n_log.c revision 1.7.42.1
1 /* $NetBSD: n_log.c,v 1.7.42.1 2014/10/13 19:34:58 martin 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 32 #if 0 33 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93"; 34 #endif 35 #endif /* not lint */ 36 37 #include "../src/namespace.h" 38 39 #include <math.h> 40 #include <errno.h> 41 42 #include "mathimpl.h" 43 44 #ifdef __weak_alias 45 __weak_alias(log, _log); 46 __weak_alias(_logl, _log); 47 __weak_alias(logf, _logf); 48 #endif 49 50 /* Table-driven natural logarithm. 51 * 52 * This code was derived, with minor modifications, from: 53 * Peter Tang, "Table-Driven Implementation of the 54 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 55 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 56 * 57 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 58 * where F = j/128 for j an integer in [0, 128]. 59 * 60 * log(2^m) = log2_hi*m + log2_tail*m 61 * since m is an integer, the dominant term is exact. 62 * m has at most 10 digits (for subnormal numbers), 63 * and log2_hi has 11 trailing zero bits. 64 * 65 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 66 * logF_hi[] + 512 is exact. 67 * 68 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 69 * the leading term is calculated to extra precision in two 70 * parts, the larger of which adds exactly to the dominant 71 * m and F terms. 72 * There are two cases: 73 * 1. when m, j are non-zero (m | j), use absolute 74 * precision for the leading term. 75 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 76 * In this case, use a relative precision of 24 bits. 77 * (This is done differently in the original paper) 78 * 79 * Special cases: 80 * 0 return signalling -Inf 81 * neg return signalling NaN 82 * +Inf return +Inf 83 */ 84 85 #if defined(__vax__) || defined(tahoe) 86 #define _IEEE 0 87 #define TRUNC(x) x = (double) (float) (x) 88 #else 89 #define _IEEE 1 90 #define endian (((*(int *) &one)) ? 1 : 0) 91 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 92 #define infnan(x) 0.0 93 #endif 94 95 #define N 128 96 97 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 98 * Used for generation of extend precision logarithms. 99 * The constant 35184372088832 is 2^45, so the divide is exact. 100 * It ensures correct reading of logF_head, even for inaccurate 101 * decimal-to-binary conversion routines. (Everybody gets the 102 * right answer for integers less than 2^53.) 103 * Values for log(F) were generated using error < 10^-57 absolute 104 * with the bc -l package. 105 */ 106 static const double A1 = .08333333333333178827; 107 static const double A2 = .01250000000377174923; 108 static const double A3 = .002232139987919447809; 109 static const double A4 = .0004348877777076145742; 110 111 static const double logF_head[N+1] = { 112 0., 113 .007782140442060381246, 114 .015504186535963526694, 115 .023167059281547608406, 116 .030771658666765233647, 117 .038318864302141264488, 118 .045809536031242714670, 119 .053244514518837604555, 120 .060624621816486978786, 121 .067950661908525944454, 122 .075223421237524235039, 123 .082443669210988446138, 124 .089612158689760690322, 125 .096729626458454731618, 126 .103796793681567578460, 127 .110814366340264314203, 128 .117783035656430001836, 129 .124703478501032805070, 130 .131576357788617315236, 131 .138402322859292326029, 132 .145182009844575077295, 133 .151916042025732167530, 134 .158605030176659056451, 135 .165249572895390883786, 136 .171850256926518341060, 137 .178407657472689606947, 138 .184922338493834104156, 139 .191394852999565046047, 140 .197825743329758552135, 141 .204215541428766300668, 142 .210564769107350002741, 143 .216873938300523150246, 144 .223143551314024080056, 145 .229374101064877322642, 146 .235566071312860003672, 147 .241719936886966024758, 148 .247836163904594286577, 149 .253915209980732470285, 150 .259957524436686071567, 151 .265963548496984003577, 152 .271933715484010463114, 153 .277868451003087102435, 154 .283768173130738432519, 155 .289633292582948342896, 156 .295464212893421063199, 157 .301261330578199704177, 158 .307025035294827830512, 159 .312755710004239517729, 160 .318453731118097493890, 161 .324119468654316733591, 162 .329753286372579168528, 163 .335355541920762334484, 164 .340926586970454081892, 165 .346466767346100823488, 166 .351976423156884266063, 167 .357455888922231679316, 168 .362905493689140712376, 169 .368325561158599157352, 170 .373716409793814818840, 171 .379078352934811846353, 172 .384411698910298582632, 173 .389716751140440464951, 174 .394993808240542421117, 175 .400243164127459749579, 176 .405465108107819105498, 177 .410659924985338875558, 178 .415827895143593195825, 179 .420969294644237379543, 180 .426084395310681429691, 181 .431173464818130014464, 182 .436236766774527495726, 183 .441274560805140936281, 184 .446287102628048160113, 185 .451274644139630254358, 186 .456237433481874177232, 187 .461175715122408291790, 188 .466089729924533457960, 189 .470979715219073113985, 190 .475845904869856894947, 191 .480688529345570714212, 192 .485507815781602403149, 193 .490303988045525329653, 194 .495077266798034543171, 195 .499827869556611403822, 196 .504556010751912253908, 197 .509261901790523552335, 198 .513945751101346104405, 199 .518607764208354637958, 200 .523248143765158602036, 201 .527867089620485785417, 202 .532464798869114019908, 203 .537041465897345915436, 204 .541597282432121573947, 205 .546132437597407260909, 206 .550647117952394182793, 207 .555141507540611200965, 208 .559615787935399566777, 209 .564070138285387656651, 210 .568504735352689749561, 211 .572919753562018740922, 212 .577315365035246941260, 213 .581691739635061821900, 214 .586049045003164792433, 215 .590387446602107957005, 216 .594707107746216934174, 217 .599008189645246602594, 218 .603290851438941899687, 219 .607555250224322662688, 220 .611801541106615331955, 221 .616029877215623855590, 222 .620240409751204424537, 223 .624433288012369303032, 224 .628608659422752680256, 225 .632766669570628437213, 226 .636907462236194987781, 227 .641031179420679109171, 228 .645137961373620782978, 229 .649227946625615004450, 230 .653301272011958644725, 231 .657358072709030238911, 232 .661398482245203922502, 233 .665422632544505177065, 234 .669430653942981734871, 235 .673422675212350441142, 236 .677398823590920073911, 237 .681359224807238206267, 238 .685304003098281100392, 239 .689233281238557538017, 240 .693147180560117703862 241 }; 242 243 static const double logF_tail[N+1] = { 244 0., 245 -.00000000000000543229938420049, 246 .00000000000000172745674997061, 247 -.00000000000001323017818229233, 248 -.00000000000001154527628289872, 249 -.00000000000000466529469958300, 250 .00000000000005148849572685810, 251 -.00000000000002532168943117445, 252 -.00000000000005213620639136504, 253 -.00000000000001819506003016881, 254 .00000000000006329065958724544, 255 .00000000000008614512936087814, 256 -.00000000000007355770219435028, 257 .00000000000009638067658552277, 258 .00000000000007598636597194141, 259 .00000000000002579999128306990, 260 -.00000000000004654729747598444, 261 -.00000000000007556920687451336, 262 .00000000000010195735223708472, 263 -.00000000000017319034406422306, 264 -.00000000000007718001336828098, 265 .00000000000010980754099855238, 266 -.00000000000002047235780046195, 267 -.00000000000008372091099235912, 268 .00000000000014088127937111135, 269 .00000000000012869017157588257, 270 .00000000000017788850778198106, 271 .00000000000006440856150696891, 272 .00000000000016132822667240822, 273 -.00000000000007540916511956188, 274 -.00000000000000036507188831790, 275 .00000000000009120937249914984, 276 .00000000000018567570959796010, 277 -.00000000000003149265065191483, 278 -.00000000000009309459495196889, 279 .00000000000017914338601329117, 280 -.00000000000001302979717330866, 281 .00000000000023097385217586939, 282 .00000000000023999540484211737, 283 .00000000000015393776174455408, 284 -.00000000000036870428315837678, 285 .00000000000036920375082080089, 286 -.00000000000009383417223663699, 287 .00000000000009433398189512690, 288 .00000000000041481318704258568, 289 -.00000000000003792316480209314, 290 .00000000000008403156304792424, 291 -.00000000000034262934348285429, 292 .00000000000043712191957429145, 293 -.00000000000010475750058776541, 294 -.00000000000011118671389559323, 295 .00000000000037549577257259853, 296 .00000000000013912841212197565, 297 .00000000000010775743037572640, 298 .00000000000029391859187648000, 299 -.00000000000042790509060060774, 300 .00000000000022774076114039555, 301 .00000000000010849569622967912, 302 -.00000000000023073801945705758, 303 .00000000000015761203773969435, 304 .00000000000003345710269544082, 305 -.00000000000041525158063436123, 306 .00000000000032655698896907146, 307 -.00000000000044704265010452446, 308 .00000000000034527647952039772, 309 -.00000000000007048962392109746, 310 .00000000000011776978751369214, 311 -.00000000000010774341461609578, 312 .00000000000021863343293215910, 313 .00000000000024132639491333131, 314 .00000000000039057462209830700, 315 -.00000000000026570679203560751, 316 .00000000000037135141919592021, 317 -.00000000000017166921336082431, 318 -.00000000000028658285157914353, 319 -.00000000000023812542263446809, 320 .00000000000006576659768580062, 321 -.00000000000028210143846181267, 322 .00000000000010701931762114254, 323 .00000000000018119346366441110, 324 .00000000000009840465278232627, 325 -.00000000000033149150282752542, 326 -.00000000000018302857356041668, 327 -.00000000000016207400156744949, 328 .00000000000048303314949553201, 329 -.00000000000071560553172382115, 330 .00000000000088821239518571855, 331 -.00000000000030900580513238244, 332 -.00000000000061076551972851496, 333 .00000000000035659969663347830, 334 .00000000000035782396591276383, 335 -.00000000000046226087001544578, 336 .00000000000062279762917225156, 337 .00000000000072838947272065741, 338 .00000000000026809646615211673, 339 -.00000000000010960825046059278, 340 .00000000000002311949383800537, 341 -.00000000000058469058005299247, 342 -.00000000000002103748251144494, 343 -.00000000000023323182945587408, 344 -.00000000000042333694288141916, 345 -.00000000000043933937969737844, 346 .00000000000041341647073835565, 347 .00000000000006841763641591466, 348 .00000000000047585534004430641, 349 .00000000000083679678674757695, 350 -.00000000000085763734646658640, 351 .00000000000021913281229340092, 352 -.00000000000062242842536431148, 353 -.00000000000010983594325438430, 354 .00000000000065310431377633651, 355 -.00000000000047580199021710769, 356 -.00000000000037854251265457040, 357 .00000000000040939233218678664, 358 .00000000000087424383914858291, 359 .00000000000025218188456842882, 360 -.00000000000003608131360422557, 361 -.00000000000050518555924280902, 362 .00000000000078699403323355317, 363 -.00000000000067020876961949060, 364 .00000000000016108575753932458, 365 .00000000000058527188436251509, 366 -.00000000000035246757297904791, 367 -.00000000000018372084495629058, 368 .00000000000088606689813494916, 369 .00000000000066486268071468700, 370 .00000000000063831615170646519, 371 .00000000000025144230728376072, 372 -.00000000000017239444525614834 373 }; 374 375 double 376 log(double x) 377 { 378 int m, j; 379 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 380 volatile double u1; 381 382 /* Catch special cases */ 383 if (x <= 0) { 384 if (_IEEE && x == zero) /* log(0) = -Inf */ 385 return (-one/zero); 386 else if (_IEEE) /* log(neg) = NaN */ 387 return (zero/zero); 388 else if (x == zero) /* NOT REACHED IF _IEEE */ 389 return (infnan(-ERANGE)); 390 else 391 return (infnan(EDOM)); 392 } else if (!finite(x)) { 393 if (_IEEE) /* x = NaN, Inf */ 394 return (x+x); 395 else 396 return (infnan(ERANGE)); 397 } 398 399 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 400 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 401 402 m = logb(x); 403 g = ldexp(x, -m); 404 if (_IEEE && m == -1022) { 405 j = logb(g), m += j; 406 g = ldexp(g, -j); 407 } 408 j = N*(g-1) + .5; 409 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 410 f = g - F; 411 412 /* Approximate expansion for log(1+f/F) ~= u + q */ 413 g = 1/(2*F+f); 414 u = 2*f*g; 415 v = u*u; 416 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 417 418 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 419 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 420 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 421 */ 422 if (m | j) 423 u1 = u + 513, u1 -= 513; 424 425 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 426 * u1 = u to 24 bits. 427 */ 428 else 429 u1 = u, TRUNC(u1); 430 u2 = (2.0*(f - F*u1) - u1*f) * g; 431 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 432 433 /* log(x) = log(2^m*F*(1+f/F)) = */ 434 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 435 /* (exact) + (tiny) */ 436 437 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 438 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 439 u2 += logF_tail[N]*m; 440 return (u1 + u2); 441 } 442 443 /* 444 * Extra precision variant, returning struct {double a, b;}; 445 * log(x) = a+b to 63 bits, with a is rounded to 26 bits. 446 */ 447 struct Double 448 __log__D(double x) 449 { 450 int m, j; 451 double F, f, g, q, u, v, u2; 452 volatile double u1; 453 struct Double r; 454 455 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 456 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 457 458 m = logb(x); 459 g = ldexp(x, -m); 460 if (_IEEE && m == -1022) { 461 j = logb(g), m += j; 462 g = ldexp(g, -j); 463 } 464 j = N*(g-1) + .5; 465 F = (1.0/N) * j + 1; 466 f = g - F; 467 468 g = 1/(2*F+f); 469 u = 2*f*g; 470 v = u*u; 471 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 472 if (m | j) 473 u1 = u + 513, u1 -= 513; 474 else 475 u1 = u, TRUNC(u1); 476 u2 = (2.0*(f - F*u1) - u1*f) * g; 477 478 u1 += m*logF_head[N] + logF_head[j]; 479 480 u2 += logF_tail[j]; u2 += q; 481 u2 += logF_tail[N]*m; 482 r.a = u1 + u2; /* Only difference is here */ 483 TRUNC(r.a); 484 r.b = (u1 - r.a) + u2; 485 return (r); 486 } 487 488 float 489 logf(float x) 490 { 491 return(log((double)x)); 492 } 493
Indexes created Fri Oct 03 02:09:57 GMT 2025