1/*	$NetBSD$	*/
2
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#if 0
39__FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_log.c 176449 2008-02-22 02:26:51Z das $");
40#else
41__RCSID("$NetBSD$");
42#endif
43
44#include <errno.h>
45#include "math.h"
46#include "math_private.h"
47
48/* Table-driven natural logarithm.
49 *
50 * This code was derived, with minor modifications, from:
51 *	Peter Tang, "Table-Driven Implementation of the
52 *	Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
53 *	Math Software, vol 16. no 4, pp 378-400, Dec 1990).
54 *
55 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
56 * where F = j/128 for j an integer in [0, 128].
57 *
58 * log(2^m) = log2_hi*m + log2_tail*m
59 * since m is an integer, the dominant term is exact.
60 * m has at most 10 digits (for subnormal numbers),
61 * and log2_hi has 11 trailing zero bits.
62 *
63 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
64 * logF_hi[] + 512 is exact.
65 *
66 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
67 * the leading term is calculated to extra precision in two
68 * parts, the larger of which adds exactly to the dominant
69 * m and F terms.
70 * There are two cases:
71 *	1. when m, j are non-zero (m | j), use absolute
72 *	   precision for the leading term.
73 *	2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
74 *	   In this case, use a relative precision of 24 bits.
75 * (This is done differently in the original paper)
76 *
77 * Special cases:
78 *	0	return signalling -Inf
79 *	neg	return signalling NaN
80 *	+Inf	return +Inf
81*/
82
83#define N 128
84
85/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
86 * Used for generation of extend precision logarithms.
87 * The constant 35184372088832 is 2^45, so the divide is exact.
88 * It ensures correct reading of logF_head, even for inaccurate
89 * decimal-to-binary conversion routines.  (Everybody gets the
90 * right answer for integers less than 2^53.)
91 * Values for log(F) were generated using error < 10^-57 absolute
92 * with the bc -l package.
93*/
94static const double	A1 = 	  .08333333333333178827;
95static const double	A2 = 	  .01250000000377174923;
96static const double	A3 =	 .002232139987919447809;
97static const double	A4 =	.0004348877777076145742;
98
99static const double logF_head[N+1] = {
100	0.,
101	.007782140442060381246,
102	.015504186535963526694,
103	.023167059281547608406,
104	.030771658666765233647,
105	.038318864302141264488,
106	.045809536031242714670,
107	.053244514518837604555,
108	.060624621816486978786,
109	.067950661908525944454,
110	.075223421237524235039,
111	.082443669210988446138,
112	.089612158689760690322,
113	.096729626458454731618,
114	.103796793681567578460,
115	.110814366340264314203,
116	.117783035656430001836,
117	.124703478501032805070,
118	.131576357788617315236,
119	.138402322859292326029,
120	.145182009844575077295,
121	.151916042025732167530,
122	.158605030176659056451,
123	.165249572895390883786,
124	.171850256926518341060,
125	.178407657472689606947,
126	.184922338493834104156,
127	.191394852999565046047,
128	.197825743329758552135,
129	.204215541428766300668,
130	.210564769107350002741,
131	.216873938300523150246,
132	.223143551314024080056,
133	.229374101064877322642,
134	.235566071312860003672,
135	.241719936886966024758,
136	.247836163904594286577,
137	.253915209980732470285,
138	.259957524436686071567,
139	.265963548496984003577,
140	.271933715484010463114,
141	.277868451003087102435,
142	.283768173130738432519,
143	.289633292582948342896,
144	.295464212893421063199,
145	.301261330578199704177,
146	.307025035294827830512,
147	.312755710004239517729,
148	.318453731118097493890,
149	.324119468654316733591,
150	.329753286372579168528,
151	.335355541920762334484,
152	.340926586970454081892,
153	.346466767346100823488,
154	.351976423156884266063,
155	.357455888922231679316,
156	.362905493689140712376,
157	.368325561158599157352,
158	.373716409793814818840,
159	.379078352934811846353,
160	.384411698910298582632,
161	.389716751140440464951,
162	.394993808240542421117,
163	.400243164127459749579,
164	.405465108107819105498,
165	.410659924985338875558,
166	.415827895143593195825,
167	.420969294644237379543,
168	.426084395310681429691,
169	.431173464818130014464,
170	.436236766774527495726,
171	.441274560805140936281,
172	.446287102628048160113,
173	.451274644139630254358,
174	.456237433481874177232,
175	.461175715122408291790,
176	.466089729924533457960,
177	.470979715219073113985,
178	.475845904869856894947,
179	.480688529345570714212,
180	.485507815781602403149,
181	.490303988045525329653,
182	.495077266798034543171,
183	.499827869556611403822,
184	.504556010751912253908,
185	.509261901790523552335,
186	.513945751101346104405,
187	.518607764208354637958,
188	.523248143765158602036,
189	.527867089620485785417,
190	.532464798869114019908,
191	.537041465897345915436,
192	.541597282432121573947,
193	.546132437597407260909,
194	.550647117952394182793,
195	.555141507540611200965,
196	.559615787935399566777,
197	.564070138285387656651,
198	.568504735352689749561,
199	.572919753562018740922,
200	.577315365035246941260,
201	.581691739635061821900,
202	.586049045003164792433,
203	.590387446602107957005,
204	.594707107746216934174,
205	.599008189645246602594,
206	.603290851438941899687,
207	.607555250224322662688,
208	.611801541106615331955,
209	.616029877215623855590,
210	.620240409751204424537,
211	.624433288012369303032,
212	.628608659422752680256,
213	.632766669570628437213,
214	.636907462236194987781,
215	.641031179420679109171,
216	.645137961373620782978,
217	.649227946625615004450,
218	.653301272011958644725,
219	.657358072709030238911,
220	.661398482245203922502,
221	.665422632544505177065,
222	.669430653942981734871,
223	.673422675212350441142,
224	.677398823590920073911,
225	.681359224807238206267,
226	.685304003098281100392,
227	.689233281238557538017,
228	.693147180560117703862
229};
230
231static const double logF_tail[N+1] = {
232	0.,
233	-.00000000000000543229938420049,
234	 .00000000000000172745674997061,
235	-.00000000000001323017818229233,
236	-.00000000000001154527628289872,
237	-.00000000000000466529469958300,
238	 .00000000000005148849572685810,
239	-.00000000000002532168943117445,
240	-.00000000000005213620639136504,
241	-.00000000000001819506003016881,
242	 .00000000000006329065958724544,
243	 .00000000000008614512936087814,
244	-.00000000000007355770219435028,
245	 .00000000000009638067658552277,
246	 .00000000000007598636597194141,
247	 .00000000000002579999128306990,
248	-.00000000000004654729747598444,
249	-.00000000000007556920687451336,
250	 .00000000000010195735223708472,
251	-.00000000000017319034406422306,
252	-.00000000000007718001336828098,
253	 .00000000000010980754099855238,
254	-.00000000000002047235780046195,
255	-.00000000000008372091099235912,
256	 .00000000000014088127937111135,
257	 .00000000000012869017157588257,
258	 .00000000000017788850778198106,
259	 .00000000000006440856150696891,
260	 .00000000000016132822667240822,
261	-.00000000000007540916511956188,
262	-.00000000000000036507188831790,
263	 .00000000000009120937249914984,
264	 .00000000000018567570959796010,
265	-.00000000000003149265065191483,
266	-.00000000000009309459495196889,
267	 .00000000000017914338601329117,
268	-.00000000000001302979717330866,
269	 .00000000000023097385217586939,
270	 .00000000000023999540484211737,
271	 .00000000000015393776174455408,
272	-.00000000000036870428315837678,
273	 .00000000000036920375082080089,
274	-.00000000000009383417223663699,
275	 .00000000000009433398189512690,
276	 .00000000000041481318704258568,
277	-.00000000000003792316480209314,
278	 .00000000000008403156304792424,
279	-.00000000000034262934348285429,
280	 .00000000000043712191957429145,
281	-.00000000000010475750058776541,
282	-.00000000000011118671389559323,
283	 .00000000000037549577257259853,
284	 .00000000000013912841212197565,
285	 .00000000000010775743037572640,
286	 .00000000000029391859187648000,
287	-.00000000000042790509060060774,
288	 .00000000000022774076114039555,
289	 .00000000000010849569622967912,
290	-.00000000000023073801945705758,
291	 .00000000000015761203773969435,
292	 .00000000000003345710269544082,
293	-.00000000000041525158063436123,
294	 .00000000000032655698896907146,
295	-.00000000000044704265010452446,
296	 .00000000000034527647952039772,
297	-.00000000000007048962392109746,
298	 .00000000000011776978751369214,
299	-.00000000000010774341461609578,
300	 .00000000000021863343293215910,
301	 .00000000000024132639491333131,
302	 .00000000000039057462209830700,
303	-.00000000000026570679203560751,
304	 .00000000000037135141919592021,
305	-.00000000000017166921336082431,
306	-.00000000000028658285157914353,
307	-.00000000000023812542263446809,
308	 .00000000000006576659768580062,
309	-.00000000000028210143846181267,
310	 .00000000000010701931762114254,
311	 .00000000000018119346366441110,
312	 .00000000000009840465278232627,
313	-.00000000000033149150282752542,
314	-.00000000000018302857356041668,
315	-.00000000000016207400156744949,
316	 .00000000000048303314949553201,
317	-.00000000000071560553172382115,
318	 .00000000000088821239518571855,
319	-.00000000000030900580513238244,
320	-.00000000000061076551972851496,
321	 .00000000000035659969663347830,
322	 .00000000000035782396591276383,
323	-.00000000000046226087001544578,
324	 .00000000000062279762917225156,
325	 .00000000000072838947272065741,
326	 .00000000000026809646615211673,
327	-.00000000000010960825046059278,
328	 .00000000000002311949383800537,
329	-.00000000000058469058005299247,
330	-.00000000000002103748251144494,
331	-.00000000000023323182945587408,
332	-.00000000000042333694288141916,
333	-.00000000000043933937969737844,
334	 .00000000000041341647073835565,
335	 .00000000000006841763641591466,
336	 .00000000000047585534004430641,
337	 .00000000000083679678674757695,
338	-.00000000000085763734646658640,
339	 .00000000000021913281229340092,
340	-.00000000000062242842536431148,
341	-.00000000000010983594325438430,
342	 .00000000000065310431377633651,
343	-.00000000000047580199021710769,
344	-.00000000000037854251265457040,
345	 .00000000000040939233218678664,
346	 .00000000000087424383914858291,
347	 .00000000000025218188456842882,
348	-.00000000000003608131360422557,
349	-.00000000000050518555924280902,
350	 .00000000000078699403323355317,
351	-.00000000000067020876961949060,
352	 .00000000000016108575753932458,
353	 .00000000000058527188436251509,
354	-.00000000000035246757297904791,
355	-.00000000000018372084495629058,
356	 .00000000000088606689813494916,
357	 .00000000000066486268071468700,
358	 .00000000000063831615170646519,
359	 .00000000000025144230728376072,
360	-.00000000000017239444525614834
361};
362
363/*
364 * Extra precision variant, returning struct {double a, b;};
365 * log(x) = a+b to 63 bits, with a rounded to 26 bits.
366 */
367struct Double
368__log__D(double x)
369{
370	int m, j;
371	double F, f, g, q, u, v, u2;
372	volatile double u1;
373	struct Double r;
374
375	/* Argument reduction: 1 <= g < 2; x/2^m = g;	*/
376	/* y = F*(1 + f/F) for |f| <= 2^-8		*/
377
378	m = logb(x);
379	g = ldexp(x, -m);
380	if (m == -1022) {
381		j = logb(g), m += j;
382		g = ldexp(g, -j);
383	}
384	j = N*(g-1) + .5;
385	F = (1.0/N) * j + 1;
386	f = g - F;
387
388	g = 1/(2*F+f);
389	u = 2*f*g;
390	v = u*u;
391	q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
392	if (m | j)
393		u1 = u + 513, u1 -= 513;
394	else
395		u1 = u, TRUNC(u1);
396	u2 = (2.0*(f - F*u1) - u1*f) * g;
397
398	u1 += m*logF_head[N] + logF_head[j];
399
400	u2 +=  logF_tail[j]; u2 += q;
401	u2 += logF_tail[N]*m;
402	r.a = u1 + u2;			/* Only difference is here */
403	TRUNC(r.a);
404	r.b = (u1 - r.a) + u2;
405	return (r);
406}
407