1 /* $NetBSD: n_log.c,v 1.9 2024/07/16 14:52:50 riastradh 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 #include <sys/cdefs.h> 32 __RCSID("$NetBSD: n_log.c,v 1.9 2024/07/16 14:52:50 riastradh Exp $"); 33 34 #ifndef lint 35 #if 0 36 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93"; 37 #endif 38 #endif /* not lint */ 39 40 #include "namespace.h" 41 42 #include <math.h> 43 #include <errno.h> 44 45 #include "mathimpl.h" 46 47 __weak_alias(logl, _logl) 48 __strong_alias(_logl, _log) 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 __weak_alias(log, _log) 376 double 377 log(double x) 378 { 379 int m, j; 380 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 381 volatile double u1; 382 383 /* Catch special cases */ 384 if (x <= 0) { 385 if (_IEEE && x == zero) /* log(0) = -Inf */ 386 return (-one/zero); 387 else if (_IEEE) /* log(neg) = NaN */ 388 return (zero/zero); 389 else if (x == zero) /* NOT REACHED IF _IEEE */ 390 return (infnan(-ERANGE)); 391 else 392 return (infnan(EDOM)); 393 } else if (!finite(x)) { 394 if (_IEEE) /* x = NaN, Inf */ 395 return (x+x); 396 else 397 return (infnan(ERANGE)); 398 } 399 400 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 401 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 402 403 m = logb(x); 404 g = ldexp(x, -m); 405 if (_IEEE && m == -1022) { 406 j = logb(g), m += j; 407 g = ldexp(g, -j); 408 } 409 j = N*(g-1) + .5; 410 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 411 f = g - F; 412 413 /* Approximate expansion for log(1+f/F) ~= u + q */ 414 g = 1/(2*F+f); 415 u = 2*f*g; 416 v = u*u; 417 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 418 419 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 420 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 421 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 422 */ 423 if (m | j) 424 u1 = u + 513, u1 -= 513; 425 426 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 427 * u1 = u to 24 bits. 428 */ 429 else 430 u1 = u, TRUNC(u1); 431 u2 = (2.0*(f - F*u1) - u1*f) * g; 432 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 433 434 /* log(x) = log(2^m*F*(1+f/F)) = */ 435 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 436 /* (exact) + (tiny) */ 437 438 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 439 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 440 u2 += logF_tail[N]*m; 441 return (u1 + u2); 442 } 443 444 /* 445 * Extra precision variant, returning struct {double a, b;}; 446 * log(x) = a+b to 63 bits, with a is rounded to 26 bits. 447 */ 448 struct Double 449 __log__D(double x) 450 { 451 int m, j; 452 double F, f, g, q, u, v, u2; 453 volatile double u1; 454 struct Double r; 455 456 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 457 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 458 459 m = logb(x); 460 g = ldexp(x, -m); 461 if (_IEEE && m == -1022) { 462 j = logb(g), m += j; 463 g = ldexp(g, -j); 464 } 465 j = N*(g-1) + .5; 466 F = (1.0/N) * j + 1; 467 f = g - F; 468 469 g = 1/(2*F+f); 470 u = 2*f*g; 471 v = u*u; 472 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 473 if (m | j) 474 u1 = u + 513, u1 -= 513; 475 else 476 u1 = u, TRUNC(u1); 477 u2 = (2.0*(f - F*u1) - u1*f) * g; 478 479 u1 += m*logF_head[N] + logF_head[j]; 480 481 u2 += logF_tail[j]; u2 += q; 482 u2 += logF_tail[N]*m; 483 r.a = u1 + u2; /* Only difference is here */ 484 TRUNC(r.a); 485 r.b = (u1 - r.a) + u2; 486 return (r); 487 } 488 489 __weak_alias(logf, _logf) 490 float 491 logf(float x) 492 { 493 return(log((double)x)); 494 } 495
Indexes created Sat Sep 20 22:09:52 GMT 2025