Index: head/lib/msun/ld128/e_rem_pio2l.h =================================================================== --- head/lib/msun/ld128/e_rem_pio2l.h (revision 336544) +++ head/lib/msun/ld128/e_rem_pio2l.h (revision 336545) @@ -1,140 +1,135 @@ /* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * * Optimized by Bruce D. Evans. */ #include __FBSDID("$FreeBSD$"); /* ld128 version of __ieee754_rem_pio2l(x,y) * * return the remainder of x rem pi/2 in y[0]+y[1] * use __kernel_rem_pio2() */ #include #include "math.h" #include "math_private.h" #include "fpmath.h" #define BIAS (LDBL_MAX_EXP - 1) /* * XXX need to verify that nonzero integer multiples of pi/2 within the * range get no closer to a long double than 2**-140, or that * ilogb(x) + ilogb(min_delta) < 45 - -140. */ /* * invpio2: 113 bits of 2/pi * pio2_1: first 68 bits of pi/2 * pio2_1t: pi/2 - pio2_1 * pio2_2: second 68 bits of pi/2 * pio2_2t: pi/2 - (pio2_1+pio2_2) * pio2_3: third 68 bits of pi/2 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) */ static const double zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ two24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */ static const long double invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */ pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */ pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */ pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */ pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */ pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */ pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */ static inline __always_inline int __ieee754_rem_pio2l(long double x, long double *y) { union IEEEl2bits u,u1; long double z,w,t,r,fn; double tx[5],ty[3]; int64_t n; int e0,ex,i,j,nx; int16_t expsign; u.e = x; expsign = u.xbits.expsign; ex = expsign & 0x7fff; if (ex < BIAS + 45 || ex == BIAS + 45 && u.bits.manh < 0x921fb54442d1LL) { /* |x| ~< 2^45*(pi/2), medium size */ - /* Use a specialized rint() to get fn. Assume round-to-nearest. */ - fn = x*invpio2+0x1.8p112; - fn = fn-0x1.8p112; -#ifdef HAVE_EFFICIENT_I64RINT + /* TODO: use only double precision for fn, as in expl(). */ + fn = rnintl(x * invpio2); n = i64rint(fn); -#else - n = fn; -#endif r = x-fn*pio2_1; w = fn*pio2_1t; /* 1st round good to 180 bit */ { union IEEEl2bits u2; int ex1; j = ex; y[0] = r-w; u2.e = y[0]; ex1 = u2.xbits.expsign & 0x7fff; i = j-ex1; if(i>51) { /* 2nd iteration needed, good to 248 */ t = r; w = fn*pio2_2; r = t-w; w = fn*pio2_2t-((t-r)-w); y[0] = r-w; u2.e = y[0]; ex1 = u2.xbits.expsign & 0x7fff; i = j-ex1; if(i>119) { /* 3rd iteration need, 316 bits acc */ t = r; /* will cover all possible cases */ w = fn*pio2_3; r = t-w; w = fn*pio2_3t-((t-r)-w); y[0] = r-w; } } } y[1] = (r-y[0])-w; return n; } /* * all other (large) arguments */ if(ex==0x7fff) { /* x is inf or NaN */ y[0]=y[1]=x-x; return 0; } /* set z = scalbn(|x|,ilogb(x)-23) */ u1.e = x; e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ u1.xbits.expsign = ex - e0; z = u1.e; for(i=0;i<4;i++) { tx[i] = (double)((int32_t)(z)); z = (z-tx[i])*two24; } tx[4] = z; nx = 5; while(tx[nx-1]==zero) nx--; /* skip zero term */ n = __kernel_rem_pio2(tx,ty,e0,nx,3); t = (long double)ty[2] + ty[1]; r = t + ty[0]; w = ty[0] - (r - t); if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} y[0] = r; y[1] = w; return n; } Index: head/lib/msun/ld128/k_expl.h =================================================================== --- head/lib/msun/ld128/k_expl.h (revision 336544) +++ head/lib/msun/ld128/k_expl.h (revision 336545) @@ -1,330 +1,322 @@ /* from: FreeBSD: head/lib/msun/ld128/s_expl.c 251345 2013-06-03 20:09:22Z kargl */ /*- * SPDX-License-Identifier: BSD-2-Clause-FreeBSD * * 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 __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) + fn = rnint((double)x * INV_L); 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 */ Index: head/lib/msun/ld128/s_expl.c =================================================================== --- head/lib/msun/ld128/s_expl.c (revision 336544) +++ head/lib/msun/ld128/s_expl.c (revision 336545) @@ -1,328 +1,323 @@ /*- * SPDX-License-Identifier: BSD-2-Clause-FreeBSD * * 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 __FBSDID("$FreeBSD$"); /* * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments. */ #include #include "fpmath.h" #include "math.h" #include "math_private.h" #include "k_expl.h" /* XXX Prevent compilers from erroneously constant folding these: */ static const volatile long double huge = 0x1p10000L, tiny = 0x1p-10000L; static const long double twom10000 = 0x1p-10000L; static const long double /* log(2**16384 - 0.5) rounded towards zero: */ /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ o_threshold = 11356.523406294143949491931077970763428L, /* log(2**(-16381-64-1)) rounded towards zero: */ u_threshold = -11433.462743336297878837243843452621503L; long double expl(long double x) { 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; ix = hx & 0x7fff; if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ if (ix == BIAS + LDBL_MAX_EXP) { if (hx & 0x8000) /* x is -Inf or -NaN */ RETURNP(-1 / x); RETURNP(x + x); /* x is +Inf or +NaN */ } if (x > o_threshold) RETURNP(huge * huge); if (x < u_threshold) RETURNP(tiny * tiny); } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ RETURN2P(1, x); /* 1 with inexact iff x != 0 */ } ENTERI(); 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 { SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); RETURNI(t * twopk * twom10000); } } /* * Our T1 and T2 are chosen to be approximately the points where method * A and method B have the same accuracy. Tang's T1 and T2 are the * points where method A's accuracy changes by a full bit. For Tang, * this drop in accuracy makes method A immediately less accurate than * method B, but our larger INTERVALS makes method A 2 bits more * accurate so it remains the most accurate method significantly * closer to the origin despite losing the full bit in our extended * range for it. * * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2]. * 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) */ T2 = 0.1659, /* ~30.625/128 * log(2) */ 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, C4 = 4.16666666666666666666666666666666645e-2L, C5 = 8.33333333333333333333333333333371638e-3L, C6 = 1.38888888888888888888888888891188658e-3L, C7 = 1.98412698412698412698412697235950394e-4L, C8 = 2.48015873015873015873015112487849040e-5L, C9 = 2.75573192239858906525606685484412005e-6L, C10 = 2.75573192239858906612966093057020362e-7L, 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 */ C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */ C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */ C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */ /* * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]: * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44 */ static const long double D3 = 1.66666666666666666666666666666682245e-1L, D4 = 4.16666666666666666666666666634228324e-2L, D5 = 8.33333333333333333333333364022244481e-3L, D6 = 1.38888888888888888888887138722762072e-3L, D7 = 1.98412698412698412699085805424661471e-4L, D8 = 2.48015873015873015687993712101479612e-5L, D9 = 2.75573192239858944101036288338208042e-6L, D10 = 2.75573192239853161148064676533754048e-7L, 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 */ D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */ D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */ long double expm1l(long double x) { union IEEEl2bits u, v; long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi; long double x_lo, x2; double dr, dx, fn, r2; int k, n, n2; uint16_t hx, ix; DOPRINT_START(&x); /* Filter out exceptional cases. */ u.e = x; hx = u.xbits.expsign; ix = hx & 0x7fff; if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ if (ix == BIAS + LDBL_MAX_EXP) { if (hx & 0x8000) /* x is -Inf or -NaN */ RETURNP(-1 / x - 1); RETURNP(x + x); /* x is +Inf or +NaN */ } if (x > o_threshold) RETURNP(huge * huge); /* * expm1l() never underflows, but it must avoid * unrepresentable large negative exponents. We used a * much smaller threshold for large |x| above than in * expl() so as to handle not so large negative exponents * in the same way as large ones here. */ if (hx & 0x8000) /* x <= -128 */ RETURN2P(tiny, -1); /* good for x < -114ln2 - eps */ } ENTERI(); if (T1 < x && x < T2) { x2 = x * x; dx = x; if (x < T3) { if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ /* x (rounded) with inexact if x != 0: */ RETURNPI(x == 0 ? x : (0x1p200 * x + fabsl(x)) * 0x1p-200); } q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + x * (C7 + x * (C8 + x * (C9 + x * (C10 + x * (C11 + x * (C12 + x * (C13 + dx * (C14 + dx * (C15 + dx * (C16 + dx * (C17 + dx * C18)))))))))))))); } else { q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 + x * (D7 + x * (D8 + x * (D9 + x * (D10 + x * (D11 + x * (D12 + x * (D13 + dx * (D14 + dx * (D15 + dx * (D16 + dx * D17))))))))))))); } x_hi = (float)x; x_lo = x - x_hi; hx2_hi = x_hi * x_hi / 2; hx2_lo = x_lo * (x + x_hi) / 2; if (ix >= BIAS - 7) RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); else RETURN2PI(x, hx2_lo + q + hx2_hi); } /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ - /* Use a specialized rint() to get fn. Assume round-to-nearest. */ - fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; -#if defined(HAVE_EFFICIENT_IRINT) + fn = rnint((double)x * INV_L); 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 factor. */ v.e = 1; v.xbits.expsign = BIAS + k; twopk = v.e; /* * Evaluate lower terms of * 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; if (k == 0) { t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1); RETURNI(t); } if (k == -1) { t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1); RETURNI(t / 2); } if (k < -7) { t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk - 1); } if (k > 2 * LDBL_MANT_DIG - 1) { 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); } v.xbits.expsign = BIAS - k; twomk = v.e; if (k > LDBL_MANT_DIG - 1) t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); else t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk); } Index: head/lib/msun/ld80/e_rem_pio2l.h =================================================================== --- head/lib/msun/ld80/e_rem_pio2l.h (revision 336544) +++ head/lib/msun/ld80/e_rem_pio2l.h (revision 336545) @@ -1,149 +1,143 @@ /* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * * Optimized by Bruce D. Evans. */ #include __FBSDID("$FreeBSD$"); /* ld80 version of __ieee754_rem_pio2l(x,y) * * return the remainder of x rem pi/2 in y[0]+y[1] * use __kernel_rem_pio2() */ #include #include "math.h" #include "math_private.h" #include "fpmath.h" #define BIAS (LDBL_MAX_EXP - 1) /* * invpio2: 64 bits of 2/pi * pio2_1: first 39 bits of pi/2 * pio2_1t: pi/2 - pio2_1 * pio2_2: second 39 bits of pi/2 * pio2_2t: pi/2 - (pio2_1+pio2_2) * pio2_3: third 39 bits of pi/2 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) */ static const double zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ #if defined(__amd64__) || defined(__i386__) /* Long double constants are slow on these arches, and broken on i386. */ static const volatile double invpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */ invpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */ pio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */ pio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */ pio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */ pio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */ pio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */ pio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */ #define invpio2 ((long double)invpio2hi + invpio2lo) #define pio2_1t ((long double)pio2_1thi + pio2_1tlo) #define pio2_2t ((long double)pio2_2thi + pio2_2tlo) #define pio2_3t ((long double)pio2_3thi + pio2_3tlo) #else static const long double invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ #endif static inline __always_inline int __ieee754_rem_pio2l(long double x, long double *y) { union IEEEl2bits u,u1; long double z,w,t,r,fn; double tx[3],ty[2]; int e0,ex,i,j,nx,n; int16_t expsign; u.e = x; expsign = u.xbits.expsign; ex = expsign & 0x7fff; if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) { /* |x| ~< 2^25*(pi/2), medium size */ - /* Use a specialized rint() to get fn. Assume round-to-nearest. */ - fn = x*invpio2+0x1.8p63; - fn = fn-0x1.8p63; -#ifdef HAVE_EFFICIENT_IRINT + fn = rnintl(x*invpio2); n = irint(fn); -#else - n = fn; -#endif r = x-fn*pio2_1; w = fn*pio2_1t; /* 1st round good to 102 bit */ { union IEEEl2bits u2; int ex1; j = ex; y[0] = r-w; u2.e = y[0]; ex1 = u2.xbits.expsign & 0x7fff; i = j-ex1; if(i>22) { /* 2nd iteration needed, good to 141 */ t = r; w = fn*pio2_2; r = t-w; w = fn*pio2_2t-((t-r)-w); y[0] = r-w; u2.e = y[0]; ex1 = u2.xbits.expsign & 0x7fff; i = j-ex1; if(i>61) { /* 3rd iteration need, 180 bits acc */ t = r; /* will cover all possible cases */ w = fn*pio2_3; r = t-w; w = fn*pio2_3t-((t-r)-w); y[0] = r-w; } } } y[1] = (r-y[0])-w; return n; } /* * all other (large) arguments */ if(ex==0x7fff) { /* x is inf or NaN */ y[0]=y[1]=x-x; return 0; } /* set z = scalbn(|x|,ilogb(x)-23) */ u1.e = x; e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ u1.xbits.expsign = ex - e0; z = u1.e; for(i=0;i<2;i++) { tx[i] = (double)((int32_t)(z)); z = (z-tx[i])*two24; } tx[2] = z; nx = 3; while(tx[nx-1]==zero) nx--; /* skip zero term */ n = __kernel_rem_pio2(tx,ty,e0,nx,2); r = (long double)ty[0] + ty[1]; w = ty[1] - (r - ty[0]); if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} y[0] = r; y[1] = w; return n; } Index: head/lib/msun/ld80/k_expl.h =================================================================== --- head/lib/msun/ld80/k_expl.h (revision 336544) +++ head/lib/msun/ld80/k_expl.h (revision 336545) @@ -1,307 +1,300 @@ /* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */ /*- * SPDX-License-Identifier: BSD-2-Clause-FreeBSD * * 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 __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; + fn = rnintl(x * INV_L); 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 (CMPLXL(cos(y) * exp_x * scale1 * scale2, sinl(y) * exp_x * scale1 * scale2)); } #endif /* _COMPLEX_H */ Index: head/lib/msun/ld80/s_expl.c =================================================================== --- head/lib/msun/ld80/s_expl.c (revision 336544) +++ head/lib/msun/ld80/s_expl.c (revision 336545) @@ -1,286 +1,279 @@ /*- * SPDX-License-Identifier: BSD-2-Clause-FreeBSD * * 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 __FBSDID("$FreeBSD$"); /** * Compute the exponential of x for Intel 80-bit format. This is based on: * * PTP Tang, "Table-driven implementation of the exponential function * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15, * 144-157 (1989). * * where the 32 table entries have been expanded to INTERVALS (see below). */ #include #ifdef __i386__ #include #endif #include "fpmath.h" #include "math.h" #include "math_private.h" #include "k_expl.h" /* XXX Prevent compilers from erroneously constant folding these: */ static const volatile long double huge = 0x1p10000L, tiny = 0x1p-10000L; static const long double twom10000 = 0x1p-10000L; static const union IEEEl2bits /* log(2**16384 - 0.5) rounded towards zero: */ /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), #define o_threshold (o_thresholdu.e) /* log(2**(-16381-64-1)) rounded towards zero: */ u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); #define u_threshold (u_thresholdu.e) long double expl(long double x) { 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; ix = hx & 0x7fff; 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 */ RETURNP(-1 / x); RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ } if (x > o_threshold) RETURNP(huge * huge); if (x < u_threshold) RETURNP(tiny * tiny); } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ RETURN2P(1, x); /* 1 with inexact iff x != 0 */ } ENTERI(); 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 { SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); RETURNI(t * twopk * twom10000); } } /** * Compute expm1l(x) for Intel 80-bit format. This is based on: * * PTP Tang, "Table-driven implementation of the Expm1 function * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, * 211-222 (1992). */ /* * Our T1 and T2 are chosen to be approximately the points where method * A and method B have the same accuracy. Tang's T1 and T2 are the * points where method A's accuracy changes by a full bit. For Tang, * this drop in accuracy makes method A immediately less accurate than * method B, but our larger INTERVALS makes method A 2 bits more * accurate so it remains the most accurate method significantly * closer to the origin despite losing the full bit in our extended * range for it. */ static const double T1 = -0.1659, /* ~-30.625/128 * log(2) */ T2 = 0.1659; /* ~30.625/128 * log(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), B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); static const double B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ long double expm1l(long double x) { union IEEEl2bits u, v; long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; long double x_lo, x2, z; long double x4; int k, n, n2; uint16_t hx, ix; DOPRINT_START(&x); /* Filter out exceptional cases. */ u.e = x; hx = u.xbits.expsign; ix = hx & 0x7fff; 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 */ RETURNP(-1 / x - 1); RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ } if (x > o_threshold) RETURNP(huge * huge); /* * expm1l() never underflows, but it must avoid * unrepresentable large negative exponents. We used a * much smaller threshold for large |x| above than in * expl() so as to handle not so large negative exponents * in the same way as large ones here. */ if (hx & 0x8000) /* x <= -64 */ RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */ } ENTERI(); if (T1 < x && x < T2) { if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ /* x (rounded) with inexact if x != 0: */ RETURNPI(x == 0 ? x : (0x1p100 * x + fabsl(x)) * 0x1p-100); } x2 = x * x; x4 = x2 * x2; q = x4 * (x2 * (x4 * /* * XXX the number of terms is no longer good for * pairwise grouping of all except B3, and the * grouping is no longer from highest down. */ (x2 * B12 + (x * B11 + B10)) + (x2 * (x * B9 + B8) + (x * B7 + B6))) + (x * B5 + B4.e)) + x2 * x * B3.e; x_hi = (float)x; x_lo = x - x_hi; hx2_hi = x_hi * x_hi / 2; hx2_lo = x_lo * (x + x_hi) / 2; if (ix >= BIAS - 7) RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); else RETURN2PI(x, hx2_lo + q + hx2_hi); } /* 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; -#if defined(HAVE_EFFICIENT_IRINTL) - n = irintl(fn); -#elif defined(HAVE_EFFICIENT_IRINT) + fn = rnintl(x * INV_L); 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 factor. */ v.e = 1; v.xbits.expsign = BIAS + k; twopk = v.e; /* * Evaluate lower terms of * 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; if (k == 0) { t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1); RETURNI(t); } if (k == -1) { t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1); RETURNI(t / 2); } if (k < -7) { t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk - 1); } if (k > 2 * LDBL_MANT_DIG - 1) { 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); } v.xbits.expsign = BIAS - k; twomk = v.e; if (k > LDBL_MANT_DIG - 1) t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); else t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk); } Index: head/lib/msun/src/e_rem_pio2.c =================================================================== --- head/lib/msun/src/e_rem_pio2.c (revision 336544) +++ head/lib/msun/src/e_rem_pio2.c (revision 336545) @@ -1,186 +1,180 @@ /* @(#)e_rem_pio2.c 1.4 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * * Optimized by Bruce D. Evans. */ #include __FBSDID("$FreeBSD$"); /* __ieee754_rem_pio2(x,y) * * return the remainder of x rem pi/2 in y[0]+y[1] * use __kernel_rem_pio2() */ #include #include "math.h" #include "math_private.h" /* * invpio2: 53 bits of 2/pi * pio2_1: first 33 bit of pi/2 * pio2_1t: pi/2 - pio2_1 * pio2_2: second 33 bit of pi/2 * pio2_2t: pi/2 - (pio2_1+pio2_2) * pio2_3: third 33 bit of pi/2 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) */ static const double zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ #ifdef INLINE_REM_PIO2 static __inline __always_inline #endif int __ieee754_rem_pio2(double x, double *y) { double z,w,t,r,fn; double tx[3],ty[2]; int32_t e0,i,j,nx,n,ix,hx; u_int32_t low; GET_HIGH_WORD(hx,x); /* high word of x */ ix = hx&0x7fffffff; #if 0 /* Must be handled in caller. */ if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ {y[0] = x; y[1] = 0; return 0;} #endif if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */ if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */ goto medium; /* cancellation -- use medium case */ if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */ if (hx > 0) { z = x - pio2_1; /* one round good to 85 bits */ y[0] = z - pio2_1t; y[1] = (z-y[0])-pio2_1t; return 1; } else { z = x + pio2_1; y[0] = z + pio2_1t; y[1] = (z-y[0])+pio2_1t; return -1; } } else { if (hx > 0) { z = x - 2*pio2_1; y[0] = z - 2*pio2_1t; y[1] = (z-y[0])-2*pio2_1t; return 2; } else { z = x + 2*pio2_1; y[0] = z + 2*pio2_1t; y[1] = (z-y[0])+2*pio2_1t; return -2; } } } if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */ if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */ if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */ goto medium; if (hx > 0) { z = x - 3*pio2_1; y[0] = z - 3*pio2_1t; y[1] = (z-y[0])-3*pio2_1t; return 3; } else { z = x + 3*pio2_1; y[0] = z + 3*pio2_1t; y[1] = (z-y[0])+3*pio2_1t; return -3; } } else { if (ix == 0x401921fb) /* |x| ~= 4pi/2 */ goto medium; if (hx > 0) { z = x - 4*pio2_1; y[0] = z - 4*pio2_1t; y[1] = (z-y[0])-4*pio2_1t; return 4; } else { z = x + 4*pio2_1; y[0] = z + 4*pio2_1t; y[1] = (z-y[0])+4*pio2_1t; return -4; } } } if(ix<0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */ medium: - /* Use a specialized rint() to get fn. Assume round-to-nearest. */ - STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52); - fn = fn-0x1.8p52; -#ifdef HAVE_EFFICIENT_IRINT + fn = rnint((double_t)x*invpio2); n = irint(fn); -#else - n = (int32_t)fn; -#endif r = x-fn*pio2_1; w = fn*pio2_1t; /* 1st round good to 85 bit */ { u_int32_t high; j = ix>>20; y[0] = r-w; GET_HIGH_WORD(high,y[0]); i = j-((high>>20)&0x7ff); if(i>16) { /* 2nd iteration needed, good to 118 */ t = r; w = fn*pio2_2; r = t-w; w = fn*pio2_2t-((t-r)-w); y[0] = r-w; GET_HIGH_WORD(high,y[0]); i = j-((high>>20)&0x7ff); if(i>49) { /* 3rd iteration need, 151 bits acc */ t = r; /* will cover all possible cases */ w = fn*pio2_3; r = t-w; w = fn*pio2_3t-((t-r)-w); y[0] = r-w; } } } y[1] = (r-y[0])-w; return n; } /* * all other (large) arguments */ if(ix>=0x7ff00000) { /* x is inf or NaN */ y[0]=y[1]=x-x; return 0; } /* set z = scalbn(|x|,ilogb(x)-23) */ GET_LOW_WORD(low,x); e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ INSERT_WORDS(z, ix - ((int32_t)(e0<<20)), low); for(i=0;i<2;i++) { tx[i] = (double)((int32_t)(z)); z = (z-tx[i])*two24; } tx[2] = z; nx = 3; while(tx[nx-1]==zero) nx--; /* skip zero term */ n = __kernel_rem_pio2(tx,ty,e0,nx,1); if(hx<0) {y[0] = -ty[0]; y[1] = -ty[1]; return -n;} y[0] = ty[0]; y[1] = ty[1]; return n; } Index: head/lib/msun/src/e_rem_pio2f.c =================================================================== --- head/lib/msun/src/e_rem_pio2f.c (revision 336544) +++ head/lib/msun/src/e_rem_pio2f.c (revision 336545) @@ -1,84 +1,78 @@ /* e_rem_pio2f.c -- float version of e_rem_pio2.c * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. * Debugged and optimized by Bruce D. Evans. */ /* * ==================================================== * 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 __FBSDID("$FreeBSD$"); /* __ieee754_rem_pio2f(x,y) * * return the remainder of x rem pi/2 in *y * use double precision for everything except passing x * use __kernel_rem_pio2() for large x */ #include #include "math.h" #include "math_private.h" /* * invpio2: 53 bits of 2/pi * pio2_1: first 25 bits of pi/2 * pio2_1t: pi/2 - pio2_1 */ static const double invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ pio2_1 = 1.57079631090164184570e+00, /* 0x3FF921FB, 0x50000000 */ pio2_1t = 1.58932547735281966916e-08; /* 0x3E5110b4, 0x611A6263 */ #ifdef INLINE_REM_PIO2F static __inline __always_inline #endif int __ieee754_rem_pio2f(float x, double *y) { double w,r,fn; double tx[1],ty[1]; float z; int32_t e0,n,ix,hx; GET_FLOAT_WORD(hx,x); ix = hx&0x7fffffff; /* 33+53 bit pi is good enough for medium size */ if(ix<0x4dc90fdb) { /* |x| ~< 2^28*(pi/2), medium size */ - /* Use a specialized rint() to get fn. Assume round-to-nearest. */ - STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52); - fn = fn-0x1.8p52; -#ifdef HAVE_EFFICIENT_IRINT + fn = rnint((float_t)x*invpio2); n = irint(fn); -#else - n = (int32_t)fn; -#endif r = x-fn*pio2_1; w = fn*pio2_1t; *y = r-w; return n; } /* * all other (large) arguments */ if(ix>=0x7f800000) { /* x is inf or NaN */ *y=x-x; return 0; } /* set z = scalbn(|x|,ilogb(|x|)-23) */ e0 = (ix>>23)-150; /* e0 = ilogb(|x|)-23; */ SET_FLOAT_WORD(z, ix - ((int32_t)(e0<<23))); tx[0] = z; n = __kernel_rem_pio2(tx,ty,e0,1,0); if(hx<0) {*y = -ty[0]; return -n;} *y = ty[0]; return n; } Index: head/lib/msun/src/math_private.h =================================================================== --- head/lib/msun/src/math_private.h (revision 336544) +++ head/lib/msun/src/math_private.h (revision 336545) @@ -1,852 +1,920 @@ /* * ==================================================== * 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. * ==================================================== */ /* * from: @(#)fdlibm.h 5.1 93/09/24 * $FreeBSD$ */ #ifndef _MATH_PRIVATE_H_ #define _MATH_PRIVATE_H_ #include #include /* * The original fdlibm code used statements like: * n0 = ((*(int*)&one)>>29)^1; * index of high word * * ix0 = *(n0+(int*)&x); * high word of x * * ix1 = *((1-n0)+(int*)&x); * low word of x * * to dig two 32 bit words out of the 64 bit IEEE floating point * value. That is non-ANSI, and, moreover, the gcc instruction * scheduler gets it wrong. We instead use the following macros. * Unlike the original code, we determine the endianness at compile * time, not at run time; I don't see much benefit to selecting * endianness at run time. */ /* * A union which permits us to convert between a double and two 32 bit * ints. */ #ifdef __arm__ #if defined(__VFP_FP__) || defined(__ARM_EABI__) #define IEEE_WORD_ORDER BYTE_ORDER #else #define IEEE_WORD_ORDER BIG_ENDIAN #endif #else /* __arm__ */ #define IEEE_WORD_ORDER BYTE_ORDER #endif /* A union which permits us to convert between a long double and four 32 bit ints. */ #if IEEE_WORD_ORDER == BIG_ENDIAN typedef union { long double value; struct { u_int32_t mswhi; u_int32_t mswlo; u_int32_t lswhi; u_int32_t lswlo; } parts32; struct { u_int64_t msw; u_int64_t lsw; } parts64; } ieee_quad_shape_type; #endif #if IEEE_WORD_ORDER == LITTLE_ENDIAN typedef union { long double value; struct { u_int32_t lswlo; u_int32_t lswhi; u_int32_t mswlo; u_int32_t mswhi; } parts32; struct { u_int64_t lsw; u_int64_t msw; } parts64; } ieee_quad_shape_type; #endif #if IEEE_WORD_ORDER == BIG_ENDIAN typedef union { double value; struct { u_int32_t msw; u_int32_t lsw; } parts; struct { u_int64_t w; } xparts; } ieee_double_shape_type; #endif #if IEEE_WORD_ORDER == LITTLE_ENDIAN typedef union { double value; struct { u_int32_t lsw; u_int32_t msw; } parts; struct { u_int64_t w; } xparts; } ieee_double_shape_type; #endif /* Get two 32 bit ints from a double. */ #define EXTRACT_WORDS(ix0,ix1,d) \ do { \ ieee_double_shape_type ew_u; \ ew_u.value = (d); \ (ix0) = ew_u.parts.msw; \ (ix1) = ew_u.parts.lsw; \ } while (0) /* Get a 64-bit int from a double. */ #define EXTRACT_WORD64(ix,d) \ do { \ ieee_double_shape_type ew_u; \ ew_u.value = (d); \ (ix) = ew_u.xparts.w; \ } while (0) /* Get the more significant 32 bit int from a double. */ #define GET_HIGH_WORD(i,d) \ do { \ ieee_double_shape_type gh_u; \ gh_u.value = (d); \ (i) = gh_u.parts.msw; \ } while (0) /* Get the less significant 32 bit int from a double. */ #define GET_LOW_WORD(i,d) \ do { \ ieee_double_shape_type gl_u; \ gl_u.value = (d); \ (i) = gl_u.parts.lsw; \ } while (0) /* Set a double from two 32 bit ints. */ #define INSERT_WORDS(d,ix0,ix1) \ do { \ ieee_double_shape_type iw_u; \ iw_u.parts.msw = (ix0); \ iw_u.parts.lsw = (ix1); \ (d) = iw_u.value; \ } while (0) /* Set a double from a 64-bit int. */ #define INSERT_WORD64(d,ix) \ do { \ ieee_double_shape_type iw_u; \ iw_u.xparts.w = (ix); \ (d) = iw_u.value; \ } while (0) /* Set the more significant 32 bits of a double from an int. */ #define SET_HIGH_WORD(d,v) \ do { \ ieee_double_shape_type sh_u; \ sh_u.value = (d); \ sh_u.parts.msw = (v); \ (d) = sh_u.value; \ } while (0) /* Set the less significant 32 bits of a double from an int. */ #define SET_LOW_WORD(d,v) \ do { \ ieee_double_shape_type sl_u; \ sl_u.value = (d); \ sl_u.parts.lsw = (v); \ (d) = sl_u.value; \ } while (0) /* * A union which permits us to convert between a float and a 32 bit * int. */ typedef union { float value; /* FIXME: Assumes 32 bit int. */ unsigned int word; } ieee_float_shape_type; /* Get a 32 bit int from a float. */ #define GET_FLOAT_WORD(i,d) \ do { \ ieee_float_shape_type gf_u; \ gf_u.value = (d); \ (i) = gf_u.word; \ } while (0) /* Set a float from a 32 bit int. */ #define SET_FLOAT_WORD(d,i) \ do { \ ieee_float_shape_type sf_u; \ sf_u.word = (i); \ (d) = sf_u.value; \ } while (0) /* * Get expsign and mantissa as 16 bit and 64 bit ints from an 80 bit long * double. */ #define EXTRACT_LDBL80_WORDS(ix0,ix1,d) \ do { \ union IEEEl2bits ew_u; \ ew_u.e = (d); \ (ix0) = ew_u.xbits.expsign; \ (ix1) = ew_u.xbits.man; \ } while (0) /* * Get expsign and mantissa as one 16 bit and two 64 bit ints from a 128 bit * long double. */ #define EXTRACT_LDBL128_WORDS(ix0,ix1,ix2,d) \ do { \ union IEEEl2bits ew_u; \ ew_u.e = (d); \ (ix0) = ew_u.xbits.expsign; \ (ix1) = ew_u.xbits.manh; \ (ix2) = ew_u.xbits.manl; \ } while (0) /* Get expsign as a 16 bit int from a long double. */ #define GET_LDBL_EXPSIGN(i,d) \ do { \ union IEEEl2bits ge_u; \ ge_u.e = (d); \ (i) = ge_u.xbits.expsign; \ } while (0) /* * Set an 80 bit long double from a 16 bit int expsign and a 64 bit int * mantissa. */ #define INSERT_LDBL80_WORDS(d,ix0,ix1) \ do { \ union IEEEl2bits iw_u; \ iw_u.xbits.expsign = (ix0); \ iw_u.xbits.man = (ix1); \ (d) = iw_u.e; \ } while (0) /* * Set a 128 bit long double from a 16 bit int expsign and two 64 bit ints * comprising the mantissa. */ #define INSERT_LDBL128_WORDS(d,ix0,ix1,ix2) \ do { \ union IEEEl2bits iw_u; \ iw_u.xbits.expsign = (ix0); \ iw_u.xbits.manh = (ix1); \ iw_u.xbits.manl = (ix2); \ (d) = iw_u.e; \ } while (0) /* Set expsign of a long double from a 16 bit int. */ #define SET_LDBL_EXPSIGN(d,v) \ do { \ union IEEEl2bits se_u; \ se_u.e = (d); \ se_u.xbits.expsign = (v); \ (d) = se_u.e; \ } while (0) #ifdef __i386__ /* Long double constants are broken on i386. */ #define LD80C(m, ex, v) { \ .xbits.man = __CONCAT(m, ULL), \ .xbits.expsign = (0x3fff + (ex)) | ((v) < 0 ? 0x8000 : 0), \ } #else /* The above works on non-i386 too, but we use this to check v. */ #define LD80C(m, ex, v) { .e = (v), } #endif #ifdef FLT_EVAL_METHOD /* * Attempt to get strict C99 semantics for assignment with non-C99 compilers. */ #if FLT_EVAL_METHOD == 0 || __GNUC__ == 0 #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval)) #else #define STRICT_ASSIGN(type, lval, rval) do { \ volatile type __lval; \ \ if (sizeof(type) >= sizeof(long double)) \ (lval) = (rval); \ else { \ __lval = (rval); \ (lval) = __lval; \ } \ } while (0) #endif #endif /* FLT_EVAL_METHOD */ /* Support switching the mode to FP_PE if necessary. */ #if defined(__i386__) && !defined(NO_FPSETPREC) #define ENTERI() ENTERIT(long double) #define ENTERIT(returntype) \ returntype __retval; \ fp_prec_t __oprec; \ \ if ((__oprec = fpgetprec()) != FP_PE) \ fpsetprec(FP_PE) #define RETURNI(x) do { \ __retval = (x); \ if (__oprec != FP_PE) \ fpsetprec(__oprec); \ RETURNF(__retval); \ } while (0) #define ENTERV() \ fp_prec_t __oprec; \ \ if ((__oprec = fpgetprec()) != FP_PE) \ fpsetprec(FP_PE) #define RETURNV() do { \ if (__oprec != FP_PE) \ fpsetprec(__oprec); \ return; \ } while (0) #else #define ENTERI() #define ENTERIT(x) #define RETURNI(x) RETURNF(x) #define ENTERV() #define RETURNV() return #endif /* Default return statement if hack*_t() is not used. */ #define RETURNF(v) return (v) /* * 2sum gives the same result as 2sumF without requiring |a| >= |b| or * a == 0, but is slower. */ #define _2sum(a, b) do { \ __typeof(a) __s, __w; \ \ __w = (a) + (b); \ __s = __w - (a); \ (b) = ((a) - (__w - __s)) + ((b) - __s); \ (a) = __w; \ } while (0) /* * 2sumF algorithm. * * "Normalize" the terms in the infinite-precision expression a + b for * the sum of 2 floating point values so that b is as small as possible * relative to 'a'. (The resulting 'a' is the value of the expression in * the same precision as 'a' and the resulting b is the rounding error.) * |a| must be >= |b| or 0, b's type must be no larger than 'a's type, and * exponent overflow or underflow must not occur. This uses a Theorem of * Dekker (1971). See Knuth (1981) 4.2.2 Theorem C. The name "TwoSum" * is apparently due to Skewchuk (1997). * * For this to always work, assignment of a + b to 'a' must not retain any * extra precision in a + b. This is required by C standards but broken * in many compilers. The brokenness cannot be worked around using * STRICT_ASSIGN() like we do elsewhere, since the efficiency of this * algorithm would be destroyed by non-null strict assignments. (The * compilers are correct to be broken -- the efficiency of all floating * point code calculations would be destroyed similarly if they forced the * conversions.) * * Fortunately, a case that works well can usually be arranged by building * any extra precision into the type of 'a' -- 'a' should have type float_t, * double_t or long double. b's type should be no larger than 'a's type. * Callers should use these types with scopes as large as possible, to * reduce their own extra-precision and efficiciency problems. In * particular, they shouldn't convert back and forth just to call here. */ #ifdef DEBUG #define _2sumF(a, b) do { \ __typeof(a) __w; \ volatile __typeof(a) __ia, __ib, __r, __vw; \ \ __ia = (a); \ __ib = (b); \ assert(__ia == 0 || fabsl(__ia) >= fabsl(__ib)); \ \ __w = (a) + (b); \ (b) = ((a) - __w) + (b); \ (a) = __w; \ \ /* The next 2 assertions are weak if (a) is already long double. */ \ assert((long double)__ia + __ib == (long double)(a) + (b)); \ __vw = __ia + __ib; \ __r = __ia - __vw; \ __r += __ib; \ assert(__vw == (a) && __r == (b)); \ } while (0) #else /* !DEBUG */ #define _2sumF(a, b) do { \ __typeof(a) __w; \ \ __w = (a) + (b); \ (b) = ((a) - __w) + (b); \ (a) = __w; \ } while (0) #endif /* DEBUG */ /* * Set x += c, where x is represented in extra precision as a + b. * x must be sufficiently normalized and sufficiently larger than c, * and the result is then sufficiently normalized. * * The details of ordering are that |a| must be >= |c| (so that (a, c) * can be normalized without extra work to swap 'a' with c). The details of * the normalization are that b must be small relative to the normalized 'a'. * Normalization of (a, c) makes the normalized c tiny relative to the * normalized a, so b remains small relative to 'a' in the result. However, * b need not ever be tiny relative to 'a'. For example, b might be about * 2**20 times smaller than 'a' to give about 20 extra bits of precision. * That is usually enough, and adding c (which by normalization is about * 2**53 times smaller than a) cannot change b significantly. However, * cancellation of 'a' with c in normalization of (a, c) may reduce 'a' * significantly relative to b. The caller must ensure that significant * cancellation doesn't occur, either by having c of the same sign as 'a', * or by having |c| a few percent smaller than |a|. Pre-normalization of * (a, b) may help. * * This is is a variant of an algorithm of Kahan (see Knuth (1981) 4.2.2 * exercise 19). We gain considerable efficiency by requiring the terms to * be sufficiently normalized and sufficiently increasing. */ #define _3sumF(a, b, c) do { \ __typeof(a) __tmp; \ \ __tmp = (c); \ _2sumF(__tmp, (a)); \ (b) += (a); \ (a) = __tmp; \ } while (0) /* * Common routine to process the arguments to nan(), nanf(), and nanl(). */ void _scan_nan(uint32_t *__words, int __num_words, const char *__s); /* * Mix 1 or 2 NaNs. First add 0 to each arg. This normally just turns * signaling NaNs into quiet NaNs by setting a quiet bit. We do this * because we want to never return a signaling NaN, and also because we * don't want the quiet bit to affect the result. Then mix the converted * args using addition. The result is typically the arg whose mantissa * bits (considered as in integer) are largest. * * Technical complications: the result in bits might depend on the precision * and/or on compiler optimizations, especially when different register sets * are used for different precisions. Try to make the result not depend on * at least the precision by always doing the main mixing step in long double * precision. Try to reduce dependencies on optimizations by adding the * the 0's in different precisions (unless everything is in long double * precision). */ #define nan_mix(x, y) (((x) + 0.0L) + ((y) + 0)) #ifdef _COMPLEX_H /* * C99 specifies that complex numbers have the same representation as * an array of two elements, where the first element is the real part * and the second element is the imaginary part. */ typedef union { float complex f; float a[2]; } float_complex; typedef union { double complex f; double a[2]; } double_complex; typedef union { long double complex f; long double a[2]; } long_double_complex; #define REALPART(z) ((z).a[0]) #define IMAGPART(z) ((z).a[1]) /* * Inline functions that can be used to construct complex values. * * The C99 standard intends x+I*y to be used for this, but x+I*y is * currently unusable in general since gcc introduces many overflow, * underflow, sign and efficiency bugs by rewriting I*y as * (0.0+I)*(y+0.0*I) and laboriously computing the full complex product. * In particular, I*Inf is corrupted to NaN+I*Inf, and I*-0 is corrupted * to -0.0+I*0.0. * * The C11 standard introduced the macros CMPLX(), CMPLXF() and CMPLXL() * to construct complex values. Compilers that conform to the C99 * standard require the following functions to avoid the above issues. */ #ifndef CMPLXF static __inline float complex CMPLXF(float x, float y) { float_complex z; REALPART(z) = x; IMAGPART(z) = y; return (z.f); } #endif #ifndef CMPLX static __inline double complex CMPLX(double x, double y) { double_complex z; REALPART(z) = x; IMAGPART(z) = y; return (z.f); } #endif #ifndef CMPLXL static __inline long double complex CMPLXL(long double x, long double y) { long_double_complex z; REALPART(z) = x; IMAGPART(z) = y; return (z.f); } #endif #endif /* _COMPLEX_H */ -#ifdef __GNUCLIKE_ASM +/* + * The rnint() family rounds to the nearest integer for a restricted range + * range of args (up to about 2**MANT_DIG). We assume that the current + * rounding mode is FE_TONEAREST so that this can be done efficiently. + * Extra precision causes more problems in practice, and we only centralize + * this here to reduce those problems, and have not solved the efficiency + * problems. The exp2() family uses a more delicate version of this that + * requires extracting bits from the intermediate value, so it is not + * centralized here and should copy any solution of the efficiency problems. + */ -/* Asm versions of some functions. */ +static inline double +rnint(__double_t x) +{ + /* + * This casts to double to kill any extra precision. This depends + * on the cast being applied to a double_t to avoid compiler bugs + * (this is a cleaner version of STRICT_ASSIGN()). This is + * inefficient if there actually is extra precision, but is hard + * to improve on. We use double_t in the API to minimise conversions + * for just calling here. Note that we cannot easily change the + * magic number to the one that works directly with double_t, since + * the rounding precision is variable at runtime on x86 so the + * magic number would need to be variable. Assuming that the + * rounding precision is always the default is too fragile. This + * and many other complications will move when the default is + * changed to FP_PE. + */ + return ((double)(x + 0x1.8p52) - 0x1.8p52); +} -#ifdef __amd64__ +static inline float +rnintf(__float_t x) +{ + /* + * As for rnint(), except we could just call that to handle the + * extra precision case, usually without losing efficiency. + */ + return ((float)(x + 0x1.8p23F) - 0x1.8p23F); +} + +#ifdef LDBL_MANT_DIG +/* + * The complications for extra precision are smaller for rnintl() since it + * can safely assume that the rounding precision has been increased from + * its default to FP_PE on x86. We don't exploit that here to get small + * optimizations from limiting the rangle to double. We just need it for + * the magic number to work with long doubles. ld128 callers should use + * rnint() instead of this if possible. ld80 callers should prefer + * rnintl() since for amd64 this avoids swapping the register set, while + * for i386 it makes no difference (assuming FP_PE), and for other arches + * it makes little difference. + */ +static inline long double +rnintl(long double x) +{ + return (x + __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2 - + __CONCAT(0x1.8p, LDBL_MANT_DIG) / 2); +} +#endif /* LDBL_MANT_DIG */ + +/* + * irint() and i64rint() give the same result as casting to their integer + * return type provided their arg is a floating point integer. They can + * sometimes be more efficient because no rounding is required. + */ +#if (defined(amd64) || defined(__i386__)) && defined(__GNUCLIKE_ASM) +#define irint(x) \ + (sizeof(x) == sizeof(float) && \ + sizeof(__float_t) == sizeof(long double) ? irintf(x) : \ + sizeof(x) == sizeof(double) && \ + sizeof(__double_t) == sizeof(long double) ? irintd(x) : \ + sizeof(x) == sizeof(long double) ? irintl(x) : (int)(x)) +#else +#define irint(x) ((int)(x)) +#endif + +#define i64rint(x) ((int64_t)(x)) /* only needed for ld128 so not opt. */ + +#if defined(__i386__) && defined(__GNUCLIKE_ASM) static __inline int -irint(double x) +irintf(float x) { int n; - asm("cvtsd2si %1,%0" : "=r" (n) : "x" (x)); + __asm("fistl %0" : "=m" (n) : "t" (x)); return (n); } -#define HAVE_EFFICIENT_IRINT -#endif -#ifdef __i386__ static __inline int -irint(double x) +irintd(double x) { int n; - asm("fistl %0" : "=m" (n) : "t" (x)); + __asm("fistl %0" : "=m" (n) : "t" (x)); return (n); } -#define HAVE_EFFICIENT_IRINT #endif -#if defined(__amd64__) || defined(__i386__) +#if (defined(__amd64__) || defined(__i386__)) && defined(__GNUCLIKE_ASM) static __inline int irintl(long double x) { int n; - asm("fistl %0" : "=m" (n) : "t" (x)); + __asm("fistl %0" : "=m" (n) : "t" (x)); return (n); } -#define HAVE_EFFICIENT_IRINTL #endif - -#endif /* __GNUCLIKE_ASM */ #ifdef DEBUG #if defined(__amd64__) || defined(__i386__) #define breakpoint() asm("int $3") #else #include #define breakpoint() raise(SIGTRAP) #endif #endif /* Write a pari script to test things externally. */ #ifdef DOPRINT #include #ifndef DOPRINT_SWIZZLE #define DOPRINT_SWIZZLE 0 #endif #ifdef DOPRINT_LD80 #define DOPRINT_START(xp) do { \ uint64_t __lx; \ uint16_t __hx; \ \ /* Hack to give more-problematic args. */ \ EXTRACT_LDBL80_WORDS(__hx, __lx, *xp); \ __lx ^= DOPRINT_SWIZZLE; \ INSERT_LDBL80_WORDS(*xp, __hx, __lx); \ printf("x = %.21Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #elif defined(DOPRINT_D64) #define DOPRINT_START(xp) do { \ uint32_t __hx, __lx; \ \ EXTRACT_WORDS(__hx, __lx, *xp); \ __lx ^= DOPRINT_SWIZZLE; \ INSERT_WORDS(*xp, __hx, __lx); \ printf("x = %.21Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #elif defined(DOPRINT_F32) #define DOPRINT_START(xp) do { \ uint32_t __hx; \ \ GET_FLOAT_WORD(__hx, *xp); \ __hx ^= DOPRINT_SWIZZLE; \ SET_FLOAT_WORD(*xp, __hx); \ printf("x = %.21Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.21Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.21Lg; z = %.21Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #else /* !DOPRINT_LD80 && !DOPRINT_D64 (LD128 only) */ #ifndef DOPRINT_SWIZZLE_HIGH #define DOPRINT_SWIZZLE_HIGH 0 #endif #define DOPRINT_START(xp) do { \ uint64_t __lx, __llx; \ uint16_t __hx; \ \ EXTRACT_LDBL128_WORDS(__hx, __lx, __llx, *xp); \ __llx ^= DOPRINT_SWIZZLE; \ __lx ^= DOPRINT_SWIZZLE_HIGH; \ INSERT_LDBL128_WORDS(*xp, __hx, __lx, __llx); \ printf("x = %.36Lg; ", (long double)*xp); \ } while (0) #define DOPRINT_END1(v) \ printf("y = %.36Lg; z = 0; show(x, y, z);\n", (long double)(v)) #define DOPRINT_END2(hi, lo) \ printf("y = %.36Lg; z = %.36Lg; show(x, y, z);\n", \ (long double)(hi), (long double)(lo)) #endif /* DOPRINT_LD80 */ #else /* !DOPRINT */ #define DOPRINT_START(xp) #define DOPRINT_END1(v) #define DOPRINT_END2(hi, lo) #endif /* DOPRINT */ #define RETURNP(x) do { \ DOPRINT_END1(x); \ RETURNF(x); \ } while (0) #define RETURNPI(x) do { \ DOPRINT_END1(x); \ RETURNI(x); \ } while (0) #define RETURN2P(x, y) do { \ DOPRINT_END2((x), (y)); \ RETURNF((x) + (y)); \ } while (0) #define RETURN2PI(x, y) do { \ DOPRINT_END2((x), (y)); \ RETURNI((x) + (y)); \ } while (0) #ifdef STRUCT_RETURN #define RETURNSP(rp) do { \ if (!(rp)->lo_set) \ RETURNP((rp)->hi); \ RETURN2P((rp)->hi, (rp)->lo); \ } while (0) #define RETURNSPI(rp) do { \ if (!(rp)->lo_set) \ RETURNPI((rp)->hi); \ RETURN2PI((rp)->hi, (rp)->lo); \ } while (0) #endif #define SUM2P(x, y) ({ \ const __typeof (x) __x = (x); \ const __typeof (y) __y = (y); \ \ DOPRINT_END2(__x, __y); \ __x + __y; \ }) /* * ieee style elementary functions * * We rename functions here to improve other sources' diffability * against fdlibm. */ #define __ieee754_sqrt sqrt #define __ieee754_acos acos #define __ieee754_acosh acosh #define __ieee754_log log #define __ieee754_log2 log2 #define __ieee754_atanh atanh #define __ieee754_asin asin #define __ieee754_atan2 atan2 #define __ieee754_exp exp #define __ieee754_cosh cosh #define __ieee754_fmod fmod #define __ieee754_pow pow #define __ieee754_lgamma lgamma #define __ieee754_gamma gamma #define __ieee754_lgamma_r lgamma_r #define __ieee754_gamma_r gamma_r #define __ieee754_log10 log10 #define __ieee754_sinh sinh #define __ieee754_hypot hypot #define __ieee754_j0 j0 #define __ieee754_j1 j1 #define __ieee754_y0 y0 #define __ieee754_y1 y1 #define __ieee754_jn jn #define __ieee754_yn yn #define __ieee754_remainder remainder #define __ieee754_scalb scalb #define __ieee754_sqrtf sqrtf #define __ieee754_acosf acosf #define __ieee754_acoshf acoshf #define __ieee754_logf logf #define __ieee754_atanhf atanhf #define __ieee754_asinf asinf #define __ieee754_atan2f atan2f #define __ieee754_expf expf #define __ieee754_coshf coshf #define __ieee754_fmodf fmodf #define __ieee754_powf powf #define __ieee754_lgammaf lgammaf #define __ieee754_gammaf gammaf #define __ieee754_lgammaf_r lgammaf_r #define __ieee754_gammaf_r gammaf_r #define __ieee754_log10f log10f #define __ieee754_log2f log2f #define __ieee754_sinhf sinhf #define __ieee754_hypotf hypotf #define __ieee754_j0f j0f #define __ieee754_j1f j1f #define __ieee754_y0f y0f #define __ieee754_y1f y1f #define __ieee754_jnf jnf #define __ieee754_ynf ynf #define __ieee754_remainderf remainderf #define __ieee754_scalbf scalbf /* fdlibm kernel function */ int __kernel_rem_pio2(double*,double*,int,int,int); /* double precision kernel functions */ #ifndef INLINE_REM_PIO2 int __ieee754_rem_pio2(double,double*); #endif double __kernel_sin(double,double,int); double __kernel_cos(double,double); double __kernel_tan(double,double,int); double __ldexp_exp(double,int); #ifdef _COMPLEX_H double complex __ldexp_cexp(double complex,int); #endif /* float precision kernel functions */ #ifndef INLINE_REM_PIO2F int __ieee754_rem_pio2f(float,double*); #endif #ifndef INLINE_KERNEL_SINDF float __kernel_sindf(double); #endif #ifndef INLINE_KERNEL_COSDF float __kernel_cosdf(double); #endif #ifndef INLINE_KERNEL_TANDF float __kernel_tandf(double,int); #endif float __ldexp_expf(float,int); #ifdef _COMPLEX_H float complex __ldexp_cexpf(float complex,int); #endif /* long double precision kernel functions */ long double __kernel_sinl(long double, long double, int); long double __kernel_cosl(long double, long double); long double __kernel_tanl(long double, long double, int); long double __p1evll(long double, void *, int); long double __polevll(long double, void *, int); #endif /* !_MATH_PRIVATE_H_ */