n_log.c revision 1.7
1 /* $NetBSD: n_log.c,v 1.7 2008/03/20 16:41:26 mhitch 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(logf, _logf); 47 #endif 48 49 /* Table-driven natural logarithm. 50 * 51 * This code was derived, with minor modifications, from: 52 * Peter Tang, "Table-Driven Implementation of the 53 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 54 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 55 * 56 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 57 * where F = j/128 for j an integer in [0, 128]. 58 * 59 * log(2^m) = log2_hi*m + log2_tail*m 60 * since m is an integer, the dominant term is exact. 61 * m has at most 10 digits (for subnormal numbers), 62 * and log2_hi has 11 trailing zero bits. 63 * 64 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 65 * logF_hi[] + 512 is exact. 66 * 67 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 68 * the leading term is calculated to extra precision in two 69 * parts, the larger of which adds exactly to the dominant 70 * m and F terms. 71 * There are two cases: 72 * 1. when m, j are non-zero (m | j), use absolute 73 * precision for the leading term. 74 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 75 * In this case, use a relative precision of 24 bits. 76 * (This is done differently in the original paper) 77 * 78 * Special cases: 79 * 0 return signalling -Inf 80 * neg return signalling NaN 81 * +Inf return +Inf 82 */ 83 84 #if defined(__vax__) || defined(tahoe) 85 #define _IEEE 0 86 #define TRUNC(x) x = (double) (float) (x) 87 #else 88 #define _IEEE 1 89 #define endian (((*(int *) &one)) ? 1 : 0) 90 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 91 #define infnan(x) 0.0 92 #endif 93 94 #define N 128 95 96 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 97 * Used for generation of extend precision logarithms. 98 * The constant 35184372088832 is 2^45, so the divide is exact. 99 * It ensures correct reading of logF_head, even for inaccurate 100 * decimal-to-binary conversion routines. (Everybody gets the 101 * right answer for integers less than 2^53.) 102 * Values for log(F) were generated using error < 10^-57 absolute 103 * with the bc -l package. 104 */ 105 static const double A1 = .08333333333333178827; 106 static const double A2 = .01250000000377174923; 107 static const double A3 = .002232139987919447809; 108 static const double A4 = .0004348877777076145742; 109 110 static const double logF_head[N+1] = { 111 0., 112 .007782140442060381246, 113 .015504186535963526694, 114 .023167059281547608406, 115 .030771658666765233647, 116 .038318864302141264488, 117 .045809536031242714670, 118 .053244514518837604555, 119 .060624621816486978786, 120 .067950661908525944454, 121 .075223421237524235039, 122 .082443669210988446138, 123 .089612158689760690322, 124 .096729626458454731618, 125 .103796793681567578460, 126 .110814366340264314203, 127 .117783035656430001836, 128 .124703478501032805070, 129 .131576357788617315236, 130 .138402322859292326029, 131 .145182009844575077295, 132 .151916042025732167530, 133 .158605030176659056451, 134 .165249572895390883786, 135 .171850256926518341060, 136 .178407657472689606947, 137 .184922338493834104156, 138 .191394852999565046047, 139 .197825743329758552135, 140 .204215541428766300668, 141 .210564769107350002741, 142 .216873938300523150246, 143 .223143551314024080056, 144 .229374101064877322642, 145 .235566071312860003672, 146 .241719936886966024758, 147 .247836163904594286577, 148 .253915209980732470285, 149 .259957524436686071567, 150 .265963548496984003577, 151 .271933715484010463114, 152 .277868451003087102435, 153 .283768173130738432519, 154 .289633292582948342896, 155 .295464212893421063199, 156 .301261330578199704177, 157 .307025035294827830512, 158 .312755710004239517729, 159 .318453731118097493890, 160 .324119468654316733591, 161 .329753286372579168528, 162 .335355541920762334484, 163 .340926586970454081892, 164 .346466767346100823488, 165 .351976423156884266063, 166 .357455888922231679316, 167 .362905493689140712376, 168 .368325561158599157352, 169 .373716409793814818840, 170 .379078352934811846353, 171 .384411698910298582632, 172 .389716751140440464951, 173 .394993808240542421117, 174 .400243164127459749579, 175 .405465108107819105498, 176 .410659924985338875558, 177 .415827895143593195825, 178 .420969294644237379543, 179 .426084395310681429691, 180 .431173464818130014464, 181 .436236766774527495726, 182 .441274560805140936281, 183 .446287102628048160113, 184 .451274644139630254358, 185 .456237433481874177232, 186 .461175715122408291790, 187 .466089729924533457960, 188 .470979715219073113985, 189 .475845904869856894947, 190 .480688529345570714212, 191 .485507815781602403149, 192 .490303988045525329653, 193 .495077266798034543171, 194 .499827869556611403822, 195 .504556010751912253908, 196 .509261901790523552335, 197 .513945751101346104405, 198 .518607764208354637958, 199 .523248143765158602036, 200 .527867089620485785417, 201 .532464798869114019908, 202 .537041465897345915436, 203 .541597282432121573947, 204 .546132437597407260909, 205 .550647117952394182793, 206 .555141507540611200965, 207 .559615787935399566777, 208 .564070138285387656651, 209 .568504735352689749561, 210 .572919753562018740922, 211 .577315365035246941260, 212 .581691739635061821900, 213 .586049045003164792433, 214 .590387446602107957005, 215 .594707107746216934174, 216 .599008189645246602594, 217 .603290851438941899687, 218 .607555250224322662688, 219 .611801541106615331955, 220 .616029877215623855590, 221 .620240409751204424537, 222 .624433288012369303032, 223 .628608659422752680256, 224 .632766669570628437213, 225 .636907462236194987781, 226 .641031179420679109171, 227 .645137961373620782978, 228 .649227946625615004450, 229 .653301272011958644725, 230 .657358072709030238911, 231 .661398482245203922502, 232 .665422632544505177065, 233 .669430653942981734871, 234 .673422675212350441142, 235 .677398823590920073911, 236 .681359224807238206267, 237 .685304003098281100392, 238 .689233281238557538017, 239 .693147180560117703862 240 }; 241 242 static const double logF_tail[N+1] = { 243 0., 244 -.00000000000000543229938420049, 245 .00000000000000172745674997061, 246 -.00000000000001323017818229233, 247 -.00000000000001154527628289872, 248 -.00000000000000466529469958300, 249 .00000000000005148849572685810, 250 -.00000000000002532168943117445, 251 -.00000000000005213620639136504, 252 -.00000000000001819506003016881, 253 .00000000000006329065958724544, 254 .00000000000008614512936087814, 255 -.00000000000007355770219435028, 256 .00000000000009638067658552277, 257 .00000000000007598636597194141, 258 .00000000000002579999128306990, 259 -.00000000000004654729747598444, 260 -.00000000000007556920687451336, 261 .00000000000010195735223708472, 262 -.00000000000017319034406422306, 263 -.00000000000007718001336828098, 264 .00000000000010980754099855238, 265 -.00000000000002047235780046195, 266 -.00000000000008372091099235912, 267 .00000000000014088127937111135, 268 .00000000000012869017157588257, 269 .00000000000017788850778198106, 270 .00000000000006440856150696891, 271 .00000000000016132822667240822, 272 -.00000000000007540916511956188, 273 -.00000000000000036507188831790, 274 .00000000000009120937249914984, 275 .00000000000018567570959796010, 276 -.00000000000003149265065191483, 277 -.00000000000009309459495196889, 278 .00000000000017914338601329117, 279 -.00000000000001302979717330866, 280 .00000000000023097385217586939, 281 .00000000000023999540484211737, 282 .00000000000015393776174455408, 283 -.00000000000036870428315837678, 284 .00000000000036920375082080089, 285 -.00000000000009383417223663699, 286 .00000000000009433398189512690, 287 .00000000000041481318704258568, 288 -.00000000000003792316480209314, 289 .00000000000008403156304792424, 290 -.00000000000034262934348285429, 291 .00000000000043712191957429145, 292 -.00000000000010475750058776541, 293 -.00000000000011118671389559323, 294 .00000000000037549577257259853, 295 .00000000000013912841212197565, 296 .00000000000010775743037572640, 297 .00000000000029391859187648000, 298 -.00000000000042790509060060774, 299 .00000000000022774076114039555, 300 .00000000000010849569622967912, 301 -.00000000000023073801945705758, 302 .00000000000015761203773969435, 303 .00000000000003345710269544082, 304 -.00000000000041525158063436123, 305 .00000000000032655698896907146, 306 -.00000000000044704265010452446, 307 .00000000000034527647952039772, 308 -.00000000000007048962392109746, 309 .00000000000011776978751369214, 310 -.00000000000010774341461609578, 311 .00000000000021863343293215910, 312 .00000000000024132639491333131, 313 .00000000000039057462209830700, 314 -.00000000000026570679203560751, 315 .00000000000037135141919592021, 316 -.00000000000017166921336082431, 317 -.00000000000028658285157914353, 318 -.00000000000023812542263446809, 319 .00000000000006576659768580062, 320 -.00000000000028210143846181267, 321 .00000000000010701931762114254, 322 .00000000000018119346366441110, 323 .00000000000009840465278232627, 324 -.00000000000033149150282752542, 325 -.00000000000018302857356041668, 326 -.00000000000016207400156744949, 327 .00000000000048303314949553201, 328 -.00000000000071560553172382115, 329 .00000000000088821239518571855, 330 -.00000000000030900580513238244, 331 -.00000000000061076551972851496, 332 .00000000000035659969663347830, 333 .00000000000035782396591276383, 334 -.00000000000046226087001544578, 335 .00000000000062279762917225156, 336 .00000000000072838947272065741, 337 .00000000000026809646615211673, 338 -.00000000000010960825046059278, 339 .00000000000002311949383800537, 340 -.00000000000058469058005299247, 341 -.00000000000002103748251144494, 342 -.00000000000023323182945587408, 343 -.00000000000042333694288141916, 344 -.00000000000043933937969737844, 345 .00000000000041341647073835565, 346 .00000000000006841763641591466, 347 .00000000000047585534004430641, 348 .00000000000083679678674757695, 349 -.00000000000085763734646658640, 350 .00000000000021913281229340092, 351 -.00000000000062242842536431148, 352 -.00000000000010983594325438430, 353 .00000000000065310431377633651, 354 -.00000000000047580199021710769, 355 -.00000000000037854251265457040, 356 .00000000000040939233218678664, 357 .00000000000087424383914858291, 358 .00000000000025218188456842882, 359 -.00000000000003608131360422557, 360 -.00000000000050518555924280902, 361 .00000000000078699403323355317, 362 -.00000000000067020876961949060, 363 .00000000000016108575753932458, 364 .00000000000058527188436251509, 365 -.00000000000035246757297904791, 366 -.00000000000018372084495629058, 367 .00000000000088606689813494916, 368 .00000000000066486268071468700, 369 .00000000000063831615170646519, 370 .00000000000025144230728376072, 371 -.00000000000017239444525614834 372 }; 373 374 double 375 log(double x) 376 { 377 int m, j; 378 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 379 volatile double u1; 380 381 /* Catch special cases */ 382 if (x <= 0) { 383 if (_IEEE && x == zero) /* log(0) = -Inf */ 384 return (-one/zero); 385 else if (_IEEE) /* log(neg) = NaN */ 386 return (zero/zero); 387 else if (x == zero) /* NOT REACHED IF _IEEE */ 388 return (infnan(-ERANGE)); 389 else 390 return (infnan(EDOM)); 391 } else if (!finite(x)) { 392 if (_IEEE) /* x = NaN, Inf */ 393 return (x+x); 394 else 395 return (infnan(ERANGE)); 396 } 397 398 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 399 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 400 401 m = logb(x); 402 g = ldexp(x, -m); 403 if (_IEEE && m == -1022) { 404 j = logb(g), m += j; 405 g = ldexp(g, -j); 406 } 407 j = N*(g-1) + .5; 408 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 409 f = g - F; 410 411 /* Approximate expansion for log(1+f/F) ~= u + q */ 412 g = 1/(2*F+f); 413 u = 2*f*g; 414 v = u*u; 415 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 416 417 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 418 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 419 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 420 */ 421 if (m | j) 422 u1 = u + 513, u1 -= 513; 423 424 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 425 * u1 = u to 24 bits. 426 */ 427 else 428 u1 = u, TRUNC(u1); 429 u2 = (2.0*(f - F*u1) - u1*f) * g; 430 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 431 432 /* log(x) = log(2^m*F*(1+f/F)) = */ 433 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 434 /* (exact) + (tiny) */ 435 436 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 437 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 438 u2 += logF_tail[N]*m; 439 return (u1 + u2); 440 } 441 442 /* 443 * Extra precision variant, returning struct {double a, b;}; 444 * log(x) = a+b to 63 bits, with a is rounded to 26 bits. 445 */ 446 struct Double 447 __log__D(double x) 448 { 449 int m, j; 450 double F, f, g, q, u, v, u2; 451 volatile double u1; 452 struct Double r; 453 454 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 455 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 456 457 m = logb(x); 458 g = ldexp(x, -m); 459 if (_IEEE && m == -1022) { 460 j = logb(g), m += j; 461 g = ldexp(g, -j); 462 } 463 j = N*(g-1) + .5; 464 F = (1.0/N) * j + 1; 465 f = g - F; 466 467 g = 1/(2*F+f); 468 u = 2*f*g; 469 v = u*u; 470 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 471 if (m | j) 472 u1 = u + 513, u1 -= 513; 473 else 474 u1 = u, TRUNC(u1); 475 u2 = (2.0*(f - F*u1) - u1*f) * g; 476 477 u1 += m*logF_head[N] + logF_head[j]; 478 479 u2 += logF_tail[j]; u2 += q; 480 u2 += logF_tail[N]*m; 481 r.a = u1 + u2; /* Only difference is here */ 482 TRUNC(r.a); 483 r.b = (u1 - r.a) + u2; 484 return (r); 485 } 486 487 float 488 logf(float x) 489 { 490 return(log((double)x)); 491 } 492
Indexes created Mon Sep 22 13:09:51 GMT 2025