1 /* $NetBSD: n_log.c,v 1.6 2003/08/07 16:44:51 agc 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 <math.h> 38 #include <errno.h> 39 40 #include "mathimpl.h" 41 42 /* Table-driven natural logarithm. 43 * 44 * This code was derived, with minor modifications, from: 45 * Peter Tang, "Table-Driven Implementation of the 46 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 47 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 48 * 49 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 50 * where F = j/128 for j an integer in [0, 128]. 51 * 52 * log(2^m) = log2_hi*m + log2_tail*m 53 * since m is an integer, the dominant term is exact. 54 * m has at most 10 digits (for subnormal numbers), 55 * and log2_hi has 11 trailing zero bits. 56 * 57 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 58 * logF_hi[] + 512 is exact. 59 * 60 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 61 * the leading term is calculated to extra precision in two 62 * parts, the larger of which adds exactly to the dominant 63 * m and F terms. 64 * There are two cases: 65 * 1. when m, j are non-zero (m | j), use absolute 66 * precision for the leading term. 67 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 68 * In this case, use a relative precision of 24 bits. 69 * (This is done differently in the original paper) 70 * 71 * Special cases: 72 * 0 return signalling -Inf 73 * neg return signalling NaN 74 * +Inf return +Inf 75 */ 76 77 #if defined(__vax__) || defined(tahoe) 78 #define _IEEE 0 79 #define TRUNC(x) x = (double) (float) (x) 80 #else 81 #define _IEEE 1 82 #define endian (((*(int *) &one)) ? 1 : 0) 83 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 84 #define infnan(x) 0.0 85 #endif 86 87 #define N 128 88 89 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 90 * Used for generation of extend precision logarithms. 91 * The constant 35184372088832 is 2^45, so the divide is exact. 92 * It ensures correct reading of logF_head, even for inaccurate 93 * decimal-to-binary conversion routines. (Everybody gets the 94 * right answer for integers less than 2^53.) 95 * Values for log(F) were generated using error < 10^-57 absolute 96 * with the bc -l package. 97 */ 98 static const double A1 = .08333333333333178827; 99 static const double A2 = .01250000000377174923; 100 static const double A3 = .002232139987919447809; 101 static const double A4 = .0004348877777076145742; 102 103 static const double logF_head[N+1] = { 104 0., 105 .007782140442060381246, 106 .015504186535963526694, 107 .023167059281547608406, 108 .030771658666765233647, 109 .038318864302141264488, 110 .045809536031242714670, 111 .053244514518837604555, 112 .060624621816486978786, 113 .067950661908525944454, 114 .075223421237524235039, 115 .082443669210988446138, 116 .089612158689760690322, 117 .096729626458454731618, 118 .103796793681567578460, 119 .110814366340264314203, 120 .117783035656430001836, 121 .124703478501032805070, 122 .131576357788617315236, 123 .138402322859292326029, 124 .145182009844575077295, 125 .151916042025732167530, 126 .158605030176659056451, 127 .165249572895390883786, 128 .171850256926518341060, 129 .178407657472689606947, 130 .184922338493834104156, 131 .191394852999565046047, 132 .197825743329758552135, 133 .204215541428766300668, 134 .210564769107350002741, 135 .216873938300523150246, 136 .223143551314024080056, 137 .229374101064877322642, 138 .235566071312860003672, 139 .241719936886966024758, 140 .247836163904594286577, 141 .253915209980732470285, 142 .259957524436686071567, 143 .265963548496984003577, 144 .271933715484010463114, 145 .277868451003087102435, 146 .283768173130738432519, 147 .289633292582948342896, 148 .295464212893421063199, 149 .301261330578199704177, 150 .307025035294827830512, 151 .312755710004239517729, 152 .318453731118097493890, 153 .324119468654316733591, 154 .329753286372579168528, 155 .335355541920762334484, 156 .340926586970454081892, 157 .346466767346100823488, 158 .351976423156884266063, 159 .357455888922231679316, 160 .362905493689140712376, 161 .368325561158599157352, 162 .373716409793814818840, 163 .379078352934811846353, 164 .384411698910298582632, 165 .389716751140440464951, 166 .394993808240542421117, 167 .400243164127459749579, 168 .405465108107819105498, 169 .410659924985338875558, 170 .415827895143593195825, 171 .420969294644237379543, 172 .426084395310681429691, 173 .431173464818130014464, 174 .436236766774527495726, 175 .441274560805140936281, 176 .446287102628048160113, 177 .451274644139630254358, 178 .456237433481874177232, 179 .461175715122408291790, 180 .466089729924533457960, 181 .470979715219073113985, 182 .475845904869856894947, 183 .480688529345570714212, 184 .485507815781602403149, 185 .490303988045525329653, 186 .495077266798034543171, 187 .499827869556611403822, 188 .504556010751912253908, 189 .509261901790523552335, 190 .513945751101346104405, 191 .518607764208354637958, 192 .523248143765158602036, 193 .527867089620485785417, 194 .532464798869114019908, 195 .537041465897345915436, 196 .541597282432121573947, 197 .546132437597407260909, 198 .550647117952394182793, 199 .555141507540611200965, 200 .559615787935399566777, 201 .564070138285387656651, 202 .568504735352689749561, 203 .572919753562018740922, 204 .577315365035246941260, 205 .581691739635061821900, 206 .586049045003164792433, 207 .590387446602107957005, 208 .594707107746216934174, 209 .599008189645246602594, 210 .603290851438941899687, 211 .607555250224322662688, 212 .611801541106615331955, 213 .616029877215623855590, 214 .620240409751204424537, 215 .624433288012369303032, 216 .628608659422752680256, 217 .632766669570628437213, 218 .636907462236194987781, 219 .641031179420679109171, 220 .645137961373620782978, 221 .649227946625615004450, 222 .653301272011958644725, 223 .657358072709030238911, 224 .661398482245203922502, 225 .665422632544505177065, 226 .669430653942981734871, 227 .673422675212350441142, 228 .677398823590920073911, 229 .681359224807238206267, 230 .685304003098281100392, 231 .689233281238557538017, 232 .693147180560117703862 233 }; 234 235 static const double logF_tail[N+1] = { 236 0., 237 -.00000000000000543229938420049, 238 .00000000000000172745674997061, 239 -.00000000000001323017818229233, 240 -.00000000000001154527628289872, 241 -.00000000000000466529469958300, 242 .00000000000005148849572685810, 243 -.00000000000002532168943117445, 244 -.00000000000005213620639136504, 245 -.00000000000001819506003016881, 246 .00000000000006329065958724544, 247 .00000000000008614512936087814, 248 -.00000000000007355770219435028, 249 .00000000000009638067658552277, 250 .00000000000007598636597194141, 251 .00000000000002579999128306990, 252 -.00000000000004654729747598444, 253 -.00000000000007556920687451336, 254 .00000000000010195735223708472, 255 -.00000000000017319034406422306, 256 -.00000000000007718001336828098, 257 .00000000000010980754099855238, 258 -.00000000000002047235780046195, 259 -.00000000000008372091099235912, 260 .00000000000014088127937111135, 261 .00000000000012869017157588257, 262 .00000000000017788850778198106, 263 .00000000000006440856150696891, 264 .00000000000016132822667240822, 265 -.00000000000007540916511956188, 266 -.00000000000000036507188831790, 267 .00000000000009120937249914984, 268 .00000000000018567570959796010, 269 -.00000000000003149265065191483, 270 -.00000000000009309459495196889, 271 .00000000000017914338601329117, 272 -.00000000000001302979717330866, 273 .00000000000023097385217586939, 274 .00000000000023999540484211737, 275 .00000000000015393776174455408, 276 -.00000000000036870428315837678, 277 .00000000000036920375082080089, 278 -.00000000000009383417223663699, 279 .00000000000009433398189512690, 280 .00000000000041481318704258568, 281 -.00000000000003792316480209314, 282 .00000000000008403156304792424, 283 -.00000000000034262934348285429, 284 .00000000000043712191957429145, 285 -.00000000000010475750058776541, 286 -.00000000000011118671389559323, 287 .00000000000037549577257259853, 288 .00000000000013912841212197565, 289 .00000000000010775743037572640, 290 .00000000000029391859187648000, 291 -.00000000000042790509060060774, 292 .00000000000022774076114039555, 293 .00000000000010849569622967912, 294 -.00000000000023073801945705758, 295 .00000000000015761203773969435, 296 .00000000000003345710269544082, 297 -.00000000000041525158063436123, 298 .00000000000032655698896907146, 299 -.00000000000044704265010452446, 300 .00000000000034527647952039772, 301 -.00000000000007048962392109746, 302 .00000000000011776978751369214, 303 -.00000000000010774341461609578, 304 .00000000000021863343293215910, 305 .00000000000024132639491333131, 306 .00000000000039057462209830700, 307 -.00000000000026570679203560751, 308 .00000000000037135141919592021, 309 -.00000000000017166921336082431, 310 -.00000000000028658285157914353, 311 -.00000000000023812542263446809, 312 .00000000000006576659768580062, 313 -.00000000000028210143846181267, 314 .00000000000010701931762114254, 315 .00000000000018119346366441110, 316 .00000000000009840465278232627, 317 -.00000000000033149150282752542, 318 -.00000000000018302857356041668, 319 -.00000000000016207400156744949, 320 .00000000000048303314949553201, 321 -.00000000000071560553172382115, 322 .00000000000088821239518571855, 323 -.00000000000030900580513238244, 324 -.00000000000061076551972851496, 325 .00000000000035659969663347830, 326 .00000000000035782396591276383, 327 -.00000000000046226087001544578, 328 .00000000000062279762917225156, 329 .00000000000072838947272065741, 330 .00000000000026809646615211673, 331 -.00000000000010960825046059278, 332 .00000000000002311949383800537, 333 -.00000000000058469058005299247, 334 -.00000000000002103748251144494, 335 -.00000000000023323182945587408, 336 -.00000000000042333694288141916, 337 -.00000000000043933937969737844, 338 .00000000000041341647073835565, 339 .00000000000006841763641591466, 340 .00000000000047585534004430641, 341 .00000000000083679678674757695, 342 -.00000000000085763734646658640, 343 .00000000000021913281229340092, 344 -.00000000000062242842536431148, 345 -.00000000000010983594325438430, 346 .00000000000065310431377633651, 347 -.00000000000047580199021710769, 348 -.00000000000037854251265457040, 349 .00000000000040939233218678664, 350 .00000000000087424383914858291, 351 .00000000000025218188456842882, 352 -.00000000000003608131360422557, 353 -.00000000000050518555924280902, 354 .00000000000078699403323355317, 355 -.00000000000067020876961949060, 356 .00000000000016108575753932458, 357 .00000000000058527188436251509, 358 -.00000000000035246757297904791, 359 -.00000000000018372084495629058, 360 .00000000000088606689813494916, 361 .00000000000066486268071468700, 362 .00000000000063831615170646519, 363 .00000000000025144230728376072, 364 -.00000000000017239444525614834 365 }; 366 367 double 368 log(double x) 369 { 370 int m, j; 371 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 372 volatile double u1; 373 374 /* Catch special cases */ 375 if (x <= 0) { 376 if (_IEEE && x == zero) /* log(0) = -Inf */ 377 return (-one/zero); 378 else if (_IEEE) /* log(neg) = NaN */ 379 return (zero/zero); 380 else if (x == zero) /* NOT REACHED IF _IEEE */ 381 return (infnan(-ERANGE)); 382 else 383 return (infnan(EDOM)); 384 } else if (!finite(x)) { 385 if (_IEEE) /* x = NaN, Inf */ 386 return (x+x); 387 else 388 return (infnan(ERANGE)); 389 } 390 391 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 392 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 393 394 m = logb(x); 395 g = ldexp(x, -m); 396 if (_IEEE && m == -1022) { 397 j = logb(g), m += j; 398 g = ldexp(g, -j); 399 } 400 j = N*(g-1) + .5; 401 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 402 f = g - F; 403 404 /* Approximate expansion for log(1+f/F) ~= u + q */ 405 g = 1/(2*F+f); 406 u = 2*f*g; 407 v = u*u; 408 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 409 410 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 411 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 412 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 413 */ 414 if (m | j) 415 u1 = u + 513, u1 -= 513; 416 417 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 418 * u1 = u to 24 bits. 419 */ 420 else 421 u1 = u, TRUNC(u1); 422 u2 = (2.0*(f - F*u1) - u1*f) * g; 423 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 424 425 /* log(x) = log(2^m*F*(1+f/F)) = */ 426 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 427 /* (exact) + (tiny) */ 428 429 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 430 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 431 u2 += logF_tail[N]*m; 432 return (u1 + u2); 433 } 434 435 /* 436 * Extra precision variant, returning struct {double a, b;}; 437 * log(x) = a+b to 63 bits, with a is rounded to 26 bits. 438 */ 439 struct Double 440 __log__D(double x) 441 { 442 int m, j; 443 double F, f, g, q, u, v, u2; 444 volatile double u1; 445 struct Double r; 446 447 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 448 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 449 450 m = logb(x); 451 g = ldexp(x, -m); 452 if (_IEEE && m == -1022) { 453 j = logb(g), m += j; 454 g = ldexp(g, -j); 455 } 456 j = N*(g-1) + .5; 457 F = (1.0/N) * j + 1; 458 f = g - F; 459 460 g = 1/(2*F+f); 461 u = 2*f*g; 462 v = u*u; 463 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 464 if (m | j) 465 u1 = u + 513, u1 -= 513; 466 else 467 u1 = u, TRUNC(u1); 468 u2 = (2.0*(f - F*u1) - u1*f) * g; 469 470 u1 += m*logF_head[N] + logF_head[j]; 471 472 u2 += logF_tail[j]; u2 += q; 473 u2 += logF_tail[N]*m; 474 r.a = u1 + u2; /* Only difference is here */ 475 TRUNC(r.a); 476 r.b = (u1 - r.a) + u2; 477 return (r); 478 } 479