1/*	$OpenBSD: s_exp2l.c,v 1.2 2014/07/21 01:51:11 guenther Exp $	*/
2/*-
3 * Copyright (c) 2005-2008 David Schultz <das@FreeBSD.ORG>
4 * All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 *    notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 *    notice, this list of conditions and the following disclaimer in the
13 *    documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
18 * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
25 * SUCH DAMAGE.
26 */
27
28#include <sys/types.h>
29#include <machine/ieee.h>
30#include <float.h>
31#include <math.h>
32#include <stdint.h>
33
34#define	TBLBITS	7
35#define	TBLSIZE	(1 << TBLBITS)
36
37#define	BIAS	(LDBL_MAX_EXP - 1)
38#define	EXPMASK	(BIAS + LDBL_MAX_EXP)
39
40#if 0 /* XXX Prevent gcc from erroneously constant folding this. */
41static const long double twom10000 = 0x1p-10000L;
42#else
43static volatile long double twom10000 = 0x1p-10000L;
44#endif
45
46static const long double
47    huge      = 0x1p10000L,
48    P1        = 0x1.62e42fefa39ef35793c7673007e6p-1L,
49    P2	      = 0x1.ebfbdff82c58ea86f16b06ec9736p-3L,
50    P3        = 0x1.c6b08d704a0bf8b33a762bad3459p-5L,
51    P4        = 0x1.3b2ab6fba4e7729ccbbe0b4f3fc2p-7L,
52    P5        = 0x1.5d87fe78a67311071dee13fd11d9p-10L,
53    P6        = 0x1.430912f86c7876f4b663b23c5fe5p-13L;
54
55static const double
56    P7        = 0x1.ffcbfc588b041p-17,
57    P8        = 0x1.62c0223a5c7c7p-20,
58    P9        = 0x1.b52541ff59713p-24,
59    P10       = 0x1.e4cf56a391e22p-28,
60    redux     = 0x1.8p112 / TBLSIZE;
61
62static const long double tbl[TBLSIZE] = {
63	0x1.6a09e667f3bcc908b2fb1366dfeap-1L,
64	0x1.6c012750bdabeed76a99800f4edep-1L,
65	0x1.6dfb23c651a2ef220e2cbe1bc0d4p-1L,
66	0x1.6ff7df9519483cf87e1b4f3e1e98p-1L,
67	0x1.71f75e8ec5f73dd2370f2ef0b148p-1L,
68	0x1.73f9a48a58173bd5c9a4e68ab074p-1L,
69	0x1.75feb564267c8bf6e9aa33a489a8p-1L,
70	0x1.780694fde5d3f619ae02808592a4p-1L,
71	0x1.7a11473eb0186d7d51023f6ccb1ap-1L,
72	0x1.7c1ed0130c1327c49334459378dep-1L,
73	0x1.7e2f336cf4e62105d02ba1579756p-1L,
74	0x1.80427543e1a11b60de67649a3842p-1L,
75	0x1.82589994cce128acf88afab34928p-1L,
76	0x1.8471a4623c7acce52f6b97c6444cp-1L,
77	0x1.868d99b4492ec80e41d90ac2556ap-1L,
78	0x1.88ac7d98a669966530bcdf2d4cc0p-1L,
79	0x1.8ace5422aa0db5ba7c55a192c648p-1L,
80	0x1.8cf3216b5448bef2aa1cd161c57ap-1L,
81	0x1.8f1ae991577362b982745c72eddap-1L,
82	0x1.9145b0b91ffc588a61b469f6b6a0p-1L,
83	0x1.93737b0cdc5e4f4501c3f2540ae8p-1L,
84	0x1.95a44cbc8520ee9b483695a0e7fep-1L,
85	0x1.97d829fde4e4f8b9e920f91e8eb6p-1L,
86	0x1.9a0f170ca07b9ba3109b8c467844p-1L,
87	0x1.9c49182a3f0901c7c46b071f28dep-1L,
88	0x1.9e86319e323231824ca78e64c462p-1L,
89	0x1.a0c667b5de564b29ada8b8cabbacp-1L,
90	0x1.a309bec4a2d3358c171f770db1f4p-1L,
91	0x1.a5503b23e255c8b424491caf88ccp-1L,
92	0x1.a799e1330b3586f2dfb2b158f31ep-1L,
93	0x1.a9e6b5579fdbf43eb243bdff53a2p-1L,
94	0x1.ac36bbfd3f379c0db966a3126988p-1L,
95	0x1.ae89f995ad3ad5e8734d17731c80p-1L,
96	0x1.b0e07298db66590842acdfc6fb4ep-1L,
97	0x1.b33a2b84f15faf6bfd0e7bd941b0p-1L,
98	0x1.b59728de559398e3881111648738p-1L,
99	0x1.b7f76f2fb5e46eaa7b081ab53ff6p-1L,
100	0x1.ba5b030a10649840cb3c6af5b74cp-1L,
101	0x1.bcc1e904bc1d2247ba0f45b3d06cp-1L,
102	0x1.bf2c25bd71e088408d7025190cd0p-1L,
103	0x1.c199bdd85529c2220cb12a0916bap-1L,
104	0x1.c40ab5fffd07a6d14df820f17deap-1L,
105	0x1.c67f12e57d14b4a2137fd20f2a26p-1L,
106	0x1.c8f6d9406e7b511acbc48805c3f6p-1L,
107	0x1.cb720dcef90691503cbd1e949d0ap-1L,
108	0x1.cdf0b555dc3f9c44f8958fac4f12p-1L,
109	0x1.d072d4a07897b8d0f22f21a13792p-1L,
110	0x1.d2f87080d89f18ade123989ea50ep-1L,
111	0x1.d5818dcfba48725da05aeb66dff8p-1L,
112	0x1.d80e316c98397bb84f9d048807a0p-1L,
113	0x1.da9e603db3285708c01a5b6d480cp-1L,
114	0x1.dd321f301b4604b695de3c0630c0p-1L,
115	0x1.dfc97337b9b5eb968cac39ed284cp-1L,
116	0x1.e264614f5a128a12761fa17adc74p-1L,
117	0x1.e502ee78b3ff6273d130153992d0p-1L,
118	0x1.e7a51fbc74c834b548b2832378a4p-1L,
119	0x1.ea4afa2a490d9858f73a18f5dab4p-1L,
120	0x1.ecf482d8e67f08db0312fb949d50p-1L,
121	0x1.efa1bee615a27771fd21a92dabb6p-1L,
122	0x1.f252b376bba974e8696fc3638f24p-1L,
123	0x1.f50765b6e4540674f84b762861a6p-1L,
124	0x1.f7bfdad9cbe138913b4bfe72bd78p-1L,
125	0x1.fa7c1819e90d82e90a7e74b26360p-1L,
126	0x1.fd3c22b8f71f10975ba4b32bd006p-1L,
127	0x1.0000000000000000000000000000p+0L,
128	0x1.0163da9fb33356d84a66ae336e98p+0L,
129	0x1.02c9a3e778060ee6f7caca4f7a18p+0L,
130	0x1.04315e86e7f84bd738f9a20da442p+0L,
131	0x1.059b0d31585743ae7c548eb68c6ap+0L,
132	0x1.0706b29ddf6ddc6dc403a9d87b1ep+0L,
133	0x1.0874518759bc808c35f25d942856p+0L,
134	0x1.09e3ecac6f3834521e060c584d5cp+0L,
135	0x1.0b5586cf9890f6298b92b7184200p+0L,
136	0x1.0cc922b7247f7407b705b893dbdep+0L,
137	0x1.0e3ec32d3d1a2020742e4f8af794p+0L,
138	0x1.0fb66affed31af232091dd8a169ep+0L,
139	0x1.11301d0125b50a4ebbf1aed9321cp+0L,
140	0x1.12abdc06c31cbfb92bad324d6f84p+0L,
141	0x1.1429aaea92ddfb34101943b2588ep+0L,
142	0x1.15a98c8a58e512480d573dd562aep+0L,
143	0x1.172b83c7d517adcdf7c8c50eb162p+0L,
144	0x1.18af9388c8de9bbbf70b9a3c269cp+0L,
145	0x1.1a35beb6fcb753cb698f692d2038p+0L,
146	0x1.1bbe084045cd39ab1e72b442810ep+0L,
147	0x1.1d4873168b9aa7805b8028990be8p+0L,
148	0x1.1ed5022fcd91cb8819ff61121fbep+0L,
149	0x1.2063b88628cd63b8eeb0295093f6p+0L,
150	0x1.21f49917ddc962552fd29294bc20p+0L,
151	0x1.2387a6e75623866c1fadb1c159c0p+0L,
152	0x1.251ce4fb2a63f3582ab7de9e9562p+0L,
153	0x1.26b4565e27cdd257a673281d3068p+0L,
154	0x1.284dfe1f5638096cf15cf03c9fa0p+0L,
155	0x1.29e9df51fdee12c25d15f5a25022p+0L,
156	0x1.2b87fd0dad98ffddea46538fca24p+0L,
157	0x1.2d285a6e4030b40091d536d0733ep+0L,
158	0x1.2ecafa93e2f5611ca0f45d5239a4p+0L,
159	0x1.306fe0a31b7152de8d5a463063bep+0L,
160	0x1.32170fc4cd8313539cf1c3009330p+0L,
161	0x1.33c08b26416ff4c9c8610d96680ep+0L,
162	0x1.356c55f929ff0c94623476373be4p+0L,
163	0x1.371a7373aa9caa7145502f45452ap+0L,
164	0x1.38cae6d05d86585a9cb0d9bed530p+0L,
165	0x1.3a7db34e59ff6ea1bc9299e0a1fep+0L,
166	0x1.3c32dc313a8e484001f228b58cf0p+0L,
167	0x1.3dea64c12342235b41223e13d7eep+0L,
168	0x1.3fa4504ac801ba0bf701aa417b9cp+0L,
169	0x1.4160a21f72e29f84325b8f3dbacap+0L,
170	0x1.431f5d950a896dc704439410b628p+0L,
171	0x1.44e086061892d03136f409df0724p+0L,
172	0x1.46a41ed1d005772512f459229f0ap+0L,
173	0x1.486a2b5c13cd013c1a3b69062f26p+0L,
174	0x1.4a32af0d7d3de672d8bcf46f99b4p+0L,
175	0x1.4bfdad5362a271d4397afec42e36p+0L,
176	0x1.4dcb299fddd0d63b36ef1a9e19dep+0L,
177	0x1.4f9b2769d2ca6ad33d8b69aa0b8cp+0L,
178	0x1.516daa2cf6641c112f52c84d6066p+0L,
179	0x1.5342b569d4f81df0a83c49d86bf4p+0L,
180	0x1.551a4ca5d920ec52ec620243540cp+0L,
181	0x1.56f4736b527da66ecb004764e61ep+0L,
182	0x1.58d12d497c7fd252bc2b7343d554p+0L,
183	0x1.5ab07dd48542958c93015191e9a8p+0L,
184	0x1.5c9268a5946b701c4b1b81697ed4p+0L,
185	0x1.5e76f15ad21486e9be4c20399d12p+0L,
186	0x1.605e1b976dc08b076f592a487066p+0L,
187	0x1.6247eb03a5584b1f0fa06fd2d9eap+0L,
188	0x1.6434634ccc31fc76f8714c4ee122p+0L,
189	0x1.66238825522249127d9e29b92ea2p+0L,
190	0x1.68155d44ca973081c57227b9f69ep+0L,
191};
192
193static const float eps[TBLSIZE] = {
194	-0x1.5c50p-101,
195	-0x1.5d00p-106,
196	 0x1.8e90p-102,
197	-0x1.5340p-103,
198	 0x1.1bd0p-102,
199	-0x1.4600p-105,
200	-0x1.7a40p-104,
201	 0x1.d590p-102,
202	-0x1.d590p-101,
203	 0x1.b100p-103,
204	-0x1.0d80p-105,
205	 0x1.6b00p-103,
206	-0x1.9f00p-105,
207	 0x1.c400p-103,
208	 0x1.e120p-103,
209	-0x1.c100p-104,
210	-0x1.9d20p-103,
211	 0x1.a800p-108,
212	 0x1.4c00p-106,
213	-0x1.9500p-106,
214	 0x1.6900p-105,
215	-0x1.29d0p-100,
216	 0x1.4c60p-103,
217	 0x1.13a0p-102,
218	-0x1.5b60p-103,
219	-0x1.1c40p-103,
220	 0x1.db80p-102,
221	 0x1.91a0p-102,
222	 0x1.dc00p-105,
223	 0x1.44c0p-104,
224	 0x1.9710p-102,
225	 0x1.8760p-103,
226	-0x1.a720p-103,
227	 0x1.ed20p-103,
228	-0x1.49c0p-102,
229	-0x1.e000p-111,
230	 0x1.86a0p-103,
231	 0x1.2b40p-103,
232	-0x1.b400p-108,
233	 0x1.1280p-99,
234	-0x1.02d8p-102,
235	-0x1.e3d0p-103,
236	-0x1.b080p-105,
237	-0x1.f100p-107,
238	-0x1.16c0p-105,
239	-0x1.1190p-103,
240	-0x1.a7d2p-100,
241	 0x1.3450p-103,
242	-0x1.67c0p-105,
243	 0x1.4b80p-104,
244	-0x1.c4e0p-103,
245	 0x1.6000p-108,
246	-0x1.3f60p-105,
247	 0x1.93f0p-104,
248	 0x1.5fe0p-105,
249	 0x1.6f80p-107,
250	-0x1.7600p-106,
251	 0x1.21e0p-106,
252	-0x1.3a40p-106,
253	-0x1.40c0p-104,
254	-0x1.9860p-105,
255	-0x1.5d40p-108,
256	-0x1.1d70p-106,
257	 0x1.2760p-105,
258	 0x0.0000p+0,
259	 0x1.21e2p-104,
260	-0x1.9520p-108,
261	-0x1.5720p-106,
262	-0x1.4810p-106,
263	-0x1.be00p-109,
264	 0x1.0080p-105,
265	-0x1.5780p-108,
266	-0x1.d460p-105,
267	-0x1.6140p-105,
268	 0x1.4630p-104,
269	 0x1.ad50p-103,
270	 0x1.82e0p-105,
271	 0x1.1d3cp-101,
272	 0x1.6100p-107,
273	 0x1.ec30p-104,
274	 0x1.f200p-108,
275	 0x1.0b40p-103,
276	 0x1.3660p-102,
277	 0x1.d9d0p-103,
278	-0x1.02d0p-102,
279	 0x1.b070p-103,
280	 0x1.b9c0p-104,
281	-0x1.01c0p-103,
282	-0x1.dfe0p-103,
283	 0x1.1b60p-104,
284	-0x1.ae94p-101,
285	-0x1.3340p-104,
286	 0x1.b3d8p-102,
287	-0x1.6e40p-105,
288	-0x1.3670p-103,
289	 0x1.c140p-104,
290	 0x1.1840p-101,
291	 0x1.1ab0p-102,
292	-0x1.a400p-104,
293	 0x1.1f00p-104,
294	-0x1.7180p-103,
295	 0x1.4ce0p-102,
296	 0x1.9200p-107,
297	-0x1.54c0p-103,
298	 0x1.1b80p-105,
299	-0x1.1828p-101,
300	 0x1.5720p-102,
301	-0x1.a060p-100,
302	 0x1.9160p-102,
303	 0x1.a280p-104,
304	 0x1.3400p-107,
305	 0x1.2b20p-102,
306	 0x1.7800p-108,
307	 0x1.cfd0p-101,
308	 0x1.2ef0p-102,
309	-0x1.2760p-99,
310	 0x1.b380p-104,
311	 0x1.0048p-101,
312	-0x1.60b0p-102,
313	 0x1.a1ccp-100,
314	-0x1.a640p-104,
315	-0x1.08a0p-101,
316	 0x1.7e60p-102,
317	 0x1.22c0p-103,
318	-0x1.7200p-106,
319	 0x1.f0f0p-102,
320	 0x1.eb4ep-99,
321	 0x1.c6e0p-103,
322};
323
324/*
325 * exp2l(x): compute the base 2 exponential of x
326 *
327 * Accuracy: Peak error < 0.502 ulp.
328 *
329 * Method: (accurate tables)
330 *
331 *   Reduce x:
332 *     x = 2**k + y, for integer k and |y| <= 1/2.
333 *     Thus we have exp2(x) = 2**k * exp2(y).
334 *
335 *   Reduce y:
336 *     y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
337 *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
338 *     with |z - eps[i]| <= 2**-8 + 2**-98 for the table used.
339 *
340 *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
341 *   a degree-10 minimax polynomial with maximum error under 2**-120.
342 *   The values in exp2t[] and eps[] are chosen such that
343 *   exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
344 *   that exp2t[i] is accurate to 2**-122.
345 *
346 *   Note that the range of i is +-TBLSIZE/2, so we actually index the tables
347 *   by i0 = i + TBLSIZE/2.
348 *
349 *   This method is due to Gal, with many details due to Gal and Bachelis:
350 *
351 *	Gal, S. and Bachelis, B.  An Accurate Elementary Mathematical Library
352 *	for the IEEE Floating Point Standard.  TOMS 17(1), 26-46 (1991).
353 */
354long double
355exp2l(long double x)
356{
357	union {
358		long double e;
359		struct ieee_ext bits;
360	} u, v;
361	long double r, t, twopk, twopkp10000, z;
362	uint32_t es, hx, ix, i0;
363	int k;
364
365	u.e = x;
366
367	/* Filter out exceptional cases. */
368	hx = (u.bits.ext_sign << 15) | u.bits.ext_exp;
369	ix = hx & EXPMASK;
370	if (ix >= BIAS + 14) {		/* |x| >= 16384 */
371		if (ix == BIAS + LDBL_MAX_EXP) {
372			if (u.bits.ext_frach != 0
373			    || u.bits.ext_frachm != 0
374			    || u.bits.ext_fraclm != 0
375			    || u.bits.ext_fracl != 0
376			    || (hx & 0x8000) == 0)
377				return (x + x);	/* x is NaN or +Inf */
378			else
379				return (0.0);	/* x is -Inf */
380		}
381		if (x >= 16384)
382			return (huge * huge); /* overflow */
383		if (x <= -16495)
384			return (twom10000 * twom10000); /* underflow */
385	} else if (ix <= BIAS - 115) {		/* |x| < 0x1p-115 */
386		return (1.0 + x);
387	}
388
389	/*
390	 * Reduce x, computing z, i0, and k. The low bits of x + redux
391	 * contain the 16-bit integer part of the exponent (k) followed by
392	 * TBLBITS fractional bits (i0). We use bit tricks to extract these
393	 * as integers, then set z to the remainder.
394	 *
395	 * Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8.
396	 * Then the low-order word of x + redux is 0x000abc12,
397	 * We split this into k = 0xabc and i0 = 0x12 (adjusted to
398	 * index into the table), then we compute z = 0x0.003456p0.
399	 *
400	 * XXX If the exponent is negative, the computation of k depends on
401	 *     '>>' doing sign extension.
402	 */
403	u.e = x + redux;
404	i0 = ((((uint64_t)u.bits.ext_fraclm << EXT_FRACLBITS)
405		| u.bits.ext_fracl) & 0xffffffff) + TBLSIZE / 2;
406	k = (int)i0 >> TBLBITS;
407	i0 = i0 & (TBLSIZE - 1);
408	u.e -= redux;
409	z = x - u.e;
410	v.bits.ext_frach = 0;
411	v.bits.ext_frachm = 0;
412	v.bits.ext_fraclm = 0;
413	v.bits.ext_fracl = 0;
414	if (k >= LDBL_MIN_EXP) {
415		es = LDBL_MAX_EXP - 1 + k;
416		v.bits.ext_exp = es & 0x7fffffff;
417		v.bits.ext_sign = u.bits.ext_sign >> 15;
418		twopk = v.e;
419	} else {
420		es = LDBL_MAX_EXP - 1 + k + 10000;
421		v.bits.ext_exp = es & 0x7fffffff;
422		v.bits.ext_sign = u.bits.ext_sign >> 15;
423		twopkp10000 = v.e;
424	}
425
426	/* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
427	t = tbl[i0];		/* exp2t[i0] */
428	z -= eps[i0];		/* eps[i0]   */
429	r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + z * (P6
430	    + z * (P7 + z * (P8 + z * (P9 + z * P10)))))))));
431
432	/* Scale by 2**k. */
433	if(k >= LDBL_MIN_EXP) {
434		if (k == LDBL_MAX_EXP)
435			return (r * 2.0 * 0x1p16383L);
436		return (r * twopk);
437	} else {
438		return (r * twopkp10000 * twom10000);
439	}
440}
441