n_log.c revision 1.5
1 /* $NetBSD: n_log.c,v 1.5 2002/06/15 00:10:17 matt 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. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 #if 0 37 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93"; 38 #endif 39 #endif /* not lint */ 40 41 #include <math.h> 42 #include <errno.h> 43 44 #include "mathimpl.h" 45 46 /* Table-driven natural logarithm. 47 * 48 * This code was derived, with minor modifications, from: 49 * Peter Tang, "Table-Driven Implementation of the 50 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 51 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 52 * 53 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 54 * where F = j/128 for j an integer in [0, 128]. 55 * 56 * log(2^m) = log2_hi*m + log2_tail*m 57 * since m is an integer, the dominant term is exact. 58 * m has at most 10 digits (for subnormal numbers), 59 * and log2_hi has 11 trailing zero bits. 60 * 61 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 62 * logF_hi[] + 512 is exact. 63 * 64 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 65 * the leading term is calculated to extra precision in two 66 * parts, the larger of which adds exactly to the dominant 67 * m and F terms. 68 * There are two cases: 69 * 1. when m, j are non-zero (m | j), use absolute 70 * precision for the leading term. 71 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 72 * In this case, use a relative precision of 24 bits. 73 * (This is done differently in the original paper) 74 * 75 * Special cases: 76 * 0 return signalling -Inf 77 * neg return signalling NaN 78 * +Inf return +Inf 79 */ 80 81 #if defined(__vax__) || defined(tahoe) 82 #define _IEEE 0 83 #define TRUNC(x) x = (double) (float) (x) 84 #else 85 #define _IEEE 1 86 #define endian (((*(int *) &one)) ? 1 : 0) 87 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 88 #define infnan(x) 0.0 89 #endif 90 91 #define N 128 92 93 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 94 * Used for generation of extend precision logarithms. 95 * The constant 35184372088832 is 2^45, so the divide is exact. 96 * It ensures correct reading of logF_head, even for inaccurate 97 * decimal-to-binary conversion routines. (Everybody gets the 98 * right answer for integers less than 2^53.) 99 * Values for log(F) were generated using error < 10^-57 absolute 100 * with the bc -l package. 101 */ 102 static const double A1 = .08333333333333178827; 103 static const double A2 = .01250000000377174923; 104 static const double A3 = .002232139987919447809; 105 static const double A4 = .0004348877777076145742; 106 107 static const double logF_head[N+1] = { 108 0., 109 .007782140442060381246, 110 .015504186535963526694, 111 .023167059281547608406, 112 .030771658666765233647, 113 .038318864302141264488, 114 .045809536031242714670, 115 .053244514518837604555, 116 .060624621816486978786, 117 .067950661908525944454, 118 .075223421237524235039, 119 .082443669210988446138, 120 .089612158689760690322, 121 .096729626458454731618, 122 .103796793681567578460, 123 .110814366340264314203, 124 .117783035656430001836, 125 .124703478501032805070, 126 .131576357788617315236, 127 .138402322859292326029, 128 .145182009844575077295, 129 .151916042025732167530, 130 .158605030176659056451, 131 .165249572895390883786, 132 .171850256926518341060, 133 .178407657472689606947, 134 .184922338493834104156, 135 .191394852999565046047, 136 .197825743329758552135, 137 .204215541428766300668, 138 .210564769107350002741, 139 .216873938300523150246, 140 .223143551314024080056, 141 .229374101064877322642, 142 .235566071312860003672, 143 .241719936886966024758, 144 .247836163904594286577, 145 .253915209980732470285, 146 .259957524436686071567, 147 .265963548496984003577, 148 .271933715484010463114, 149 .277868451003087102435, 150 .283768173130738432519, 151 .289633292582948342896, 152 .295464212893421063199, 153 .301261330578199704177, 154 .307025035294827830512, 155 .312755710004239517729, 156 .318453731118097493890, 157 .324119468654316733591, 158 .329753286372579168528, 159 .335355541920762334484, 160 .340926586970454081892, 161 .346466767346100823488, 162 .351976423156884266063, 163 .357455888922231679316, 164 .362905493689140712376, 165 .368325561158599157352, 166 .373716409793814818840, 167 .379078352934811846353, 168 .384411698910298582632, 169 .389716751140440464951, 170 .394993808240542421117, 171 .400243164127459749579, 172 .405465108107819105498, 173 .410659924985338875558, 174 .415827895143593195825, 175 .420969294644237379543, 176 .426084395310681429691, 177 .431173464818130014464, 178 .436236766774527495726, 179 .441274560805140936281, 180 .446287102628048160113, 181 .451274644139630254358, 182 .456237433481874177232, 183 .461175715122408291790, 184 .466089729924533457960, 185 .470979715219073113985, 186 .475845904869856894947, 187 .480688529345570714212, 188 .485507815781602403149, 189 .490303988045525329653, 190 .495077266798034543171, 191 .499827869556611403822, 192 .504556010751912253908, 193 .509261901790523552335, 194 .513945751101346104405, 195 .518607764208354637958, 196 .523248143765158602036, 197 .527867089620485785417, 198 .532464798869114019908, 199 .537041465897345915436, 200 .541597282432121573947, 201 .546132437597407260909, 202 .550647117952394182793, 203 .555141507540611200965, 204 .559615787935399566777, 205 .564070138285387656651, 206 .568504735352689749561, 207 .572919753562018740922, 208 .577315365035246941260, 209 .581691739635061821900, 210 .586049045003164792433, 211 .590387446602107957005, 212 .594707107746216934174, 213 .599008189645246602594, 214 .603290851438941899687, 215 .607555250224322662688, 216 .611801541106615331955, 217 .616029877215623855590, 218 .620240409751204424537, 219 .624433288012369303032, 220 .628608659422752680256, 221 .632766669570628437213, 222 .636907462236194987781, 223 .641031179420679109171, 224 .645137961373620782978, 225 .649227946625615004450, 226 .653301272011958644725, 227 .657358072709030238911, 228 .661398482245203922502, 229 .665422632544505177065, 230 .669430653942981734871, 231 .673422675212350441142, 232 .677398823590920073911, 233 .681359224807238206267, 234 .685304003098281100392, 235 .689233281238557538017, 236 .693147180560117703862 237 }; 238 239 static const double logF_tail[N+1] = { 240 0., 241 -.00000000000000543229938420049, 242 .00000000000000172745674997061, 243 -.00000000000001323017818229233, 244 -.00000000000001154527628289872, 245 -.00000000000000466529469958300, 246 .00000000000005148849572685810, 247 -.00000000000002532168943117445, 248 -.00000000000005213620639136504, 249 -.00000000000001819506003016881, 250 .00000000000006329065958724544, 251 .00000000000008614512936087814, 252 -.00000000000007355770219435028, 253 .00000000000009638067658552277, 254 .00000000000007598636597194141, 255 .00000000000002579999128306990, 256 -.00000000000004654729747598444, 257 -.00000000000007556920687451336, 258 .00000000000010195735223708472, 259 -.00000000000017319034406422306, 260 -.00000000000007718001336828098, 261 .00000000000010980754099855238, 262 -.00000000000002047235780046195, 263 -.00000000000008372091099235912, 264 .00000000000014088127937111135, 265 .00000000000012869017157588257, 266 .00000000000017788850778198106, 267 .00000000000006440856150696891, 268 .00000000000016132822667240822, 269 -.00000000000007540916511956188, 270 -.00000000000000036507188831790, 271 .00000000000009120937249914984, 272 .00000000000018567570959796010, 273 -.00000000000003149265065191483, 274 -.00000000000009309459495196889, 275 .00000000000017914338601329117, 276 -.00000000000001302979717330866, 277 .00000000000023097385217586939, 278 .00000000000023999540484211737, 279 .00000000000015393776174455408, 280 -.00000000000036870428315837678, 281 .00000000000036920375082080089, 282 -.00000000000009383417223663699, 283 .00000000000009433398189512690, 284 .00000000000041481318704258568, 285 -.00000000000003792316480209314, 286 .00000000000008403156304792424, 287 -.00000000000034262934348285429, 288 .00000000000043712191957429145, 289 -.00000000000010475750058776541, 290 -.00000000000011118671389559323, 291 .00000000000037549577257259853, 292 .00000000000013912841212197565, 293 .00000000000010775743037572640, 294 .00000000000029391859187648000, 295 -.00000000000042790509060060774, 296 .00000000000022774076114039555, 297 .00000000000010849569622967912, 298 -.00000000000023073801945705758, 299 .00000000000015761203773969435, 300 .00000000000003345710269544082, 301 -.00000000000041525158063436123, 302 .00000000000032655698896907146, 303 -.00000000000044704265010452446, 304 .00000000000034527647952039772, 305 -.00000000000007048962392109746, 306 .00000000000011776978751369214, 307 -.00000000000010774341461609578, 308 .00000000000021863343293215910, 309 .00000000000024132639491333131, 310 .00000000000039057462209830700, 311 -.00000000000026570679203560751, 312 .00000000000037135141919592021, 313 -.00000000000017166921336082431, 314 -.00000000000028658285157914353, 315 -.00000000000023812542263446809, 316 .00000000000006576659768580062, 317 -.00000000000028210143846181267, 318 .00000000000010701931762114254, 319 .00000000000018119346366441110, 320 .00000000000009840465278232627, 321 -.00000000000033149150282752542, 322 -.00000000000018302857356041668, 323 -.00000000000016207400156744949, 324 .00000000000048303314949553201, 325 -.00000000000071560553172382115, 326 .00000000000088821239518571855, 327 -.00000000000030900580513238244, 328 -.00000000000061076551972851496, 329 .00000000000035659969663347830, 330 .00000000000035782396591276383, 331 -.00000000000046226087001544578, 332 .00000000000062279762917225156, 333 .00000000000072838947272065741, 334 .00000000000026809646615211673, 335 -.00000000000010960825046059278, 336 .00000000000002311949383800537, 337 -.00000000000058469058005299247, 338 -.00000000000002103748251144494, 339 -.00000000000023323182945587408, 340 -.00000000000042333694288141916, 341 -.00000000000043933937969737844, 342 .00000000000041341647073835565, 343 .00000000000006841763641591466, 344 .00000000000047585534004430641, 345 .00000000000083679678674757695, 346 -.00000000000085763734646658640, 347 .00000000000021913281229340092, 348 -.00000000000062242842536431148, 349 -.00000000000010983594325438430, 350 .00000000000065310431377633651, 351 -.00000000000047580199021710769, 352 -.00000000000037854251265457040, 353 .00000000000040939233218678664, 354 .00000000000087424383914858291, 355 .00000000000025218188456842882, 356 -.00000000000003608131360422557, 357 -.00000000000050518555924280902, 358 .00000000000078699403323355317, 359 -.00000000000067020876961949060, 360 .00000000000016108575753932458, 361 .00000000000058527188436251509, 362 -.00000000000035246757297904791, 363 -.00000000000018372084495629058, 364 .00000000000088606689813494916, 365 .00000000000066486268071468700, 366 .00000000000063831615170646519, 367 .00000000000025144230728376072, 368 -.00000000000017239444525614834 369 }; 370 371 double 372 log(double x) 373 { 374 int m, j; 375 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 376 volatile double u1; 377 378 /* Catch special cases */ 379 if (x <= 0) { 380 if (_IEEE && x == zero) /* log(0) = -Inf */ 381 return (-one/zero); 382 else if (_IEEE) /* log(neg) = NaN */ 383 return (zero/zero); 384 else if (x == zero) /* NOT REACHED IF _IEEE */ 385 return (infnan(-ERANGE)); 386 else 387 return (infnan(EDOM)); 388 } else if (!finite(x)) { 389 if (_IEEE) /* x = NaN, Inf */ 390 return (x+x); 391 else 392 return (infnan(ERANGE)); 393 } 394 395 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 396 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 397 398 m = logb(x); 399 g = ldexp(x, -m); 400 if (_IEEE && m == -1022) { 401 j = logb(g), m += j; 402 g = ldexp(g, -j); 403 } 404 j = N*(g-1) + .5; 405 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 406 f = g - F; 407 408 /* Approximate expansion for log(1+f/F) ~= u + q */ 409 g = 1/(2*F+f); 410 u = 2*f*g; 411 v = u*u; 412 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 413 414 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 415 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 416 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 417 */ 418 if (m | j) 419 u1 = u + 513, u1 -= 513; 420 421 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 422 * u1 = u to 24 bits. 423 */ 424 else 425 u1 = u, TRUNC(u1); 426 u2 = (2.0*(f - F*u1) - u1*f) * g; 427 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 428 429 /* log(x) = log(2^m*F*(1+f/F)) = */ 430 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 431 /* (exact) + (tiny) */ 432 433 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 434 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 435 u2 += logF_tail[N]*m; 436 return (u1 + u2); 437 } 438 439 /* 440 * Extra precision variant, returning struct {double a, b;}; 441 * log(x) = a+b to 63 bits, with a is rounded to 26 bits. 442 */ 443 struct Double 444 __log__D(double x) 445 { 446 int m, j; 447 double F, f, g, q, u, v, u2; 448 volatile double u1; 449 struct Double r; 450 451 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 452 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 453 454 m = logb(x); 455 g = ldexp(x, -m); 456 if (_IEEE && m == -1022) { 457 j = logb(g), m += j; 458 g = ldexp(g, -j); 459 } 460 j = N*(g-1) + .5; 461 F = (1.0/N) * j + 1; 462 f = g - F; 463 464 g = 1/(2*F+f); 465 u = 2*f*g; 466 v = u*u; 467 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 468 if (m | j) 469 u1 = u + 513, u1 -= 513; 470 else 471 u1 = u, TRUNC(u1); 472 u2 = (2.0*(f - F*u1) - u1*f) * g; 473 474 u1 += m*logF_head[N] + logF_head[j]; 475 476 u2 += logF_tail[j]; u2 += q; 477 u2 += logF_tail[N]*m; 478 r.a = u1 + u2; /* Only difference is here */ 479 TRUNC(r.a); 480 r.b = (u1 - r.a) + u2; 481 return (r); 482 } 483
Indexes created Tue Sep 23 14:10:03 GMT 2025