xref: /freebsd/lib/msun/bsdsrc/b_log.c (revision fd45b686) 
1 /*-
2  * SPDX-License-Identifier: BSD-3-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. Neither the name of the University nor the names of its contributors
16  *    may be used to endorse or promote products derived from this software
17  *    without specific prior written permission.
18  *
19  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
20  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
21  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
22  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
23  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
24  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
25  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
26  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
27  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
28  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
29  * SUCH DAMAGE.
30  */
31 
32 /* Table-driven natural logarithm.
33  *
34  * This code was derived, with minor modifications, from:
35  *	Peter Tang, "Table-Driven Implementation of the
36  *	Logarithm in IEEE Floating-Point arithmetic." ACM Trans.
37  *	Math Software, vol 16. no 4, pp 378-400, Dec 1990).
38  *
39  * Calculates log(2^m*F*(1+f/F)), |f/F| <= 1/256,
40  * where F = j/128 for j an integer in [0, 128].
41  *
42  * log(2^m) = log2_hi*m + log2_tail*m
43  * The leading term is exact, because m is an integer,
44  * m has at most 10 digits (for subnormal numbers),
45  * and log2_hi has 11 trailing zero bits.
46  *
47  * log(F) = logF_hi[j] + logF_lo[j] is in table below.
48  * logF_hi[] + 512 is exact.
49  *
50  * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ...
51  *
52  * The leading term is calculated to extra precision in two
53  * parts, the larger of which adds exactly to the dominant
54  * m and F terms.
55  *
56  * There are two cases:
57  *	1. When m and j are non-zero (m | j), use absolute
58  *	   precision for the leading term.
59  *	2. When m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1).
60  *	   In this case, use a relative precision of 24 bits.
61  * (This is done differently in the original paper)
62  *
63  * Special cases:
64  *	0	return signalling -Inf
65  *	neg	return signalling NaN
66  *	+Inf	return +Inf
67  */
68 
69 #define N 128
70 
71 /*
72  * Coefficients in the polynomial approximation of log(1+f/F).
73  * Domain of x is [0,1./256] with 2**(-64.187) precision.
74  */
75 static const double
76     A1 =  8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */
77     A2 =  1.2499999999943598e-02, /* 0x3f899999, 0x99991a98 */
78     A3 =  2.2321527525957776e-03; /* 0x3f624929, 0xe24e70be */
79 
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  */
90 static double logF_head[N+1] = {
91 	0.,
92 	.007782140442060381246,
93 	.015504186535963526694,
94 	.023167059281547608406,
95 	.030771658666765233647,
96 	.038318864302141264488,
97 	.045809536031242714670,
98 	.053244514518837604555,
99 	.060624621816486978786,
100 	.067950661908525944454,
101 	.075223421237524235039,
102 	.082443669210988446138,
103 	.089612158689760690322,
104 	.096729626458454731618,
105 	.103796793681567578460,
106 	.110814366340264314203,
107 	.117783035656430001836,
108 	.124703478501032805070,
109 	.131576357788617315236,
110 	.138402322859292326029,
111 	.145182009844575077295,
112 	.151916042025732167530,
113 	.158605030176659056451,
114 	.165249572895390883786,
115 	.171850256926518341060,
116 	.178407657472689606947,
117 	.184922338493834104156,
118 	.191394852999565046047,
119 	.197825743329758552135,
120 	.204215541428766300668,
121 	.210564769107350002741,
122 	.216873938300523150246,
123 	.223143551314024080056,
124 	.229374101064877322642,
125 	.235566071312860003672,
126 	.241719936886966024758,
127 	.247836163904594286577,
128 	.253915209980732470285,
129 	.259957524436686071567,
130 	.265963548496984003577,
131 	.271933715484010463114,
132 	.277868451003087102435,
133 	.283768173130738432519,
134 	.289633292582948342896,
135 	.295464212893421063199,
136 	.301261330578199704177,
137 	.307025035294827830512,
138 	.312755710004239517729,
139 	.318453731118097493890,
140 	.324119468654316733591,
141 	.329753286372579168528,
142 	.335355541920762334484,
143 	.340926586970454081892,
144 	.346466767346100823488,
145 	.351976423156884266063,
146 	.357455888922231679316,
147 	.362905493689140712376,
148 	.368325561158599157352,
149 	.373716409793814818840,
150 	.379078352934811846353,
151 	.384411698910298582632,
152 	.389716751140440464951,
153 	.394993808240542421117,
154 	.400243164127459749579,
155 	.405465108107819105498,
156 	.410659924985338875558,
157 	.415827895143593195825,
158 	.420969294644237379543,
159 	.426084395310681429691,
160 	.431173464818130014464,
161 	.436236766774527495726,
162 	.441274560805140936281,
163 	.446287102628048160113,
164 	.451274644139630254358,
165 	.456237433481874177232,
166 	.461175715122408291790,
167 	.466089729924533457960,
168 	.470979715219073113985,
169 	.475845904869856894947,
170 	.480688529345570714212,
171 	.485507815781602403149,
172 	.490303988045525329653,
173 	.495077266798034543171,
174 	.499827869556611403822,
175 	.504556010751912253908,
176 	.509261901790523552335,
177 	.513945751101346104405,
178 	.518607764208354637958,
179 	.523248143765158602036,
180 	.527867089620485785417,
181 	.532464798869114019908,
182 	.537041465897345915436,
183 	.541597282432121573947,
184 	.546132437597407260909,
185 	.550647117952394182793,
186 	.555141507540611200965,
187 	.559615787935399566777,
188 	.564070138285387656651,
189 	.568504735352689749561,
190 	.572919753562018740922,
191 	.577315365035246941260,
192 	.581691739635061821900,
193 	.586049045003164792433,
194 	.590387446602107957005,
195 	.594707107746216934174,
196 	.599008189645246602594,
197 	.603290851438941899687,
198 	.607555250224322662688,
199 	.611801541106615331955,
200 	.616029877215623855590,
201 	.620240409751204424537,
202 	.624433288012369303032,
203 	.628608659422752680256,
204 	.632766669570628437213,
205 	.636907462236194987781,
206 	.641031179420679109171,
207 	.645137961373620782978,
208 	.649227946625615004450,
209 	.653301272011958644725,
210 	.657358072709030238911,
211 	.661398482245203922502,
212 	.665422632544505177065,
213 	.669430653942981734871,
214 	.673422675212350441142,
215 	.677398823590920073911,
216 	.681359224807238206267,
217 	.685304003098281100392,
218 	.689233281238557538017,
219 	.693147180560117703862
220 };
221 
222 static double logF_tail[N+1] = {
223 	0.,
224 	-.00000000000000543229938420049,
225 	 .00000000000000172745674997061,
226 	-.00000000000001323017818229233,
227 	-.00000000000001154527628289872,
228 	-.00000000000000466529469958300,
229 	 .00000000000005148849572685810,
230 	-.00000000000002532168943117445,
231 	-.00000000000005213620639136504,
232 	-.00000000000001819506003016881,
233 	 .00000000000006329065958724544,
234 	 .00000000000008614512936087814,
235 	-.00000000000007355770219435028,
236 	 .00000000000009638067658552277,
237 	 .00000000000007598636597194141,
238 	 .00000000000002579999128306990,
239 	-.00000000000004654729747598444,
240 	-.00000000000007556920687451336,
241 	 .00000000000010195735223708472,
242 	-.00000000000017319034406422306,
243 	-.00000000000007718001336828098,
244 	 .00000000000010980754099855238,
245 	-.00000000000002047235780046195,
246 	-.00000000000008372091099235912,
247 	 .00000000000014088127937111135,
248 	 .00000000000012869017157588257,
249 	 .00000000000017788850778198106,
250 	 .00000000000006440856150696891,
251 	 .00000000000016132822667240822,
252 	-.00000000000007540916511956188,
253 	-.00000000000000036507188831790,
254 	 .00000000000009120937249914984,
255 	 .00000000000018567570959796010,
256 	-.00000000000003149265065191483,
257 	-.00000000000009309459495196889,
258 	 .00000000000017914338601329117,
259 	-.00000000000001302979717330866,
260 	 .00000000000023097385217586939,
261 	 .00000000000023999540484211737,
262 	 .00000000000015393776174455408,
263 	-.00000000000036870428315837678,
264 	 .00000000000036920375082080089,
265 	-.00000000000009383417223663699,
266 	 .00000000000009433398189512690,
267 	 .00000000000041481318704258568,
268 	-.00000000000003792316480209314,
269 	 .00000000000008403156304792424,
270 	-.00000000000034262934348285429,
271 	 .00000000000043712191957429145,
272 	-.00000000000010475750058776541,
273 	-.00000000000011118671389559323,
274 	 .00000000000037549577257259853,
275 	 .00000000000013912841212197565,
276 	 .00000000000010775743037572640,
277 	 .00000000000029391859187648000,
278 	-.00000000000042790509060060774,
279 	 .00000000000022774076114039555,
280 	 .00000000000010849569622967912,
281 	-.00000000000023073801945705758,
282 	 .00000000000015761203773969435,
283 	 .00000000000003345710269544082,
284 	-.00000000000041525158063436123,
285 	 .00000000000032655698896907146,
286 	-.00000000000044704265010452446,
287 	 .00000000000034527647952039772,
288 	-.00000000000007048962392109746,
289 	 .00000000000011776978751369214,
290 	-.00000000000010774341461609578,
291 	 .00000000000021863343293215910,
292 	 .00000000000024132639491333131,
293 	 .00000000000039057462209830700,
294 	-.00000000000026570679203560751,
295 	 .00000000000037135141919592021,
296 	-.00000000000017166921336082431,
297 	-.00000000000028658285157914353,
298 	-.00000000000023812542263446809,
299 	 .00000000000006576659768580062,
300 	-.00000000000028210143846181267,
301 	 .00000000000010701931762114254,
302 	 .00000000000018119346366441110,
303 	 .00000000000009840465278232627,
304 	-.00000000000033149150282752542,
305 	-.00000000000018302857356041668,
306 	-.00000000000016207400156744949,
307 	 .00000000000048303314949553201,
308 	-.00000000000071560553172382115,
309 	 .00000000000088821239518571855,
310 	-.00000000000030900580513238244,
311 	-.00000000000061076551972851496,
312 	 .00000000000035659969663347830,
313 	 .00000000000035782396591276383,
314 	-.00000000000046226087001544578,
315 	 .00000000000062279762917225156,
316 	 .00000000000072838947272065741,
317 	 .00000000000026809646615211673,
318 	-.00000000000010960825046059278,
319 	 .00000000000002311949383800537,
320 	-.00000000000058469058005299247,
321 	-.00000000000002103748251144494,
322 	-.00000000000023323182945587408,
323 	-.00000000000042333694288141916,
324 	-.00000000000043933937969737844,
325 	 .00000000000041341647073835565,
326 	 .00000000000006841763641591466,
327 	 .00000000000047585534004430641,
328 	 .00000000000083679678674757695,
329 	-.00000000000085763734646658640,
330 	 .00000000000021913281229340092,
331 	-.00000000000062242842536431148,
332 	-.00000000000010983594325438430,
333 	 .00000000000065310431377633651,
334 	-.00000000000047580199021710769,
335 	-.00000000000037854251265457040,
336 	 .00000000000040939233218678664,
337 	 .00000000000087424383914858291,
338 	 .00000000000025218188456842882,
339 	-.00000000000003608131360422557,
340 	-.00000000000050518555924280902,
341 	 .00000000000078699403323355317,
342 	-.00000000000067020876961949060,
343 	 .00000000000016108575753932458,
344 	 .00000000000058527188436251509,
345 	-.00000000000035246757297904791,
346 	-.00000000000018372084495629058,
347 	 .00000000000088606689813494916,
348 	 .00000000000066486268071468700,
349 	 .00000000000063831615170646519,
350 	 .00000000000025144230728376072,
351 	-.00000000000017239444525614834
352 };
353 /*
354  * Extra precision variant, returning struct {double a, b;};
355  * log(x) = a+b to 63 bits, with 'a' rounded to 24 bits.
356  */
357 static struct Double
358 __log__D(double x)
359 {
360 	int m, j;
361 	double F, f, g, q, u, v, u1, u2;
362 	struct Double r;
363 
364 	/*
365 	 * Argument reduction: 1 <= g < 2; x/2^m = g;
366 	 * y = F*(1 + f/F) for |f| <= 2^-8
367 	 */
368 	g = frexp(x, &m);
369 	g *= 2;
370 	m--;
371 	if (m == -1022) {
372 		j = ilogb(g);
373 		m += j;
374 		g = ldexp(g, -j);
375 	}
376 	j = N * (g - 1) + 0.5;
377 	F = (1. / N) * j + 1;
378 	f = g - F;
379 
380 	g = 1 / (2 * F + f);
381 	u = 2 * f * g;
382 	v = u * u;
383 	q = u * v * (A1 + v * (A2 + v * A3));
384 	if (m | j) {
385 		u1 = u + 513;
386 		u1 -= 513;
387 	} else {
388 		u1 = (float)u;
389 	}
390 	u2 = (2 * (f - F * u1) - u1 * f) * g;
391 
392 	u1 += m * logF_head[N] + logF_head[j];
393 
394 	u2 += logF_tail[j];
395 	u2 += q;
396 	u2 += logF_tail[N] * m;
397 	r.a = (float)(u1 + u2);		/* Only difference is here. */
398 	r.b = (u1 - r.a) + u2;
399 	return (r);
400 }
401