 */ R=ra0+s*(ra1+s*(ra2+s*ra3)); - S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4))); - } else { /* |x| >= 1/0.35 */ - R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4))); - S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4))); + S=one+s*(sa1+s*(sa2+s*sa3)); + } else { /* |x| >= 2.85715 ~ 1/0.35 */ + R=rb0+s*(rb1+s*rb2); + S=one+s*(sb1+s*(sb2+s*sb3)); } SET_FLOAT_WORD(z,hx&0xffffe000); r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S); @@ -132,8 +131,8 @@ erfcf(float x) float R,S,P,Q,s,y,z,r; GET_FLOAT_WORD(hx,x); ix = hx&0x7fffffff; - if(ix>=0x7f800000) { /* erfc(nan)=nan */ - /* erfc(+-inf)=0,2 */ + if(ix>=0x7f800000) { /* erfcf(nan)=nan */ + /* erfcf(+-inf)=0,2 */ return (float)(((u_int32_t)hx>>31)<<1)+one/x; } @@ -155,7 +154,7 @@ erfcf(float x) if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ s = fabsf(x)-one; P = pa0+s*(pa1+s*(pa2+s*pa3)); - Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4))); + Q = one+s*(qa1+s*(qa2+s*qa3)); if(hx>=0) { z = one-erx; return z - P/Q; } else { @@ -165,13 +164,13 @@ erfcf(float x) if (ix < 0x41300000) { /* |x|<11 */ x = fabsf(x); s = one/(x*x); - if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/ - R=ra0+s*(ra1+s*(ra2+s*ra3)); - S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4))); - } else { /* |x| >= 1/.35 ~ 2.857143 */ + if(ix< 0x4036db8c) { /* |x| < 2.85715 ~ 1/.35 */ + R=ra0+s*(ra1+s*(ra2+s*ra3)); + S=one+s*(sa1+s*(sa2+s*sa3)); + } else { /* |x| >= 2.85715 ~ 1/.35 */ if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */ - R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4))); - S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4))); + R=rb0+s*(rb1+s*rb2); + S=one+s*(sb1+s*(sb2+s*sb3)); } SET_FLOAT_WORD(z,hx&0xffffe000); r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S); diff --git a/lib/msun/src/s_round.c b/lib/msun/src/s_round.c index 65de31b..fab3019 100644 --- a/lib/msun/src/s_round.c +++ b/lib/msun/src/s_round.c @@ -27,25 +27,34 @@ #include <sys/cdefs.h> __FBSDID("$FreeBSD$"); -#include <math.h> +#include <float.h> + +#include "math.h" +#include "math_private.h" double round(double x) { double t; + uint32_t hx; - if (!isfinite(x)) - return (x); + GET_HIGH_WORD(hx, x); + if ((hx & 0x7fffffff) == 0x7ff00000) + return (x + x); - if (x >= 0.0) { + if (!(hx & 0x80000000)) { t = floor(x); if (t - x <= -0.5) - t += 1.0; + t += 1; return (t); } else { t = floor(-x); if (t + x <= -0.5) - t += 1.0; + t += 1; return (-t); } } + +#if (LDBL_MANT_DIG == 53) +__weak_reference(round, roundl); +#endif diff --git a/lib/msun/src/s_roundf.c b/lib/msun/src/s_roundf.c index 952e8e7..e7e2eb9 100644 --- a/lib/msun/src/s_roundf.c +++ b/lib/msun/src/s_roundf.c @@ -27,25 +27,28 @@ #include <sys/cdefs.h> __FBSDID("$FreeBSD$"); -#include <math.h> +#include "math.h" +#include "math_private.h" float roundf(float x) { float t; + uint32_t hx; - if (!isfinite(x)) - return (x); + GET_FLOAT_WORD(hx, x); + if ((hx & 0x7fffffff) == 0x7f800000) + return (x + x); - if (x >= 0.0) { + if (!(hx & 0x80000000)) { t = floorf(x); - if (t - x <= -0.5) - t += 1.0; + if (t - x <= -0.5F) + t += 1; return (t); } else { t = floorf(-x); - if (t + x <= -0.5) - t += 1.0; + if (t + x <= -0.5F) + t += 1; return (-t); } } diff --git a/lib/msun/src/s_roundl.c b/lib/msun/src/s_roundl.c index a70b617..2d15e13 100644 --- a/lib/msun/src/s_roundl.c +++ b/lib/msun/src/s_roundl.c @@ -27,25 +27,36 @@ #include <sys/cdefs.h> __FBSDID("$FreeBSD$"); -#include <math.h> +#include <float.h> +#ifdef __i386__ +#include <ieeefp.h> +#endif + +#include "fpmath.h" +#include "math.h" +#include "math_private.h" long double roundl(long double x) { long double t; + uint16_t hx; + + GET_LDBL_EXPSIGN(hx, x); + if ((hx & 0x7fff) == 0x7fff) + return (x + x); - if (!isfinite(x)) - return (x); + ENTERI(); - if (x >= 0.0) { + if (!(hx & 0x8000)) { t = floorl(x); - if (t - x <= -0.5) - t += 1.0; - return (t); + if (t - x <= -0.5L) + t += 1; + RETURNI(t); } else { t = floorl(-x); - if (t + x <= -0.5) - t += 1.0; - return (-t); + if (t + x <= -0.5L) + t += 1; + RETURNI(-t); } } diff --git a/lib/msun/src/s_tanh.c b/lib/msun/src/s_tanh.c index 96e3565..6d26c69 100644 --- a/lib/msun/src/s_tanh.c +++ b/lib/msun/src/s_tanh.c @@ -37,10 +37,13 @@ __FBSDID("$FreeBSD$"); * only tanh(0)=0 is exact for finite argument. */ +#include <float.h> + #include "math.h" #include "math_private.h" -static const double one = 1.0, two = 2.0, tiny = 1.0e-300, huge = 1.0e300; +static const volatile double tiny = 1.0e-300; +static const double one = 1.0, two = 2.0, huge = 1.0e300; double tanh(double x) @@ -75,3 +78,7 @@ tanh(double x) } return (jx>=0)? z: -z; } + +#if (LDBL_MANT_DIG == 53) +__weak_reference(tanh, tanhl); +#endif diff --git a/lib/msun/src/s_tanhf.c b/lib/msun/src/s_tanhf.c index 04f09c6..f537be4 100644 --- a/lib/msun/src/s_tanhf.c +++ b/lib/msun/src/s_tanhf.c @@ -19,7 +19,9 @@ __FBSDID("$FreeBSD$"); #include "math.h" #include "math_private.h" -static const float one=1.0, two=2.0, tiny = 1.0e-30, huge = 1.0e30; +static const volatile float tiny = 1.0e-30; +static const float one=1.0, two=2.0, huge = 1.0e30; + float tanhf(float x) { diff --git a/lib/msun/src/s_tanhl.c b/lib/msun/src/s_tanhl.c new file mode 100644 index 0000000..886158b --- /dev/null +++ b/lib/msun/src/s_tanhl.c @@ -0,0 +1,172 @@ +/* from: FreeBSD: head/lib/msun/src/s_tanhl.c XXX */ + +/* @(#)s_tanh.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +#include <sys/cdefs.h> +__FBSDID("$FreeBSD$"); + +/* + * See s_tanh.c for complete comments. + * + * Converted to long double by Bruce D. Evans. + */ + +#include <float.h> +#ifdef __i386__ +#include <ieeefp.h> +#endif + +#include "math.h" +#include "math_private.h" +#include "fpmath.h" +#include "k_expl.h" + +#if LDBL_MAX_EXP != 0x4000 +/* We also require the usual expsign encoding. */ +#error "Unsupported long double format" +#endif + +#define BIAS (LDBL_MAX_EXP - 1) + +static const volatile double tiny = 1.0e-300; +static const double one = 1.0; +#if LDBL_MANT_DIG == 64 +/* + * Domain [-0.25, 0.25], range ~[-1.6304e-22, 1.6304e-22]: + * |tanh(x)/x - t(x)| < 2**-72.3 + */ +static const union IEEEl2bits +T3u = LD80C(0xaaaaaaaaaaaaaa9f, -2, -3.33333333333333333017e-1L); +#define T3 T3u.e +static const double +T5 = 1.3333333333333314e-1, /* 0x1111111111110a.0p-55 */ +T7 = -5.3968253968210485e-2, /* -0x1ba1ba1ba1a1a1.0p-57 */ +T9 = 2.1869488531393817e-2, /* 0x1664f488172022.0p-58 */ +T11 = -8.8632352345964591e-3, /* -0x1226e34bc138d5.0p-59 */ +T13 = 3.5921169709993771e-3, /* 0x1d6d371d3e400f.0p-61 */ +T15 = -1.4555786415756001e-3, /* -0x17d923aa63814d.0p-62 */ +T17 = 5.8645267876296793e-4, /* 0x13378589b85aa7.0p-63 */ +T19 = -2.1121033571392224e-4; /* -0x1baf0af80c4090.0p-65 */ +#elif LDBL_MANT_DIG == 113 +/* + * Domain [-0.25, 0.25], range ~[-2.4211e-37, 2.4211e-37]: + * |tanh(x)/x - t(x)| < 2**121.6 + */ +static const long double +T3 = -3.33333333333333333333333333333332980e-1L, /* -0x1555555555555555555555555554e.0p-114L */ +T5 = 1.33333333333333333333333333332707260e-1L, /* 0x1111111111111111111111110ab7b.0p-115L */ +T7 = -5.39682539682539682539682535723482314e-2L, /* -0x1ba1ba1ba1ba1ba1ba1ba17b5fc98.0p-117L */ +T9 = 2.18694885361552028218693591149061717e-2L, /* 0x1664f4882c10f9f32d6b1a12a25e5.0p-118L */ +T11 = -8.86323552990219656883762347736381851e-3L, /* -0x1226e355e6c23c8f5a5a0f386cb4d.0p-119L */ +T13 = 3.59212803657248101358314398220822722e-3L, /* 0x1d6d3d0e157ddfb403ad3637442c6.0p-121L */ +T15 = -1.45583438705131796512568010348874662e-3L; /* -0x17da36452b75e150c44cc34253b34.0p-122L */ +static const double +T17 = 5.9002744094556621e-4, /* 0x1355824803668e.0p-63 */ +T19 = -2.3912911424260516e-4, /* -0x1f57d7734c8dde.0p-65 */ +T21 = 9.6915379535512898e-5, /* 0x1967e18ad6a6ca.0p-66 */ +T23 = -3.9278322983156353e-5, /* -0x1497d8e6b75729.0p-67 */ +T25 = 1.5918887220143869e-5, /* 0x10b1319998cafa.0p-68 */ +T27 = -6.4514295231630956e-6, /* -0x1b0f2b71b218eb.0p-70 */ +T29 = 2.6120754043964365e-6, /* 0x15e963a3cf3a39.0p-71 */ +T31 = -1.0407567231003314e-6, /* -0x1176041e656869.0p-72 */ +T33 = 3.4744117554063574e-7; /* 0x1750fe732cab9c.0p-74 */ +#endif /* LDBL_MANT_DIG == 64 */ + +static inline long double +divl(long double a, long double b, long double c, long double d, + long double e, long double f) +{ + long double inv, r; + float fr, fw; + + _2sumF(a, c); + b = b + c; + _2sumF(d, f); + e = e + f; + + inv = 1 / (d + e); + + r = (a + b) * inv; + fr = r; + r = fr; + + fw = d + e; + e = d - fw + e; + d = fw; + + r = r + (a - d * r + b - e * r) * inv; + + return r; +} + +long double +tanhl(long double x) +{ + long double hi,lo,s,x2,x4,z; + double dx2; + int16_t jx,ix; + + GET_LDBL_EXPSIGN(jx,x); + ix = jx&0x7fff; + + /* x is INF or NaN */ + if(ix>=0x7fff) { + if (jx>=0) return one/x+one; /* tanh(+-inf)=+-1 */ + else return one/x-one; /* tanh(NaN) = NaN */ + } + + ENTERI(); + + /* |x| < 40 */ + if (ix < 0x4004 || fabsl(x) < 40) { /* |x|<40 */ + if (__predict_false(ix<BIAS-(LDBL_MANT_DIG+1)/2)) { /* |x|<TINY */ + /* tanh(+-0) = +0; tanh(tiny) = tiny(-+) with inexact: */ + return (x == 0 ? x : (0x1p200 * x - x) * 0x1p-200); + } + if (ix<0x3ffd) { /* |x|<0.25 */ + x2 = x*x; +#if LDBL_MANT_DIG == 64 + x4 = x2*x2; + RETURNI(((T19*x2 + T17)*x4 + (T15*x2 + T13))*(x2*x*x2*x4*x4) + + ((T11*x2 + T9)*x4 + (T7*x2 + T5))*(x2*x*x2) + + T3*(x2*x) + x); +#elif LDBL_MANT_DIG == 113 + dx2 = x2; +#if 0 + RETURNI(((((((((((((((T33*dx2 + T31)*dx2 + T29)*dx2 + T27)*dx2 + + T25)*x2 + T23)*x2 + T21)*x2 + T19)*x2 + T17)*x2 + + T15)*x2 + T13)*x2 + T11)*x2 + T9)*x2 + T7)*x2 + T5)* + (x2*x*x2) + + T3*(x2*x) + x); +#else + long double q = ((((((((((((((T33*dx2 + T31)*dx2 + T29)*dx2 + T27)*dx2 + + T25)*x2 + T23)*x2 + T21)*x2 + T19)*x2 + T17)*x2 + + T15)*x2 + T13)*x2 + T11)*x2 + T9)*x2 + T7)*x2 + T5)* + (x2*x*x2); + RETURNI(q + T3*(x2*x) + x); +#endif +#endif + } + k_hexpl(2*fabsl(x), &hi, &lo); + if (ix<0x4001 && fabsl(x) < 1.5) /* |x|<1.5 */ + z = divl(hi, lo, -0.5, hi, lo, 0.5); + else + z = one - one/(lo+0.5+hi); + /* |x| >= 40, return +-1 */ + } else { + z = one - tiny; /* raise inexact flag */ + } + s = 1; + if (jx<0) s = -1; + RETURNI(s*z); +} |
