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      1 /* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */
      2 
      3 /*-
      4  * SPDX-License-Identifier: BSD-2-Clause
      5  *
      6  * Copyright (c) 2009-2013 Steven G. Kargl
      7  * All rights reserved.
      8  *
      9  * Redistribution and use in source and binary forms, with or without
     10  * modification, are permitted provided that the following conditions
     11  * are met:
     12  * 1. Redistributions of source code must retain the above copyright
     13  *    notice unmodified, this list of conditions, and the following
     14  *    disclaimer.
     15  * 2. Redistributions in binary form must reproduce the above copyright
     16  *    notice, this list of conditions and the following disclaimer in the
     17  *    documentation and/or other materials provided with the distribution.
     18  *
     19  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
     20  * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
     21  * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
     22  * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
     23  * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
     24  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
     25  * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
     26  * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
     27  * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
     28  * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
     29  *
     30  * Optimized by Bruce D. Evans.
     31  */
     32 
     33 #include <sys/cdefs.h>
     34 /*
     35  * See s_expl.c for more comments about __k_expl().
     36  *
     37  * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments
     38  * about the secondary kernels.
     39  */
     40 
     41 #define	INTERVALS	128
     42 #define	LOG2_INTERVALS	7
     43 #define	BIAS	(LDBL_MAX_EXP - 1)
     44 
     45 static const double
     46 /*
     47  * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication).  L1 must
     48  * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest
     49  * bits zero so that multiplication of it by n is exact.
     50  */
     51 INV_L = 1.8466496523378731e+2,		/*  0x171547652b82fe.0p-45 */
     52 L1 =  5.4152123484527692e-3,		/*  0x162e42ff000000.0p-60 */
     53 L2 = -3.2819649005320973e-13,		/* -0x1718432a1b0e26.0p-94 */
     54 /*
     55  * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]:
     56  * |exp(x) - p(x)| < 2**-77.2
     57  * (0.002708 is ln2/(2*INTERVALS) rounded up a little).
     58  */
     59 A2 =  0.5,
     60 A3 =  1.6666666666666119e-1,		/*  0x15555555555490.0p-55 */
     61 A4 =  4.1666666666665887e-2,		/*  0x155555555554e5.0p-57 */
     62 A5 =  8.3333354987869413e-3,		/*  0x1111115b789919.0p-59 */
     63 A6 =  1.3888891738560272e-3;		/*  0x16c16c651633ae.0p-62 */
     64 
     65 /*
     66  * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where
     67  * the first 53 bits of the significand are stored in hi and the next 53
     68  * bits are in lo.  Tang's paper states that the trailing 6 bits of hi must
     69  * be zero for his algorithm in both single and double precision, because
     70  * the table is re-used in the implementation of expm1() where a floating
     71  * point addition involving hi must be exact.  Here hi is double, so
     72  * converting it to long double gives 11 trailing zero bits.
     73  */
     74 static const struct {
     75 	double	hi;
     76 	double	lo;
     77 } tbl[INTERVALS] = {
     78 	{ 0x1p+0, 0x0p+0 },
     79 	/*
     80 	 * XXX hi is rounded down, and the formatting is not quite normal.
     81 	 * But I rather like both.  The 0x1.*p format is good for 4N+1
     82 	 * significand bits.  Rounding down makes the lo terms positive,
     83 	 * so that the columnar formatting can be simpler.
     84 	 */
     85 	{ 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54 },
     86 	{ 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53 },
     87 	{ 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53 },
     88 	{ 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55 },
     89 	{ 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53 },
     90 	{ 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57 },
     91 	{ 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54 },
     92 	{ 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54 },
     93 	{ 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54 },
     94 	{ 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59 },
     95 	{ 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53 },
     96 	{ 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53 },
     97 	{ 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53 },
     98 	{ 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53 },
     99 	{ 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55 },
    100 	{ 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53 },
    101 	{ 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53 },
    102 	{ 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55 },
    103 	{ 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53 },
    104 	{ 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54 },
    105 	{ 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53 },
    106 	{ 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55 },
    107 	{ 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55 },
    108 	{ 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54 },
    109 	{ 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55 },
    110 	{ 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55 },
    111 	{ 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53 },
    112 	{ 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55 },
    113 	{ 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53 },
    114 	{ 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54 },
    115 	{ 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56 },
    116 	{ 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55 },
    117 	{ 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55 },
    118 	{ 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54 },
    119 	{ 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53 },
    120 	{ 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53 },
    121 	{ 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53 },
    122 	{ 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53 },
    123 	{ 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53 },
    124 	{ 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55 },
    125 	{ 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53 },
    126 	{ 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53 },
    127 	{ 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53 },
    128 	{ 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59 },
    129 	{ 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54 },
    130 	{ 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56 },
    131 	{ 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54 },
    132 	{ 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56 },
    133 	{ 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54 },
    134 	{ 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53 },
    135 	{ 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53 },
    136 	{ 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53 },
    137 	{ 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53 },
    138 	{ 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54 },
    139 	{ 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55 },
    140 	{ 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54 },
    141 	{ 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60 },
    142 	{ 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54 },
    143 	{ 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53 },
    144 	{ 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53 },
    145 	{ 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53 },
    146 	{ 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53 },
    147 	{ 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57 },
    148 	{ 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53 },
    149 	{ 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53 },
    150 	{ 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53 },
    151 	{ 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53 },
    152 	{ 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53 },
    153 	{ 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53 },
    154 	{ 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53 },
    155 	{ 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54 },
    156 	{ 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53 },
    157 	{ 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54 },
    158 	{ 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56 },
    159 	{ 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53 },
    160 	{ 0x1.82589994cce12p+0, 0x1.159f115f56694p-53 },
    161 	{ 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53 },
    162 	{ 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53 },
    163 	{ 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54 },
    164 	{ 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54 },
    165 	{ 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53 },
    166 	{ 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55 },
    167 	{ 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53 },
    168 	{ 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53 },
    169 	{ 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53 },
    170 	{ 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53 },
    171 	{ 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53 },
    172 	{ 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56 },
    173 	{ 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56 },
    174 	{ 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53 },
    175 	{ 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54 },
    176 	{ 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53 },
    177 	{ 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54 },
    178 	{ 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54 },
    179 	{ 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53 },
    180 	{ 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54 },
    181 	{ 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53 },
    182 	{ 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53 },
    183 	{ 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53 },
    184 	{ 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53 },
    185 	{ 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53 },
    186 	{ 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55 },
    187 	{ 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53 },
    188 	{ 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55 },
    189 	{ 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54 },
    190 	{ 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54 },
    191 	{ 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56 },
    192 	{ 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56 },
    193 	{ 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53 },
    194 	{ 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53 },
    195 	{ 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53 },
    196 	{ 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55 },
    197 	{ 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53 },
    198 	{ 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54 },
    199 	{ 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54 },
    200 	{ 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53 },
    201 	{ 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53 },
    202 	{ 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55 },
    203 	{ 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54 },
    204 	{ 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53 },
    205 	{ 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53 },
    206 	{ 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54 },
    207 	{ 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54 },
    208 	{ 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54 },
    209 	{ 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53 },
    210 	{ 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55 },
    211 	{ 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 }
    212 };
    213 
    214 /*
    215  * Kernel for expl(x).  x must be finite and not tiny or huge.
    216  * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN).
    217  * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2).
    218  */
    219 static inline void
    220 __k_expl(long double x, long double *hip, long double *lop, int *kp)
    221 {
    222 	long double fn, q, r, r1, r2, t, z;
    223 	int n, n2;
    224 
    225 	/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
    226 	fn = rnintl(x * INV_L);
    227 	r = x - fn * L1 - fn * L2;	/* r = r1 + r2 done independently. */
    228 	n = irint(fn);
    229 	n2 = (unsigned)n % INTERVALS;
    230 	/* Depend on the sign bit being propagated: */
    231 	*kp = n >> LOG2_INTERVALS;
    232 	r1 = x - fn * L1;
    233 	r2 = fn * -L2;
    234 
    235 	/* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */
    236 	z = r * r;
    237 #if 0
    238 	q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
    239 #else
    240 	q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6));
    241 #endif
    242 	t = (long double)tbl[n2].lo + tbl[n2].hi;
    243 	*hip = tbl[n2].hi;
    244 	*lop = tbl[n2].lo + t * (q + r1);
    245 }
    246 
    247 static inline void
    248 k_hexpl(long double x, long double *hip, long double *lop)
    249 {
    250 	float twopkm1;
    251 	int k;
    252 
    253 	__k_expl(x, hip, lop, &k);
    254 	SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23));
    255 	*hip *= twopkm1;
    256 	*lop *= twopkm1;
    257 }
    258 
    259 static inline long double
    260 hexpl(long double x)
    261 {
    262 	long double hi, lo, twopkm2;
    263 	int k;
    264 
    265 	twopkm2 = 1;
    266 	__k_expl(x, &hi, &lo, &k);
    267 	SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2);
    268 	return (lo + hi) * 2 * twopkm2;
    269 }
    270 
    271 #ifdef _COMPLEX_H
    272 /*
    273  * See ../src/k_exp.c for details.
    274  */
    275 static inline long double complex
    276 __ldexp_cexpl(long double complex z, int expt)
    277 {
    278 	long double c, exp_x, hi, lo, s;
    279 	long double x, y, scale1, scale2;
    280 	int half_expt, k;
    281 
    282 	x = creall(z);
    283 	y = cimagl(z);
    284 	__k_expl(x, &hi, &lo, &k);
    285 
    286 	exp_x = (lo + hi) * 0x1p16382L;
    287 	expt += k - 16382;
    288 
    289 	scale1 = 1;
    290 	half_expt = expt / 2;
    291 	SET_LDBL_EXPSIGN(scale1, BIAS + half_expt);
    292 	scale2 = 1;
    293 	SET_LDBL_EXPSIGN(scale2, BIAS + expt - half_expt);
    294 
    295 	sincosl(y, &s, &c);
    296 	return (CMPLXL(c * exp_x * scale1 * scale2,
    297 	    s * exp_x * scale1 * scale2));
    298 }
    299 #endif /* _COMPLEX_H */
    300