b_log__D.c revision 1.3 
1/*	$OpenBSD: b_log__D.c,v 1.3 2008/07/16 15:25:51 martynas Exp $	*/
2/*
3 * Copyright (c) 1992, 1993
4 *	The Regents of the University of California.  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 * 3. Neither the name of the University nor the names of its contributors
15 *    may be used to endorse or promote products derived from this software
16 *    without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31#ifndef lint
32static char sccsid[] = "@(#)log.c	8.2 (Berkeley) 11/30/93";
33#endif /* not lint */
34
35#include "math.h"
36#include "math_private.h"
37
38/* Table-driven natural logarithm.
39 *
40 * This code was derived, with minor modifications, from:
41 *	Peter Tang, "Table-Driven Implementation of the
42 *	Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
43 *	Math Software, vol 16. no 4, pp 378-400, Dec 1990).
44 *
45 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256,
46 * where F = j/128 for j an integer in [0, 128].
47 *
48 * log(2^m) = log2_hi*m + log2_tail*m
49 * since m is an integer, the dominant term is exact.
50 * m has at most 10 digits (for subnormal numbers),
51 * and log2_hi has 11 trailing zero bits.
52 *
53 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
54 * logF_hi[] + 512 is exact.
55 *
56 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
57 * the leading term is calculated to extra precision in two
58 * parts, the larger of which adds exactly to the dominant
59 * m and F terms.
60 * There are two cases:
61 *	1. when m, j are non-zero (m | j), use absolute
62 *	   precision for the leading term.
63 *	2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
64 *	   In this case, use a relative precision of 24 bits.
65 * (This is done differently in the original paper)
66 *
67 * Special cases:
68 *	0	return signalling -Inf
69 *	neg	return signalling NaN
70 *	+Inf	return +Inf
71*/
72
73#define N 128
74
75/* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
76 * Used for generation of extend precision logarithms.
77 * The constant 35184372088832 is 2^45, so the divide is exact.
78 * It ensures correct reading of logF_head, even for inaccurate
79 * decimal-to-binary conversion routines.  (Everybody gets the
80 * right answer for integers less than 2^53.)
81 * Values for log(F) were generated using error < 10^-57 absolute
82 * with the bc -l package.
83*/
84static const double	A1 = 	  .08333333333333178827;
85static const double	A2 = 	  .01250000000377174923;
86static const double	A3 =	 .002232139987919447809;
87static const double	A4 =	.0004348877777076145742;
88
89static const double logF_head[N+1] = {
90	0.,
91	.007782140442060381246,
92	.015504186535963526694,
93	.023167059281547608406,
94	.030771658666765233647,
95	.038318864302141264488,
96	.045809536031242714670,
97	.053244514518837604555,
98	.060624621816486978786,
99	.067950661908525944454,
100	.075223421237524235039,
101	.082443669210988446138,
102	.089612158689760690322,
103	.096729626458454731618,
104	.103796793681567578460,
105	.110814366340264314203,
106	.117783035656430001836,
107	.124703478501032805070,
108	.131576357788617315236,
109	.138402322859292326029,
110	.145182009844575077295,
111	.151916042025732167530,
112	.158605030176659056451,
113	.165249572895390883786,
114	.171850256926518341060,
115	.178407657472689606947,
116	.184922338493834104156,
117	.191394852999565046047,
118	.197825743329758552135,
119	.204215541428766300668,
120	.210564769107350002741,
121	.216873938300523150246,
122	.223143551314024080056,
123	.229374101064877322642,
124	.235566071312860003672,
125	.241719936886966024758,
126	.247836163904594286577,
127	.253915209980732470285,
128	.259957524436686071567,
129	.265963548496984003577,
130	.271933715484010463114,
131	.277868451003087102435,
132	.283768173130738432519,
133	.289633292582948342896,
134	.295464212893421063199,
135	.301261330578199704177,
136	.307025035294827830512,
137	.312755710004239517729,
138	.318453731118097493890,
139	.324119468654316733591,
140	.329753286372579168528,
141	.335355541920762334484,
142	.340926586970454081892,
143	.346466767346100823488,
144	.351976423156884266063,
145	.357455888922231679316,
146	.362905493689140712376,
147	.368325561158599157352,
148	.373716409793814818840,
149	.379078352934811846353,
150	.384411698910298582632,
151	.389716751140440464951,
152	.394993808240542421117,
153	.400243164127459749579,
154	.405465108107819105498,
155	.410659924985338875558,
156	.415827895143593195825,
157	.420969294644237379543,
158	.426084395310681429691,
159	.431173464818130014464,
160	.436236766774527495726,
161	.441274560805140936281,
162	.446287102628048160113,
163	.451274644139630254358,
164	.456237433481874177232,
165	.461175715122408291790,
166	.466089729924533457960,
167	.470979715219073113985,
168	.475845904869856894947,
169	.480688529345570714212,
170	.485507815781602403149,
171	.490303988045525329653,
172	.495077266798034543171,
173	.499827869556611403822,
174	.504556010751912253908,
175	.509261901790523552335,
176	.513945751101346104405,
177	.518607764208354637958,
178	.523248143765158602036,
179	.527867089620485785417,
180	.532464798869114019908,
181	.537041465897345915436,
182	.541597282432121573947,
183	.546132437597407260909,
184	.550647117952394182793,
185	.555141507540611200965,
186	.559615787935399566777,
187	.564070138285387656651,
188	.568504735352689749561,
189	.572919753562018740922,
190	.577315365035246941260,
191	.581691739635061821900,
192	.586049045003164792433,
193	.590387446602107957005,
194	.594707107746216934174,
195	.599008189645246602594,
196	.603290851438941899687,
197	.607555250224322662688,
198	.611801541106615331955,
199	.616029877215623855590,
200	.620240409751204424537,
201	.624433288012369303032,
202	.628608659422752680256,
203	.632766669570628437213,
204	.636907462236194987781,
205	.641031179420679109171,
206	.645137961373620782978,
207	.649227946625615004450,
208	.653301272011958644725,
209	.657358072709030238911,
210	.661398482245203922502,
211	.665422632544505177065,
212	.669430653942981734871,
213	.673422675212350441142,
214	.677398823590920073911,
215	.681359224807238206267,
216	.685304003098281100392,
217	.689233281238557538017,
218	.693147180560117703862
219};
220
221static const double logF_tail[N+1] = {
222	0.,
223	-.00000000000000543229938420049,
224	 .00000000000000172745674997061,
225	-.00000000000001323017818229233,
226	-.00000000000001154527628289872,
227	-.00000000000000466529469958300,
228	 .00000000000005148849572685810,
229	-.00000000000002532168943117445,
230	-.00000000000005213620639136504,
231	-.00000000000001819506003016881,
232	 .00000000000006329065958724544,
233	 .00000000000008614512936087814,
234	-.00000000000007355770219435028,
235	 .00000000000009638067658552277,
236	 .00000000000007598636597194141,
237	 .00000000000002579999128306990,
238	-.00000000000004654729747598444,
239	-.00000000000007556920687451336,
240	 .00000000000010195735223708472,
241	-.00000000000017319034406422306,
242	-.00000000000007718001336828098,
243	 .00000000000010980754099855238,
244	-.00000000000002047235780046195,
245	-.00000000000008372091099235912,
246	 .00000000000014088127937111135,
247	 .00000000000012869017157588257,
248	 .00000000000017788850778198106,
249	 .00000000000006440856150696891,
250	 .00000000000016132822667240822,
251	-.00000000000007540916511956188,
252	-.00000000000000036507188831790,
253	 .00000000000009120937249914984,
254	 .00000000000018567570959796010,
255	-.00000000000003149265065191483,
256	-.00000000000009309459495196889,
257	 .00000000000017914338601329117,
258	-.00000000000001302979717330866,
259	 .00000000000023097385217586939,
260	 .00000000000023999540484211737,
261	 .00000000000015393776174455408,
262	-.00000000000036870428315837678,
263	 .00000000000036920375082080089,
264	-.00000000000009383417223663699,
265	 .00000000000009433398189512690,
266	 .00000000000041481318704258568,
267	-.00000000000003792316480209314,
268	 .00000000000008403156304792424,
269	-.00000000000034262934348285429,
270	 .00000000000043712191957429145,
271	-.00000000000010475750058776541,
272	-.00000000000011118671389559323,
273	 .00000000000037549577257259853,
274	 .00000000000013912841212197565,
275	 .00000000000010775743037572640,
276	 .00000000000029391859187648000,
277	-.00000000000042790509060060774,
278	 .00000000000022774076114039555,
279	 .00000000000010849569622967912,
280	-.00000000000023073801945705758,
281	 .00000000000015761203773969435,
282	 .00000000000003345710269544082,
283	-.00000000000041525158063436123,
284	 .00000000000032655698896907146,
285	-.00000000000044704265010452446,
286	 .00000000000034527647952039772,
287	-.00000000000007048962392109746,
288	 .00000000000011776978751369214,
289	-.00000000000010774341461609578,
290	 .00000000000021863343293215910,
291	 .00000000000024132639491333131,
292	 .00000000000039057462209830700,
293	-.00000000000026570679203560751,
294	 .00000000000037135141919592021,
295	-.00000000000017166921336082431,
296	-.00000000000028658285157914353,
297	-.00000000000023812542263446809,
298	 .00000000000006576659768580062,
299	-.00000000000028210143846181267,
300	 .00000000000010701931762114254,
301	 .00000000000018119346366441110,
302	 .00000000000009840465278232627,
303	-.00000000000033149150282752542,
304	-.00000000000018302857356041668,
305	-.00000000000016207400156744949,
306	 .00000000000048303314949553201,
307	-.00000000000071560553172382115,
308	 .00000000000088821239518571855,
309	-.00000000000030900580513238244,
310	-.00000000000061076551972851496,
311	 .00000000000035659969663347830,
312	 .00000000000035782396591276383,
313	-.00000000000046226087001544578,
314	 .00000000000062279762917225156,
315	 .00000000000072838947272065741,
316	 .00000000000026809646615211673,
317	-.00000000000010960825046059278,
318	 .00000000000002311949383800537,
319	-.00000000000058469058005299247,
320	-.00000000000002103748251144494,
321	-.00000000000023323182945587408,
322	-.00000000000042333694288141916,
323	-.00000000000043933937969737844,
324	 .00000000000041341647073835565,
325	 .00000000000006841763641591466,
326	 .00000000000047585534004430641,
327	 .00000000000083679678674757695,
328	-.00000000000085763734646658640,
329	 .00000000000021913281229340092,
330	-.00000000000062242842536431148,
331	-.00000000000010983594325438430,
332	 .00000000000065310431377633651,
333	-.00000000000047580199021710769,
334	-.00000000000037854251265457040,
335	 .00000000000040939233218678664,
336	 .00000000000087424383914858291,
337	 .00000000000025218188456842882,
338	-.00000000000003608131360422557,
339	-.00000000000050518555924280902,
340	 .00000000000078699403323355317,
341	-.00000000000067020876961949060,
342	 .00000000000016108575753932458,
343	 .00000000000058527188436251509,
344	-.00000000000035246757297904791,
345	-.00000000000018372084495629058,
346	 .00000000000088606689813494916,
347	 .00000000000066486268071468700,
348	 .00000000000063831615170646519,
349	 .00000000000025144230728376072,
350	-.00000000000017239444525614834
351};
352
353/*
354 * Extra precision variant, returning struct {double a, b;};
355 * log(x) = a+b to 63 bits, with a rounded to 26 bits.
356 */
357struct Double
358__log__D(double x)
359{
360	int m, j;
361	double F, f, g, q, u, v, u2;
362	volatile double u1;
363	struct Double r;
364
365	/* Argument reduction: 1 <= g < 2; x/2^m = g;	*/
366	/* y = F*(1 + f/F) for |f| <= 2^-8		*/
367
368	m = logb(x);
369	g = ldexp(x, -m);
370	if (m == -1022) {
371		j = logb(g);
372		m += j;
373		g = ldexp(g, -j);
374	}
375	j = N*(g-1) + .5;
376	F = (1.0/N) * j + 1;
377	f = g - F;
378
379	g = 1/(2*F+f);
380	u = 2*f*g;
381	v = u*u;
382	q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
383	if (m | j) {
384		u1 = u + 513;
385		u1 -= 513;
386	}
387	else {
388		u1 = u;
389		TRUNC(u1);
390	}
391	u2 = (2.0*(f - F*u1) - u1*f) * g;
392
393	u1 += m*logF_head[N] + logF_head[j];
394
395	u2 +=  logF_tail[j]; u2 += q;
396	u2 += logF_tail[N]*m;
397	r.a = u1 + u2;			/* Only difference is here */
398	TRUNC(r.a);
399	r.b = (u1 - r.a) + u2;
400	return (r);
401}
402