1 /* 2 * Copyright (c) 1992, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 3. All advertising materials mentioning features or use of this software 14 * must display the following acknowledgement: 15 * This product includes software developed by the University of 16 * California, Berkeley and its contributors. 17 * 4. Neither the name of the University nor the names of its contributors 18 * may be used to endorse or promote products derived from this software 19 * without specific prior written permission. 20 * 21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 31 * SUCH DAMAGE. 32 */ 33 34 #ifndef lint 35 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93"; 36 #endif /* not lint */ 37 38 #include <math.h> 39 #include <errno.h> 40 41 #include "mathimpl.h" 42 43 /* Table-driven natural logarithm. 44 * 45 * This code was derived, with minor modifications, from: 46 * Peter Tang, "Table-Driven Implementation of the 47 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 48 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 49 * 50 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 51 * where F = j/128 for j an integer in [0, 128]. 52 * 53 * log(2^m) = log2_hi*m + log2_tail*m 54 * since m is an integer, the dominant term is exact. 55 * m has at most 10 digits (for subnormal numbers), 56 * and log2_hi has 11 trailing zero bits. 57 * 58 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 59 * logF_hi[] + 512 is exact. 60 * 61 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 62 * the leading term is calculated to extra precision in two 63 * parts, the larger of which adds exactly to the dominant 64 * m and F terms. 65 * There are two cases: 66 * 1. when m, j are non-zero (m | j), use absolute 67 * precision for the leading term. 68 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 69 * In this case, use a relative precision of 24 bits. 70 * (This is done differently in the original paper) 71 * 72 * Special cases: 73 * 0 return signalling -Inf 74 * neg return signalling NaN 75 * +Inf return +Inf 76 */ 77 78 #if defined(vax) || defined(tahoe) 79 #define _IEEE 0 80 #define TRUNC(x) x = (double) (float) (x) 81 #else 82 #define _IEEE 1 83 #define endian (((*(int *) &one)) ? 1 : 0) 84 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 85 #define infnan(x) 0.0 86 #endif 87 88 #define N 128 89 90 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 91 * Used for generation of extend precision logarithms. 92 * The constant 35184372088832 is 2^45, so the divide is exact. 93 * It ensures correct reading of logF_head, even for inaccurate 94 * decimal-to-binary conversion routines. (Everybody gets the 95 * right answer for integers less than 2^53.) 96 * Values for log(F) were generated using error < 10^-57 absolute 97 * with the bc -l package. 98 */ 99 static double A1 = .08333333333333178827; 100 static double A2 = .01250000000377174923; 101 static double A3 = .002232139987919447809; 102 static double A4 = .0004348877777076145742; 103 104 static double logF_head[N+1] = { 105 0., 106 .007782140442060381246, 107 .015504186535963526694, 108 .023167059281547608406, 109 .030771658666765233647, 110 .038318864302141264488, 111 .045809536031242714670, 112 .053244514518837604555, 113 .060624621816486978786, 114 .067950661908525944454, 115 .075223421237524235039, 116 .082443669210988446138, 117 .089612158689760690322, 118 .096729626458454731618, 119 .103796793681567578460, 120 .110814366340264314203, 121 .117783035656430001836, 122 .124703478501032805070, 123 .131576357788617315236, 124 .138402322859292326029, 125 .145182009844575077295, 126 .151916042025732167530, 127 .158605030176659056451, 128 .165249572895390883786, 129 .171850256926518341060, 130 .178407657472689606947, 131 .184922338493834104156, 132 .191394852999565046047, 133 .197825743329758552135, 134 .204215541428766300668, 135 .210564769107350002741, 136 .216873938300523150246, 137 .223143551314024080056, 138 .229374101064877322642, 139 .235566071312860003672, 140 .241719936886966024758, 141 .247836163904594286577, 142 .253915209980732470285, 143 .259957524436686071567, 144 .265963548496984003577, 145 .271933715484010463114, 146 .277868451003087102435, 147 .283768173130738432519, 148 .289633292582948342896, 149 .295464212893421063199, 150 .301261330578199704177, 151 .307025035294827830512, 152 .312755710004239517729, 153 .318453731118097493890, 154 .324119468654316733591, 155 .329753286372579168528, 156 .335355541920762334484, 157 .340926586970454081892, 158 .346466767346100823488, 159 .351976423156884266063, 160 .357455888922231679316, 161 .362905493689140712376, 162 .368325561158599157352, 163 .373716409793814818840, 164 .379078352934811846353, 165 .384411698910298582632, 166 .389716751140440464951, 167 .394993808240542421117, 168 .400243164127459749579, 169 .405465108107819105498, 170 .410659924985338875558, 171 .415827895143593195825, 172 .420969294644237379543, 173 .426084395310681429691, 174 .431173464818130014464, 175 .436236766774527495726, 176 .441274560805140936281, 177 .446287102628048160113, 178 .451274644139630254358, 179 .456237433481874177232, 180 .461175715122408291790, 181 .466089729924533457960, 182 .470979715219073113985, 183 .475845904869856894947, 184 .480688529345570714212, 185 .485507815781602403149, 186 .490303988045525329653, 187 .495077266798034543171, 188 .499827869556611403822, 189 .504556010751912253908, 190 .509261901790523552335, 191 .513945751101346104405, 192 .518607764208354637958, 193 .523248143765158602036, 194 .527867089620485785417, 195 .532464798869114019908, 196 .537041465897345915436, 197 .541597282432121573947, 198 .546132437597407260909, 199 .550647117952394182793, 200 .555141507540611200965, 201 .559615787935399566777, 202 .564070138285387656651, 203 .568504735352689749561, 204 .572919753562018740922, 205 .577315365035246941260, 206 .581691739635061821900, 207 .586049045003164792433, 208 .590387446602107957005, 209 .594707107746216934174, 210 .599008189645246602594, 211 .603290851438941899687, 212 .607555250224322662688, 213 .611801541106615331955, 214 .616029877215623855590, 215 .620240409751204424537, 216 .624433288012369303032, 217 .628608659422752680256, 218 .632766669570628437213, 219 .636907462236194987781, 220 .641031179420679109171, 221 .645137961373620782978, 222 .649227946625615004450, 223 .653301272011958644725, 224 .657358072709030238911, 225 .661398482245203922502, 226 .665422632544505177065, 227 .669430653942981734871, 228 .673422675212350441142, 229 .677398823590920073911, 230 .681359224807238206267, 231 .685304003098281100392, 232 .689233281238557538017, 233 .693147180560117703862 234 }; 235 236 static double logF_tail[N+1] = { 237 0., 238 -.00000000000000543229938420049, 239 .00000000000000172745674997061, 240 -.00000000000001323017818229233, 241 -.00000000000001154527628289872, 242 -.00000000000000466529469958300, 243 .00000000000005148849572685810, 244 -.00000000000002532168943117445, 245 -.00000000000005213620639136504, 246 -.00000000000001819506003016881, 247 .00000000000006329065958724544, 248 .00000000000008614512936087814, 249 -.00000000000007355770219435028, 250 .00000000000009638067658552277, 251 .00000000000007598636597194141, 252 .00000000000002579999128306990, 253 -.00000000000004654729747598444, 254 -.00000000000007556920687451336, 255 .00000000000010195735223708472, 256 -.00000000000017319034406422306, 257 -.00000000000007718001336828098, 258 .00000000000010980754099855238, 259 -.00000000000002047235780046195, 260 -.00000000000008372091099235912, 261 .00000000000014088127937111135, 262 .00000000000012869017157588257, 263 .00000000000017788850778198106, 264 .00000000000006440856150696891, 265 .00000000000016132822667240822, 266 -.00000000000007540916511956188, 267 -.00000000000000036507188831790, 268 .00000000000009120937249914984, 269 .00000000000018567570959796010, 270 -.00000000000003149265065191483, 271 -.00000000000009309459495196889, 272 .00000000000017914338601329117, 273 -.00000000000001302979717330866, 274 .00000000000023097385217586939, 275 .00000000000023999540484211737, 276 .00000000000015393776174455408, 277 -.00000000000036870428315837678, 278 .00000000000036920375082080089, 279 -.00000000000009383417223663699, 280 .00000000000009433398189512690, 281 .00000000000041481318704258568, 282 -.00000000000003792316480209314, 283 .00000000000008403156304792424, 284 -.00000000000034262934348285429, 285 .00000000000043712191957429145, 286 -.00000000000010475750058776541, 287 -.00000000000011118671389559323, 288 .00000000000037549577257259853, 289 .00000000000013912841212197565, 290 .00000000000010775743037572640, 291 .00000000000029391859187648000, 292 -.00000000000042790509060060774, 293 .00000000000022774076114039555, 294 .00000000000010849569622967912, 295 -.00000000000023073801945705758, 296 .00000000000015761203773969435, 297 .00000000000003345710269544082, 298 -.00000000000041525158063436123, 299 .00000000000032655698896907146, 300 -.00000000000044704265010452446, 301 .00000000000034527647952039772, 302 -.00000000000007048962392109746, 303 .00000000000011776978751369214, 304 -.00000000000010774341461609578, 305 .00000000000021863343293215910, 306 .00000000000024132639491333131, 307 .00000000000039057462209830700, 308 -.00000000000026570679203560751, 309 .00000000000037135141919592021, 310 -.00000000000017166921336082431, 311 -.00000000000028658285157914353, 312 -.00000000000023812542263446809, 313 .00000000000006576659768580062, 314 -.00000000000028210143846181267, 315 .00000000000010701931762114254, 316 .00000000000018119346366441110, 317 .00000000000009840465278232627, 318 -.00000000000033149150282752542, 319 -.00000000000018302857356041668, 320 -.00000000000016207400156744949, 321 .00000000000048303314949553201, 322 -.00000000000071560553172382115, 323 .00000000000088821239518571855, 324 -.00000000000030900580513238244, 325 -.00000000000061076551972851496, 326 .00000000000035659969663347830, 327 .00000000000035782396591276383, 328 -.00000000000046226087001544578, 329 .00000000000062279762917225156, 330 .00000000000072838947272065741, 331 .00000000000026809646615211673, 332 -.00000000000010960825046059278, 333 .00000000000002311949383800537, 334 -.00000000000058469058005299247, 335 -.00000000000002103748251144494, 336 -.00000000000023323182945587408, 337 -.00000000000042333694288141916, 338 -.00000000000043933937969737844, 339 .00000000000041341647073835565, 340 .00000000000006841763641591466, 341 .00000000000047585534004430641, 342 .00000000000083679678674757695, 343 -.00000000000085763734646658640, 344 .00000000000021913281229340092, 345 -.00000000000062242842536431148, 346 -.00000000000010983594325438430, 347 .00000000000065310431377633651, 348 -.00000000000047580199021710769, 349 -.00000000000037854251265457040, 350 .00000000000040939233218678664, 351 .00000000000087424383914858291, 352 .00000000000025218188456842882, 353 -.00000000000003608131360422557, 354 -.00000000000050518555924280902, 355 .00000000000078699403323355317, 356 -.00000000000067020876961949060, 357 .00000000000016108575753932458, 358 .00000000000058527188436251509, 359 -.00000000000035246757297904791, 360 -.00000000000018372084495629058, 361 .00000000000088606689813494916, 362 .00000000000066486268071468700, 363 .00000000000063831615170646519, 364 .00000000000025144230728376072, 365 -.00000000000017239444525614834 366 }; 367 368 double 369 #ifdef _ANSI_SOURCE 370 log(double x) 371 #else 372 log(x) double x; 373 #endif 374 { 375 int m, j; 376 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 377 volatile double u1; 378 379 /* Catch special cases */ 380 if (x <= 0) 381 if (_IEEE && x == zero) /* log(0) = -Inf */ 382 return (-one/zero); 383 else if (_IEEE) /* log(neg) = NaN */ 384 return (zero/zero); 385 else if (x == zero) /* NOT REACHED IF _IEEE */ 386 return (infnan(-ERANGE)); 387 else 388 return (infnan(EDOM)); 389 else if (!finite(x)) 390 if (_IEEE) /* x = NaN, Inf */ 391 return (x+x); 392 else 393 return (infnan(ERANGE)); 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 #ifdef _ANSI_SOURCE 445 __log__D(double x) 446 #else 447 __log__D(x) double x; 448 #endif 449 { 450 int m, j; 451 double F, f, g, q, u, v, u2, one = 1.0; 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