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 #include <sys/cdefs.h> 35 __FBSDID("$FreeBSD$"); 36 37 #ifndef lint 38 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93"; 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 double A1 = .08333333333333178827; 103 static double A2 = .01250000000377174923; 104 static double A3 = .002232139987919447809; 105 static double A4 = .0004348877777076145742; 106 107 static 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 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 #ifdef _ANSI_SOURCE 373 log(double x) 374 #else 375 log(x) double x; 376 #endif 377 { 378 int m, j; 379 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 380 volatile double u1; 381 382 /* Catch special cases */ 383 if (x <= 0) 384 if (_IEEE && x == zero) /* log(0) = -Inf */ 385 return (-one/zero); 386 else if (_IEEE) /* log(neg) = NaN */ 387 return (zero/zero); 388 else if (x == zero) /* NOT REACHED IF _IEEE */ 389 return (infnan(-ERANGE)); 390 else 391 return (infnan(EDOM)); 392 else if (!finite(x)) 393 if (_IEEE) /* x = NaN, Inf */ 394 return (x+x); 395 else 396 return (infnan(ERANGE)); 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 #ifdef _ANSI_SOURCE 448 __log__D(double x) 449 #else 450 __log__D(x) double x; 451 #endif 452 { 453 int m, j; 454 double F, f, g, q, u, v, u2, one = 1.0; 455 volatile double u1; 456 struct Double r; 457 458 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 459 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 460 461 m = logb(x); 462 g = ldexp(x, -m); 463 if (_IEEE && m == -1022) { 464 j = logb(g), m += j; 465 g = ldexp(g, -j); 466 } 467 j = N*(g-1) + .5; 468 F = (1.0/N) * j + 1; 469 f = g - F; 470 471 g = 1/(2*F+f); 472 u = 2*f*g; 473 v = u*u; 474 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 475 if (m | j) 476 u1 = u + 513, u1 -= 513; 477 else 478 u1 = u, TRUNC(u1); 479 u2 = (2.0*(f - F*u1) - u1*f) * g; 480 481 u1 += m*logF_head[N] + logF_head[j]; 482 483 u2 += logF_tail[j]; u2 += q; 484 u2 += logF_tail[N]*m; 485 r.a = u1 + u2; /* Only difference is here */ 486 TRUNC(r.a); 487 r.b = (u1 - r.a) + u2; 488 return (r); 489 } 490