b_log.c revision 330897 
1/*-
2 * SPDX-License-Identifier: BSD-4-Clause
3 *
4 * Copyright (c) 1992, 1993
5 *	The Regents of the University of California.  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, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 *    notice, this list of conditions and the following disclaimer in the
14 *    documentation and/or other materials provided with the distribution.
15 * 3. All advertising materials mentioning features or use of this software
16 *    must display the following acknowledgement:
17 *	This product includes software developed by the University of
18 *	California, Berkeley and its contributors.
19 * 4. Neither the name of the University nor the names of its contributors
20 *    may be used to endorse or promote products derived from this software
21 *    without specific prior written permission.
22 *
23 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
24 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
25 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
26 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
27 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
28 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
29 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
30 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
31 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
32 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33 * SUCH DAMAGE.
34 */
35
36/* @(#)log.c	8.2 (Berkeley) 11/30/93 */
37#include <sys/cdefs.h>
38__FBSDID("$FreeBSD: stable/11/lib/msun/bsdsrc/b_log.c 330897 2018-03-14 03:19:51Z eadler $");
39
40#include <math.h>
41
42#include "mathimpl.h"
43
44/* Table-driven natural logarithm.
45 *
46 * This code was derived, with minor modifications, from:
47 *	Peter Tang, "Table-Driven Implementation of the
48 *	Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
49 *	Math Software, vol 16. no 4, pp 378-400, Dec 1990).
50 *
51 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
52 * where F = j/128 for j an integer in [0, 128].
53 *
54 * log(2^m) = log2_hi*m + log2_tail*m
55 * since m is an integer, the dominant term is exact.
56 * m has at most 10 digits (for subnormal numbers),
57 * and log2_hi has 11 trailing zero bits.
58 *
59 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
60 * logF_hi[] + 512 is exact.
61 *
62 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
63 * the leading term is calculated to extra precision in two
64 * parts, the larger of which adds exactly to the dominant
65 * m and F terms.
66 * There are two cases:
67 *	1. when m, j are non-zero (m | j), use absolute
68 *	   precision for the leading term.
69 *	2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
70 *	   In this case, use a relative precision of 24 bits.
71 * (This is done differently in the original paper)
72 *
73 * Special cases:
74 *	0	return signalling -Inf
75 *	neg	return signalling NaN
76 *	+Inf	return +Inf
77*/
78
79#define N 128
80
81/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
82 * Used for generation of extend precision logarithms.
83 * The constant 35184372088832 is 2^45, so the divide is exact.
84 * It ensures correct reading of logF_head, even for inaccurate
85 * decimal-to-binary conversion routines.  (Everybody gets the
86 * right answer for integers less than 2^53.)
87 * Values for log(F) were generated using error < 10^-57 absolute
88 * with the bc -l package.
89*/
90static double	A1 = 	  .08333333333333178827;
91static double	A2 = 	  .01250000000377174923;
92static double	A3 =	 .002232139987919447809;
93static double	A4 =	.0004348877777076145742;
94
95static double logF_head[N+1] = {
96	0.,
97	.007782140442060381246,
98	.015504186535963526694,
99	.023167059281547608406,
100	.030771658666765233647,
101	.038318864302141264488,
102	.045809536031242714670,
103	.053244514518837604555,
104	.060624621816486978786,
105	.067950661908525944454,
106	.075223421237524235039,
107	.082443669210988446138,
108	.089612158689760690322,
109	.096729626458454731618,
110	.103796793681567578460,
111	.110814366340264314203,
112	.117783035656430001836,
113	.124703478501032805070,
114	.131576357788617315236,
115	.138402322859292326029,
116	.145182009844575077295,
117	.151916042025732167530,
118	.158605030176659056451,
119	.165249572895390883786,
120	.171850256926518341060,
121	.178407657472689606947,
122	.184922338493834104156,
123	.191394852999565046047,
124	.197825743329758552135,
125	.204215541428766300668,
126	.210564769107350002741,
127	.216873938300523150246,
128	.223143551314024080056,
129	.229374101064877322642,
130	.235566071312860003672,
131	.241719936886966024758,
132	.247836163904594286577,
133	.253915209980732470285,
134	.259957524436686071567,
135	.265963548496984003577,
136	.271933715484010463114,
137	.277868451003087102435,
138	.283768173130738432519,
139	.289633292582948342896,
140	.295464212893421063199,
141	.301261330578199704177,
142	.307025035294827830512,
143	.312755710004239517729,
144	.318453731118097493890,
145	.324119468654316733591,
146	.329753286372579168528,
147	.335355541920762334484,
148	.340926586970454081892,
149	.346466767346100823488,
150	.351976423156884266063,
151	.357455888922231679316,
152	.362905493689140712376,
153	.368325561158599157352,
154	.373716409793814818840,
155	.379078352934811846353,
156	.384411698910298582632,
157	.389716751140440464951,
158	.394993808240542421117,
159	.400243164127459749579,
160	.405465108107819105498,
161	.410659924985338875558,
162	.415827895143593195825,
163	.420969294644237379543,
164	.426084395310681429691,
165	.431173464818130014464,
166	.436236766774527495726,
167	.441274560805140936281,
168	.446287102628048160113,
169	.451274644139630254358,
170	.456237433481874177232,
171	.461175715122408291790,
172	.466089729924533457960,
173	.470979715219073113985,
174	.475845904869856894947,
175	.480688529345570714212,
176	.485507815781602403149,
177	.490303988045525329653,
178	.495077266798034543171,
179	.499827869556611403822,
180	.504556010751912253908,
181	.509261901790523552335,
182	.513945751101346104405,
183	.518607764208354637958,
184	.523248143765158602036,
185	.527867089620485785417,
186	.532464798869114019908,
187	.537041465897345915436,
188	.541597282432121573947,
189	.546132437597407260909,
190	.550647117952394182793,
191	.555141507540611200965,
192	.559615787935399566777,
193	.564070138285387656651,
194	.568504735352689749561,
195	.572919753562018740922,
196	.577315365035246941260,
197	.581691739635061821900,
198	.586049045003164792433,
199	.590387446602107957005,
200	.594707107746216934174,
201	.599008189645246602594,
202	.603290851438941899687,
203	.607555250224322662688,
204	.611801541106615331955,
205	.616029877215623855590,
206	.620240409751204424537,
207	.624433288012369303032,
208	.628608659422752680256,
209	.632766669570628437213,
210	.636907462236194987781,
211	.641031179420679109171,
212	.645137961373620782978,
213	.649227946625615004450,
214	.653301272011958644725,
215	.657358072709030238911,
216	.661398482245203922502,
217	.665422632544505177065,
218	.669430653942981734871,
219	.673422675212350441142,
220	.677398823590920073911,
221	.681359224807238206267,
222	.685304003098281100392,
223	.689233281238557538017,
224	.693147180560117703862
225};
226
227static double logF_tail[N+1] = {
228	0.,
229	-.00000000000000543229938420049,
230	 .00000000000000172745674997061,
231	-.00000000000001323017818229233,
232	-.00000000000001154527628289872,
233	-.00000000000000466529469958300,
234	 .00000000000005148849572685810,
235	-.00000000000002532168943117445,
236	-.00000000000005213620639136504,
237	-.00000000000001819506003016881,
238	 .00000000000006329065958724544,
239	 .00000000000008614512936087814,
240	-.00000000000007355770219435028,
241	 .00000000000009638067658552277,
242	 .00000000000007598636597194141,
243	 .00000000000002579999128306990,
244	-.00000000000004654729747598444,
245	-.00000000000007556920687451336,
246	 .00000000000010195735223708472,
247	-.00000000000017319034406422306,
248	-.00000000000007718001336828098,
249	 .00000000000010980754099855238,
250	-.00000000000002047235780046195,
251	-.00000000000008372091099235912,
252	 .00000000000014088127937111135,
253	 .00000000000012869017157588257,
254	 .00000000000017788850778198106,
255	 .00000000000006440856150696891,
256	 .00000000000016132822667240822,
257	-.00000000000007540916511956188,
258	-.00000000000000036507188831790,
259	 .00000000000009120937249914984,
260	 .00000000000018567570959796010,
261	-.00000000000003149265065191483,
262	-.00000000000009309459495196889,
263	 .00000000000017914338601329117,
264	-.00000000000001302979717330866,
265	 .00000000000023097385217586939,
266	 .00000000000023999540484211737,
267	 .00000000000015393776174455408,
268	-.00000000000036870428315837678,
269	 .00000000000036920375082080089,
270	-.00000000000009383417223663699,
271	 .00000000000009433398189512690,
272	 .00000000000041481318704258568,
273	-.00000000000003792316480209314,
274	 .00000000000008403156304792424,
275	-.00000000000034262934348285429,
276	 .00000000000043712191957429145,
277	-.00000000000010475750058776541,
278	-.00000000000011118671389559323,
279	 .00000000000037549577257259853,
280	 .00000000000013912841212197565,
281	 .00000000000010775743037572640,
282	 .00000000000029391859187648000,
283	-.00000000000042790509060060774,
284	 .00000000000022774076114039555,
285	 .00000000000010849569622967912,
286	-.00000000000023073801945705758,
287	 .00000000000015761203773969435,
288	 .00000000000003345710269544082,
289	-.00000000000041525158063436123,
290	 .00000000000032655698896907146,
291	-.00000000000044704265010452446,
292	 .00000000000034527647952039772,
293	-.00000000000007048962392109746,
294	 .00000000000011776978751369214,
295	-.00000000000010774341461609578,
296	 .00000000000021863343293215910,
297	 .00000000000024132639491333131,
298	 .00000000000039057462209830700,
299	-.00000000000026570679203560751,
300	 .00000000000037135141919592021,
301	-.00000000000017166921336082431,
302	-.00000000000028658285157914353,
303	-.00000000000023812542263446809,
304	 .00000000000006576659768580062,
305	-.00000000000028210143846181267,
306	 .00000000000010701931762114254,
307	 .00000000000018119346366441110,
308	 .00000000000009840465278232627,
309	-.00000000000033149150282752542,
310	-.00000000000018302857356041668,
311	-.00000000000016207400156744949,
312	 .00000000000048303314949553201,
313	-.00000000000071560553172382115,
314	 .00000000000088821239518571855,
315	-.00000000000030900580513238244,
316	-.00000000000061076551972851496,
317	 .00000000000035659969663347830,
318	 .00000000000035782396591276383,
319	-.00000000000046226087001544578,
320	 .00000000000062279762917225156,
321	 .00000000000072838947272065741,
322	 .00000000000026809646615211673,
323	-.00000000000010960825046059278,
324	 .00000000000002311949383800537,
325	-.00000000000058469058005299247,
326	-.00000000000002103748251144494,
327	-.00000000000023323182945587408,
328	-.00000000000042333694288141916,
329	-.00000000000043933937969737844,
330	 .00000000000041341647073835565,
331	 .00000000000006841763641591466,
332	 .00000000000047585534004430641,
333	 .00000000000083679678674757695,
334	-.00000000000085763734646658640,
335	 .00000000000021913281229340092,
336	-.00000000000062242842536431148,
337	-.00000000000010983594325438430,
338	 .00000000000065310431377633651,
339	-.00000000000047580199021710769,
340	-.00000000000037854251265457040,
341	 .00000000000040939233218678664,
342	 .00000000000087424383914858291,
343	 .00000000000025218188456842882,
344	-.00000000000003608131360422557,
345	-.00000000000050518555924280902,
346	 .00000000000078699403323355317,
347	-.00000000000067020876961949060,
348	 .00000000000016108575753932458,
349	 .00000000000058527188436251509,
350	-.00000000000035246757297904791,
351	-.00000000000018372084495629058,
352	 .00000000000088606689813494916,
353	 .00000000000066486268071468700,
354	 .00000000000063831615170646519,
355	 .00000000000025144230728376072,
356	-.00000000000017239444525614834
357};
358
359#if 0
360double
361#ifdef _ANSI_SOURCE
362log(double x)
363#else
364log(x) double x;
365#endif
366{
367	int m, j;
368	double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
369	volatile double u1;
370
371	/* Catch special cases */
372	if (x <= 0)
373		if (x == zero)	/* log(0) = -Inf */
374			return (-one/zero);
375		else		/* log(neg) = NaN */
376			return (zero/zero);
377	else if (!finite(x))
378		return (x+x);		/* x = NaN, Inf */
379
380	/* Argument reduction: 1 <= g < 2; x/2^m = g;	*/
381	/* y = F*(1 + f/F) for |f| <= 2^-8		*/
382
383	m = logb(x);
384	g = ldexp(x, -m);
385	if (m == -1022) {
386		j = logb(g), m += j;
387		g = ldexp(g, -j);
388	}
389	j = N*(g-1) + .5;
390	F = (1.0/N) * j + 1;	/* F*128 is an integer in [128, 512] */
391	f = g - F;
392
393	/* Approximate expansion for log(1+f/F) ~= u + q */
394	g = 1/(2*F+f);
395	u = 2*f*g;
396	v = u*u;
397	q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
398
399    /* case 1: u1 = u rounded to 2^-43 absolute.  Since u < 2^-8,
400     * 	       u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits.
401     *         It also adds exactly to |m*log2_hi + log_F_head[j] | < 750
402    */
403	if (m | j)
404		u1 = u + 513, u1 -= 513;
405
406    /* case 2:	|1-x| < 1/256. The m- and j- dependent terms are zero;
407     * 		u1 = u to 24 bits.
408    */
409	else
410		u1 = u, TRUNC(u1);
411	u2 = (2.0*(f - F*u1) - u1*f) * g;
412			/* u1 + u2 = 2f/(2F+f) to extra precision.	*/
413
414	/* log(x) = log(2^m*F*(1+f/F)) =				*/
415	/* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q);	*/
416	/* (exact) + (tiny)						*/
417
418	u1 += m*logF_head[N] + logF_head[j];		/* exact */
419	u2 = (u2 + logF_tail[j]) + q;			/* tiny */
420	u2 += logF_tail[N]*m;
421	return (u1 + u2);
422}
423#endif
424
425/*
426 * Extra precision variant, returning struct {double a, b;};
427 * log(x) = a+b to 63 bits, with a rounded to 26 bits.
428 */
429struct Double
430#ifdef _ANSI_SOURCE
431__log__D(double x)
432#else
433__log__D(x) double x;
434#endif
435{
436	int m, j;
437	double F, f, g, q, u, v, u2;
438	volatile double u1;
439	struct Double r;
440
441	/* Argument reduction: 1 <= g < 2; x/2^m = g;	*/
442	/* y = F*(1 + f/F) for |f| <= 2^-8		*/
443
444	m = logb(x);
445	g = ldexp(x, -m);
446	if (m == -1022) {
447		j = logb(g), m += j;
448		g = ldexp(g, -j);
449	}
450	j = N*(g-1) + .5;
451	F = (1.0/N) * j + 1;
452	f = g - F;
453
454	g = 1/(2*F+f);
455	u = 2*f*g;
456	v = u*u;
457	q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
458	if (m | j)
459		u1 = u + 513, u1 -= 513;
460	else
461		u1 = u, TRUNC(u1);
462	u2 = (2.0*(f - F*u1) - u1*f) * g;
463
464	u1 += m*logF_head[N] + logF_head[j];
465
466	u2 +=  logF_tail[j]; u2 += q;
467	u2 += logF_tail[N]*m;
468	r.a = u1 + u2;			/* Only difference is here */
469	TRUNC(r.a);
470	r.b = (u1 - r.a) + u2;
471	return (r);
472}
473