libm: Sync with FreeBSD (gains 6 long double functions)
[dragonfly.git] / lib / libm / ld80 / k_expl.h
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2fedfd5c
JM
1/* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */
2
3/*-
4 * Copyright (c) 2009-2013 Steven G. Kargl
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice unmodified, this list of conditions, and the following
12 * disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in the
15 * documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 *
28 * $FreeBSD$
29 *
30 * Optimized by Bruce D. Evans.
31 */
32
33
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
45static 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 */
51INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
52L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */
53L2 = -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 */
59A2 = 0.5,
60A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
61A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
62A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
63A6 = 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 */
74static 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 * mantissa 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 */
219static 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 /* Use a specialized rint() to get fn. Assume round-to-nearest. */
227 fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
228 r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */
229#if defined(HAVE_EFFICIENT_IRINTL)
230 n = irintl(fn);
231#elif defined(HAVE_EFFICIENT_IRINT)
232 n = irint(fn);
233#else
234 n = (int)fn;
235#endif
236 n2 = (unsigned)n % INTERVALS;
237 /* Depend on the sign bit being propagated: */
238 *kp = n >> LOG2_INTERVALS;
239 r1 = x - fn * L1;
240 r2 = fn * -L2;
241
242 /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */
243 z = r * r;
244#if 0
245 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
246#else
247 q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6));
248#endif
249 t = (long double)tbl[n2].lo + tbl[n2].hi;
250 *hip = tbl[n2].hi;
251 *lop = tbl[n2].lo + t * (q + r1);
252}
253
254static inline void
255k_hexpl(long double x, long double *hip, long double *lop)
256{
257 float twopkm1;
258 int k;
259
260 __k_expl(x, hip, lop, &k);
261 SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23));
262 *hip *= twopkm1;
263 *lop *= twopkm1;
264}
265
266static inline long double
267hexpl(long double x)
268{
269 long double hi, lo, twopkm2;
270 int k;
271
272 twopkm2 = 1;
273 __k_expl(x, &hi, &lo, &k);
274 SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2);
275 return (lo + hi) * 2 * twopkm2;
276}
277
278#ifdef _COMPLEX_H
279/*
280 * See ../src/k_exp.c for details.
281 */
282static inline long double complex
283__ldexp_cexpl(long double complex z, int expt)
284{
285 long double exp_x, hi, lo;
286 long double x, y, scale1, scale2;
287 int half_expt, k;
288
289 x = creall(z);
290 y = cimagl(z);
291 __k_expl(x, &hi, &lo, &k);
292
293 exp_x = (lo + hi) * 0x1p16382;
294 expt += k - 16382;
295
296 scale1 = 1;
297 half_expt = expt / 2;
298 SET_LDBL_EXPSIGN(scale1, BIAS + half_expt);
299 scale2 = 1;
300 SET_LDBL_EXPSIGN(scale1, BIAS + expt - half_expt);
301
302 return (cpackl(cos(y) * exp_x * scale1 * scale2,
303 sinl(y) * exp_x * scale1 * scale2));
304}
305#endif /* _COMPLEX_H */