1 /*-
2 * Copyright (c) 2005-2008 David Schultz <das@FreeBSD.ORG>
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE.
25 */
26
27
28
29
30 #define TBLBITS 7
31 #define TBLSIZE (1 << TBLBITS)
32
33 #define BIAS (LDBL_MAX_EXP - 1)
34 #define EXPMASK (BIAS + LDBL_MAX_EXP)
35
36 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */
37 static const long double twom10000 = 0x1p-10000L;
38 #else
39 static volatile long double twom10000 = 0x1p-10000L;
40 #endif
41
42 static const long double
43 huge = 0x1p10000L,
44 P1 = 0x1.62e42fefa39ef35793c7673007e6p-1L,
45 P2 = 0x1.ebfbdff82c58ea86f16b06ec9736p-3L,
46 P3 = 0x1.c6b08d704a0bf8b33a762bad3459p-5L,
47 P4 = 0x1.3b2ab6fba4e7729ccbbe0b4f3fc2p-7L,
48 P5 = 0x1.5d87fe78a67311071dee13fd11d9p-10L,
49 P6 = 0x1.430912f86c7876f4b663b23c5fe5p-13L;
50
51 static const double
52 P7 = 0x1.ffcbfc588b041p-17,
53 P8 = 0x1.62c0223a5c7c7p-20,
54 P9 = 0x1.b52541ff59713p-24,
55 P10 = 0x1.e4cf56a391e22p-28,
56 redux = 0x1.8p112 / TBLSIZE;
57
58 static const long double tbl[TBLSIZE] = {
59 0x1.6a09e667f3bcc908b2fb1366dfeap-1L,
60 0x1.6c012750bdabeed76a99800f4edep-1L,
61 0x1.6dfb23c651a2ef220e2cbe1bc0d4p-1L,
62 0x1.6ff7df9519483cf87e1b4f3e1e98p-1L,
63 0x1.71f75e8ec5f73dd2370f2ef0b148p-1L,
64 0x1.73f9a48a58173bd5c9a4e68ab074p-1L,
65 0x1.75feb564267c8bf6e9aa33a489a8p-1L,
66 0x1.780694fde5d3f619ae02808592a4p-1L,
67 0x1.7a11473eb0186d7d51023f6ccb1ap-1L,
68 0x1.7c1ed0130c1327c49334459378dep-1L,
69 0x1.7e2f336cf4e62105d02ba1579756p-1L,
70 0x1.80427543e1a11b60de67649a3842p-1L,
71 0x1.82589994cce128acf88afab34928p-1L,
72 0x1.8471a4623c7acce52f6b97c6444cp-1L,
73 0x1.868d99b4492ec80e41d90ac2556ap-1L,
74 0x1.88ac7d98a669966530bcdf2d4cc0p-1L,
75 0x1.8ace5422aa0db5ba7c55a192c648p-1L,
76 0x1.8cf3216b5448bef2aa1cd161c57ap-1L,
77 0x1.8f1ae991577362b982745c72eddap-1L,
78 0x1.9145b0b91ffc588a61b469f6b6a0p-1L,
79 0x1.93737b0cdc5e4f4501c3f2540ae8p-1L,
80 0x1.95a44cbc8520ee9b483695a0e7fep-1L,
81 0x1.97d829fde4e4f8b9e920f91e8eb6p-1L,
82 0x1.9a0f170ca07b9ba3109b8c467844p-1L,
83 0x1.9c49182a3f0901c7c46b071f28dep-1L,
84 0x1.9e86319e323231824ca78e64c462p-1L,
85 0x1.a0c667b5de564b29ada8b8cabbacp-1L,
86 0x1.a309bec4a2d3358c171f770db1f4p-1L,
87 0x1.a5503b23e255c8b424491caf88ccp-1L,
88 0x1.a799e1330b3586f2dfb2b158f31ep-1L,
89 0x1.a9e6b5579fdbf43eb243bdff53a2p-1L,
90 0x1.ac36bbfd3f379c0db966a3126988p-1L,
91 0x1.ae89f995ad3ad5e8734d17731c80p-1L,
92 0x1.b0e07298db66590842acdfc6fb4ep-1L,
93 0x1.b33a2b84f15faf6bfd0e7bd941b0p-1L,
94 0x1.b59728de559398e3881111648738p-1L,
95 0x1.b7f76f2fb5e46eaa7b081ab53ff6p-1L,
96 0x1.ba5b030a10649840cb3c6af5b74cp-1L,
97 0x1.bcc1e904bc1d2247ba0f45b3d06cp-1L,
98 0x1.bf2c25bd71e088408d7025190cd0p-1L,
99 0x1.c199bdd85529c2220cb12a0916bap-1L,
100 0x1.c40ab5fffd07a6d14df820f17deap-1L,
101 0x1.c67f12e57d14b4a2137fd20f2a26p-1L,
102 0x1.c8f6d9406e7b511acbc48805c3f6p-1L,
103 0x1.cb720dcef90691503cbd1e949d0ap-1L,
104 0x1.cdf0b555dc3f9c44f8958fac4f12p-1L,
105 0x1.d072d4a07897b8d0f22f21a13792p-1L,
106 0x1.d2f87080d89f18ade123989ea50ep-1L,
107 0x1.d5818dcfba48725da05aeb66dff8p-1L,
108 0x1.d80e316c98397bb84f9d048807a0p-1L,
109 0x1.da9e603db3285708c01a5b6d480cp-1L,
110 0x1.dd321f301b4604b695de3c0630c0p-1L,
111 0x1.dfc97337b9b5eb968cac39ed284cp-1L,
112 0x1.e264614f5a128a12761fa17adc74p-1L,
113 0x1.e502ee78b3ff6273d130153992d0p-1L,
114 0x1.e7a51fbc74c834b548b2832378a4p-1L,
115 0x1.ea4afa2a490d9858f73a18f5dab4p-1L,
116 0x1.ecf482d8e67f08db0312fb949d50p-1L,
117 0x1.efa1bee615a27771fd21a92dabb6p-1L,
118 0x1.f252b376bba974e8696fc3638f24p-1L,
119 0x1.f50765b6e4540674f84b762861a6p-1L,
120 0x1.f7bfdad9cbe138913b4bfe72bd78p-1L,
121 0x1.fa7c1819e90d82e90a7e74b26360p-1L,
122 0x1.fd3c22b8f71f10975ba4b32bd006p-1L,
123 0x1.0000000000000000000000000000p+0L,
124 0x1.0163da9fb33356d84a66ae336e98p+0L,
125 0x1.02c9a3e778060ee6f7caca4f7a18p+0L,
126 0x1.04315e86e7f84bd738f9a20da442p+0L,
127 0x1.059b0d31585743ae7c548eb68c6ap+0L,
128 0x1.0706b29ddf6ddc6dc403a9d87b1ep+0L,
129 0x1.0874518759bc808c35f25d942856p+0L,
130 0x1.09e3ecac6f3834521e060c584d5cp+0L,
131 0x1.0b5586cf9890f6298b92b7184200p+0L,
132 0x1.0cc922b7247f7407b705b893dbdep+0L,
133 0x1.0e3ec32d3d1a2020742e4f8af794p+0L,
134 0x1.0fb66affed31af232091dd8a169ep+0L,
135 0x1.11301d0125b50a4ebbf1aed9321cp+0L,
136 0x1.12abdc06c31cbfb92bad324d6f84p+0L,
137 0x1.1429aaea92ddfb34101943b2588ep+0L,
138 0x1.15a98c8a58e512480d573dd562aep+0L,
139 0x1.172b83c7d517adcdf7c8c50eb162p+0L,
140 0x1.18af9388c8de9bbbf70b9a3c269cp+0L,
141 0x1.1a35beb6fcb753cb698f692d2038p+0L,
142 0x1.1bbe084045cd39ab1e72b442810ep+0L,
143 0x1.1d4873168b9aa7805b8028990be8p+0L,
144 0x1.1ed5022fcd91cb8819ff61121fbep+0L,
145 0x1.2063b88628cd63b8eeb0295093f6p+0L,
146 0x1.21f49917ddc962552fd29294bc20p+0L,
147 0x1.2387a6e75623866c1fadb1c159c0p+0L,
148 0x1.251ce4fb2a63f3582ab7de9e9562p+0L,
149 0x1.26b4565e27cdd257a673281d3068p+0L,
150 0x1.284dfe1f5638096cf15cf03c9fa0p+0L,
151 0x1.29e9df51fdee12c25d15f5a25022p+0L,
152 0x1.2b87fd0dad98ffddea46538fca24p+0L,
153 0x1.2d285a6e4030b40091d536d0733ep+0L,
154 0x1.2ecafa93e2f5611ca0f45d5239a4p+0L,
155 0x1.306fe0a31b7152de8d5a463063bep+0L,
156 0x1.32170fc4cd8313539cf1c3009330p+0L,
157 0x1.33c08b26416ff4c9c8610d96680ep+0L,
158 0x1.356c55f929ff0c94623476373be4p+0L,
159 0x1.371a7373aa9caa7145502f45452ap+0L,
160 0x1.38cae6d05d86585a9cb0d9bed530p+0L,
161 0x1.3a7db34e59ff6ea1bc9299e0a1fep+0L,
162 0x1.3c32dc313a8e484001f228b58cf0p+0L,
163 0x1.3dea64c12342235b41223e13d7eep+0L,
164 0x1.3fa4504ac801ba0bf701aa417b9cp+0L,
165 0x1.4160a21f72e29f84325b8f3dbacap+0L,
166 0x1.431f5d950a896dc704439410b628p+0L,
167 0x1.44e086061892d03136f409df0724p+0L,
168 0x1.46a41ed1d005772512f459229f0ap+0L,
169 0x1.486a2b5c13cd013c1a3b69062f26p+0L,
170 0x1.4a32af0d7d3de672d8bcf46f99b4p+0L,
171 0x1.4bfdad5362a271d4397afec42e36p+0L,
172 0x1.4dcb299fddd0d63b36ef1a9e19dep+0L,
173 0x1.4f9b2769d2ca6ad33d8b69aa0b8cp+0L,
174 0x1.516daa2cf6641c112f52c84d6066p+0L,
175 0x1.5342b569d4f81df0a83c49d86bf4p+0L,
176 0x1.551a4ca5d920ec52ec620243540cp+0L,
177 0x1.56f4736b527da66ecb004764e61ep+0L,
178 0x1.58d12d497c7fd252bc2b7343d554p+0L,
179 0x1.5ab07dd48542958c93015191e9a8p+0L,
180 0x1.5c9268a5946b701c4b1b81697ed4p+0L,
181 0x1.5e76f15ad21486e9be4c20399d12p+0L,
182 0x1.605e1b976dc08b076f592a487066p+0L,
183 0x1.6247eb03a5584b1f0fa06fd2d9eap+0L,
184 0x1.6434634ccc31fc76f8714c4ee122p+0L,
185 0x1.66238825522249127d9e29b92ea2p+0L,
186 0x1.68155d44ca973081c57227b9f69ep+0L,
187 };
188
189 static const float eps[TBLSIZE] = {
190 -0x1.5c50p-101,
191 -0x1.5d00p-106,
192 0x1.8e90p-102,
193 -0x1.5340p-103,
194 0x1.1bd0p-102,
195 -0x1.4600p-105,
196 -0x1.7a40p-104,
197 0x1.d590p-102,
198 -0x1.d590p-101,
199 0x1.b100p-103,
200 -0x1.0d80p-105,
201 0x1.6b00p-103,
202 -0x1.9f00p-105,
203 0x1.c400p-103,
204 0x1.e120p-103,
205 -0x1.c100p-104,
206 -0x1.9d20p-103,
207 0x1.a800p-108,
208 0x1.4c00p-106,
209 -0x1.9500p-106,
210 0x1.6900p-105,
211 -0x1.29d0p-100,
212 0x1.4c60p-103,
213 0x1.13a0p-102,
214 -0x1.5b60p-103,
215 -0x1.1c40p-103,
216 0x1.db80p-102,
217 0x1.91a0p-102,
218 0x1.dc00p-105,
219 0x1.44c0p-104,
220 0x1.9710p-102,
221 0x1.8760p-103,
222 -0x1.a720p-103,
223 0x1.ed20p-103,
224 -0x1.49c0p-102,
225 -0x1.e000p-111,
226 0x1.86a0p-103,
227 0x1.2b40p-103,
228 -0x1.b400p-108,
229 0x1.1280p-99,
230 -0x1.02d8p-102,
231 -0x1.e3d0p-103,
232 -0x1.b080p-105,
233 -0x1.f100p-107,
234 -0x1.16c0p-105,
235 -0x1.1190p-103,
236 -0x1.a7d2p-100,
237 0x1.3450p-103,
238 -0x1.67c0p-105,
239 0x1.4b80p-104,
240 -0x1.c4e0p-103,
241 0x1.6000p-108,
242 -0x1.3f60p-105,
243 0x1.93f0p-104,
244 0x1.5fe0p-105,
245 0x1.6f80p-107,
246 -0x1.7600p-106,
247 0x1.21e0p-106,
248 -0x1.3a40p-106,
249 -0x1.40c0p-104,
250 -0x1.9860p-105,
251 -0x1.5d40p-108,
252 -0x1.1d70p-106,
253 0x1.2760p-105,
254 0x0.0000p+0,
255 0x1.21e2p-104,
256 -0x1.9520p-108,
257 -0x1.5720p-106,
258 -0x1.4810p-106,
259 -0x1.be00p-109,
260 0x1.0080p-105,
261 -0x1.5780p-108,
262 -0x1.d460p-105,
263 -0x1.6140p-105,
264 0x1.4630p-104,
265 0x1.ad50p-103,
266 0x1.82e0p-105,
267 0x1.1d3cp-101,
268 0x1.6100p-107,
269 0x1.ec30p-104,
270 0x1.f200p-108,
271 0x1.0b40p-103,
272 0x1.3660p-102,
273 0x1.d9d0p-103,
274 -0x1.02d0p-102,
275 0x1.b070p-103,
276 0x1.b9c0p-104,
277 -0x1.01c0p-103,
278 -0x1.dfe0p-103,
279 0x1.1b60p-104,
280 -0x1.ae94p-101,
281 -0x1.3340p-104,
282 0x1.b3d8p-102,
283 -0x1.6e40p-105,
284 -0x1.3670p-103,
285 0x1.c140p-104,
286 0x1.1840p-101,
287 0x1.1ab0p-102,
288 -0x1.a400p-104,
289 0x1.1f00p-104,
290 -0x1.7180p-103,
291 0x1.4ce0p-102,
292 0x1.9200p-107,
293 -0x1.54c0p-103,
294 0x1.1b80p-105,
295 -0x1.1828p-101,
296 0x1.5720p-102,
297 -0x1.a060p-100,
298 0x1.9160p-102,
299 0x1.a280p-104,
300 0x1.3400p-107,
301 0x1.2b20p-102,
302 0x1.7800p-108,
303 0x1.cfd0p-101,
304 0x1.2ef0p-102,
305 -0x1.2760p-99,
306 0x1.b380p-104,
307 0x1.0048p-101,
308 -0x1.60b0p-102,
309 0x1.a1ccp-100,
310 -0x1.a640p-104,
311 -0x1.08a0p-101,
312 0x1.7e60p-102,
313 0x1.22c0p-103,
314 -0x1.7200p-106,
315 0x1.f0f0p-102,
316 0x1.eb4ep-99,
317 0x1.c6e0p-103,
318 };
319
320 /*
321 * exp2l(x): compute the base 2 exponential of x
322 *
323 * Accuracy: Peak error < 0.502 ulp.
324 *
325 * Method: (accurate tables)
326 *
327 * Reduce x:
328 * x = 2**k + y, for integer k and |y| <= 1/2.
329 * Thus we have exp2(x) = 2**k * exp2(y).
330 *
331 * Reduce y:
332 * y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
333 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
334 * with |z - eps[i]| <= 2**-8 + 2**-98 for the table used.
335 *
336 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
337 * a degree-10 minimax polynomial with maximum error under 2**-120.
338 * The values in exp2t[] and eps[] are chosen such that
339 * exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
340 * that exp2t[i] is accurate to 2**-122.
341 *
342 * Note that the range of i is +-TBLSIZE/2, so we actually index the tables
343 * by i0 = i + TBLSIZE/2.
344 *
345 * This method is due to Gal, with many details due to Gal and Bachelis:
346 *
347 * Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library
348 * for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991).
349 */
350 long double
exp2l(long double x)351 exp2l(long double x)
352 {
353 union IEEEl2bits u, v;
354 long double r, t, twopk, twopkp10000, z;
355 uint32_t hx, ix, i0;
356 int k;
357
358 u.e = x;
359
360 /* Filter out exceptional cases. */
361 hx = u.xbits.expsign;
362 ix = hx & EXPMASK;
363 if (ix >= BIAS + 14) { /* |x| >= 16384 */
364 if (ix == BIAS + LDBL_MAX_EXP) {
365 if (u.xbits.manh != 0
366 || u.xbits.manl != 0
367 || (hx & 0x8000) == 0)
368 return (x + x); /* x is NaN or +Inf */
369 else
370 return (0.0L); /* x is -Inf */
371 }
372 if (x >= 16384)
373 return __math_oflowl(0); /* overflow */
374 if (x <= -16495)
375 return __math_uflowl(0); /* underflow */
376 } else if (ix <= BIAS - 115) { /* |x| < 0x1p-115 */
377 return (1.0L + x);
378 }
379
380 /*
381 * Reduce x, computing z, i0, and k. The low bits of x + redux
382 * contain the 16-bit integer part of the exponent (k) followed by
383 * TBLBITS fractional bits (i0). We use bit tricks to extract these
384 * as integers, then set z to the remainder.
385 *
386 * Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8.
387 * Then the low-order word of x + redux is 0x000abc12,
388 * We split this into k = 0xabc and i0 = 0x12 (adjusted to
389 * index into the table), then we compute z = 0x0.003456p0.
390 *
391 * XXX If the exponent is negative, the computation of k depends on
392 * '>>' doing sign extension.
393 */
394 u.e = x + (long double)redux;
395 i0 = (u.bits.manl & 0xffffffff) + TBLSIZE / 2;
396 k = (int)i0 >> TBLBITS;
397 i0 = i0 & (TBLSIZE - 1);
398 u.e -= (long double)redux;
399 z = x - u.e;
400 v.xbits.manh = 0;
401 v.xbits.manl = 0;
402 if (k >= LDBL_MIN_EXP) {
403 v.xbits.expsign = LDBL_MAX_EXP - 1 + k;
404 twopk = v.e;
405 } else {
406 v.xbits.expsign = LDBL_MAX_EXP - 1 + k + 10000;
407 twopkp10000 = v.e;
408 }
409
410 /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
411 t = tbl[i0]; /* exp2t[i0] */
412 z -= (long double)eps[i0]; /* eps[i0] */
413 r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + z * (P6
414 + z * ((long double)P7 + z * ((long double)P8 + z * ((long double)P9 + z * (long double)P10)))))))));
415
416 /* Scale by 2**k. */
417 if(k >= LDBL_MIN_EXP) {
418 if (k == LDBL_MAX_EXP)
419 return (r * 2.0L * 0x1p16383L);
420 return (r * twopk);
421 } else {
422 return (r * twopkp10000 * twom10000);
423 }
424 }
425
