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bbae23a725
HardenedBSD/lib/msun/ld128/k_expl.h
Ed Schouten 2cec876a59 Rename cpack*() to CMPLX*(). 
The C11 standard introduced a set of macros (CMPLX, CMPLXF, CMPLXL) that
can be used to construct complex numbers from a pair of real and
imaginary numbers. Unfortunately, they require some compiler support,
which is why we only define them for Clang and GCC>=4.7.

The cpack() function in libm performs the same task as CMPLX(), but
cannot be used to generate compile-time constants. This means that all
invocations of cpack() can safely be replaced by C11's CMPLX(). To keep
the code building with GCC 4.2, provide copies of CMPLX() that can at
least be used to generate run-time complex numbers.

This makes it easier to build some of the functions outside of libm.
2014-12-16 09:21:56 +00:00

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/* 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 (CMPLXL(cos(y) * exp_x * scale1 * scale2,
sinl(y) * exp_x * scale1 * scale2));
}
#endif /* _COMPLEX_H */
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