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 #include <sys/cdefs.h> 38 __FBSDID("$FreeBSD$"); 39 40 #include <math.h> 41 #include <errno.h> 42 43 #include "mathimpl.h" 44 45 /* Table-driven natural logarithm. 46 * 47 * This code was derived, with minor modifications, from: 48 * Peter Tang, "Table-Driven Implementation of the 49 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 50 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 51 * 52 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 53 * where F = j/128 for j an integer in [0, 128]. 54 * 55 * log(2^m) = log2_hi*m + log2_tail*m 56 * since m is an integer, the dominant term is exact. 57 * m has at most 10 digits (for subnormal numbers), 58 * and log2_hi has 11 trailing zero bits. 59 * 60 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 61 * logF_hi[] + 512 is exact. 62 * 63 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 64 * the leading term is calculated to extra precision in two 65 * parts, the larger of which adds exactly to the dominant 66 * m and F terms. 67 * There are two cases: 68 * 1. when m, j are non-zero (m | j), use absolute 69 * precision for the leading term. 70 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 71 * In this case, use a relative precision of 24 bits. 72 * (This is done differently in the original paper) 73 * 74 * Special cases: 75 * 0 return signalling -Inf 76 * neg return signalling NaN 77 * +Inf return +Inf 78 */ 79 80 #if defined(vax) || defined(tahoe) 81 #define _IEEE 0 82 #define TRUNC(x) x = (double) (float) (x) 83 #else 84 #define _IEEE 1 85 #define endian (((*(int *) &one)) ? 1 : 0) 86 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 87 #define infnan(x) 0.0 88 #endif 89 90 #define N 128 91 92 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 93 * Used for generation of extend precision logarithms. 94 * The constant 35184372088832 is 2^45, so the divide is exact. 95 * It ensures correct reading of logF_head, even for inaccurate 96 * decimal-to-binary conversion routines. (Everybody gets the 97 * right answer for integers less than 2^53.) 98 * Values for log(F) were generated using error < 10^-57 absolute 99 * with the bc -l package. 100 */ 101 static double A1 = .08333333333333178827; 102 static double A2 = .01250000000377174923; 103 static double A3 = .002232139987919447809; 104 static double A4 = .0004348877777076145742; 105 106 static double logF_head[N+1] = { 107 0., 108 .007782140442060381246, 109 .015504186535963526694, 110 .023167059281547608406, 111 .030771658666765233647, 112 .038318864302141264488, 113 .045809536031242714670, 114 .053244514518837604555, 115 .060624621816486978786, 116 .067950661908525944454, 117 .075223421237524235039, 118 .082443669210988446138, 119 .089612158689760690322, 120 .096729626458454731618, 121 .103796793681567578460, 122 .110814366340264314203, 123 .117783035656430001836, 124 .124703478501032805070, 125 .131576357788617315236, 126 .138402322859292326029, 127 .145182009844575077295, 128 .151916042025732167530, 129 .158605030176659056451, 130 .165249572895390883786, 131 .171850256926518341060, 132 .178407657472689606947, 133 .184922338493834104156, 134 .191394852999565046047, 135 .197825743329758552135, 136 .204215541428766300668, 137 .210564769107350002741, 138 .216873938300523150246, 139 .223143551314024080056, 140 .229374101064877322642, 141 .235566071312860003672, 142 .241719936886966024758, 143 .247836163904594286577, 144 .253915209980732470285, 145 .259957524436686071567, 146 .265963548496984003577, 147 .271933715484010463114, 148 .277868451003087102435, 149 .283768173130738432519, 150 .289633292582948342896, 151 .295464212893421063199, 152 .301261330578199704177, 153 .307025035294827830512, 154 .312755710004239517729, 155 .318453731118097493890, 156 .324119468654316733591, 157 .329753286372579168528, 158 .335355541920762334484, 159 .340926586970454081892, 160 .346466767346100823488, 161 .351976423156884266063, 162 .357455888922231679316, 163 .362905493689140712376, 164 .368325561158599157352, 165 .373716409793814818840, 166 .379078352934811846353, 167 .384411698910298582632, 168 .389716751140440464951, 169 .394993808240542421117, 170 .400243164127459749579, 171 .405465108107819105498, 172 .410659924985338875558, 173 .415827895143593195825, 174 .420969294644237379543, 175 .426084395310681429691, 176 .431173464818130014464, 177 .436236766774527495726, 178 .441274560805140936281, 179 .446287102628048160113, 180 .451274644139630254358, 181 .456237433481874177232, 182 .461175715122408291790, 183 .466089729924533457960, 184 .470979715219073113985, 185 .475845904869856894947, 186 .480688529345570714212, 187 .485507815781602403149, 188 .490303988045525329653, 189 .495077266798034543171, 190 .499827869556611403822, 191 .504556010751912253908, 192 .509261901790523552335, 193 .513945751101346104405, 194 .518607764208354637958, 195 .523248143765158602036, 196 .527867089620485785417, 197 .532464798869114019908, 198 .537041465897345915436, 199 .541597282432121573947, 200 .546132437597407260909, 201 .550647117952394182793, 202 .555141507540611200965, 203 .559615787935399566777, 204 .564070138285387656651, 205 .568504735352689749561, 206 .572919753562018740922, 207 .577315365035246941260, 208 .581691739635061821900, 209 .586049045003164792433, 210 .590387446602107957005, 211 .594707107746216934174, 212 .599008189645246602594, 213 .603290851438941899687, 214 .607555250224322662688, 215 .611801541106615331955, 216 .616029877215623855590, 217 .620240409751204424537, 218 .624433288012369303032, 219 .628608659422752680256, 220 .632766669570628437213, 221 .636907462236194987781, 222 .641031179420679109171, 223 .645137961373620782978, 224 .649227946625615004450, 225 .653301272011958644725, 226 .657358072709030238911, 227 .661398482245203922502, 228 .665422632544505177065, 229 .669430653942981734871, 230 .673422675212350441142, 231 .677398823590920073911, 232 .681359224807238206267, 233 .685304003098281100392, 234 .689233281238557538017, 235 .693147180560117703862 236 }; 237 238 static double logF_tail[N+1] = { 239 0., 240 -.00000000000000543229938420049, 241 .00000000000000172745674997061, 242 -.00000000000001323017818229233, 243 -.00000000000001154527628289872, 244 -.00000000000000466529469958300, 245 .00000000000005148849572685810, 246 -.00000000000002532168943117445, 247 -.00000000000005213620639136504, 248 -.00000000000001819506003016881, 249 .00000000000006329065958724544, 250 .00000000000008614512936087814, 251 -.00000000000007355770219435028, 252 .00000000000009638067658552277, 253 .00000000000007598636597194141, 254 .00000000000002579999128306990, 255 -.00000000000004654729747598444, 256 -.00000000000007556920687451336, 257 .00000000000010195735223708472, 258 -.00000000000017319034406422306, 259 -.00000000000007718001336828098, 260 .00000000000010980754099855238, 261 -.00000000000002047235780046195, 262 -.00000000000008372091099235912, 263 .00000000000014088127937111135, 264 .00000000000012869017157588257, 265 .00000000000017788850778198106, 266 .00000000000006440856150696891, 267 .00000000000016132822667240822, 268 -.00000000000007540916511956188, 269 -.00000000000000036507188831790, 270 .00000000000009120937249914984, 271 .00000000000018567570959796010, 272 -.00000000000003149265065191483, 273 -.00000000000009309459495196889, 274 .00000000000017914338601329117, 275 -.00000000000001302979717330866, 276 .00000000000023097385217586939, 277 .00000000000023999540484211737, 278 .00000000000015393776174455408, 279 -.00000000000036870428315837678, 280 .00000000000036920375082080089, 281 -.00000000000009383417223663699, 282 .00000000000009433398189512690, 283 .00000000000041481318704258568, 284 -.00000000000003792316480209314, 285 .00000000000008403156304792424, 286 -.00000000000034262934348285429, 287 .00000000000043712191957429145, 288 -.00000000000010475750058776541, 289 -.00000000000011118671389559323, 290 .00000000000037549577257259853, 291 .00000000000013912841212197565, 292 .00000000000010775743037572640, 293 .00000000000029391859187648000, 294 -.00000000000042790509060060774, 295 .00000000000022774076114039555, 296 .00000000000010849569622967912, 297 -.00000000000023073801945705758, 298 .00000000000015761203773969435, 299 .00000000000003345710269544082, 300 -.00000000000041525158063436123, 301 .00000000000032655698896907146, 302 -.00000000000044704265010452446, 303 .00000000000034527647952039772, 304 -.00000000000007048962392109746, 305 .00000000000011776978751369214, 306 -.00000000000010774341461609578, 307 .00000000000021863343293215910, 308 .00000000000024132639491333131, 309 .00000000000039057462209830700, 310 -.00000000000026570679203560751, 311 .00000000000037135141919592021, 312 -.00000000000017166921336082431, 313 -.00000000000028658285157914353, 314 -.00000000000023812542263446809, 315 .00000000000006576659768580062, 316 -.00000000000028210143846181267, 317 .00000000000010701931762114254, 318 .00000000000018119346366441110, 319 .00000000000009840465278232627, 320 -.00000000000033149150282752542, 321 -.00000000000018302857356041668, 322 -.00000000000016207400156744949, 323 .00000000000048303314949553201, 324 -.00000000000071560553172382115, 325 .00000000000088821239518571855, 326 -.00000000000030900580513238244, 327 -.00000000000061076551972851496, 328 .00000000000035659969663347830, 329 .00000000000035782396591276383, 330 -.00000000000046226087001544578, 331 .00000000000062279762917225156, 332 .00000000000072838947272065741, 333 .00000000000026809646615211673, 334 -.00000000000010960825046059278, 335 .00000000000002311949383800537, 336 -.00000000000058469058005299247, 337 -.00000000000002103748251144494, 338 -.00000000000023323182945587408, 339 -.00000000000042333694288141916, 340 -.00000000000043933937969737844, 341 .00000000000041341647073835565, 342 .00000000000006841763641591466, 343 .00000000000047585534004430641, 344 .00000000000083679678674757695, 345 -.00000000000085763734646658640, 346 .00000000000021913281229340092, 347 -.00000000000062242842536431148, 348 -.00000000000010983594325438430, 349 .00000000000065310431377633651, 350 -.00000000000047580199021710769, 351 -.00000000000037854251265457040, 352 .00000000000040939233218678664, 353 .00000000000087424383914858291, 354 .00000000000025218188456842882, 355 -.00000000000003608131360422557, 356 -.00000000000050518555924280902, 357 .00000000000078699403323355317, 358 -.00000000000067020876961949060, 359 .00000000000016108575753932458, 360 .00000000000058527188436251509, 361 -.00000000000035246757297904791, 362 -.00000000000018372084495629058, 363 .00000000000088606689813494916, 364 .00000000000066486268071468700, 365 .00000000000063831615170646519, 366 .00000000000025144230728376072, 367 -.00000000000017239444525614834 368 }; 369 370 double 371 #ifdef _ANSI_SOURCE 372 log(double x) 373 #else 374 log(x) double x; 375 #endif 376 { 377 int m, j; 378 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 379 volatile double u1; 380 381 /* Catch special cases */ 382 if (x <= 0) 383 if (_IEEE && x == zero) /* log(0) = -Inf */ 384 return (-one/zero); 385 else if (_IEEE) /* log(neg) = NaN */ 386 return (zero/zero); 387 else if (x == zero) /* NOT REACHED IF _IEEE */ 388 return (infnan(-ERANGE)); 389 else 390 return (infnan(EDOM)); 391 else if (!finite(x)) 392 if (_IEEE) /* x = NaN, Inf */ 393 return (x+x); 394 else 395 return (infnan(ERANGE)); 396 397 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 398 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 399 400 m = logb(x); 401 g = ldexp(x, -m); 402 if (_IEEE && m == -1022) { 403 j = logb(g), m += j; 404 g = ldexp(g, -j); 405 } 406 j = N*(g-1) + .5; 407 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 408 f = g - F; 409 410 /* Approximate expansion for log(1+f/F) ~= u + q */ 411 g = 1/(2*F+f); 412 u = 2*f*g; 413 v = u*u; 414 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 415 416 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 417 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 418 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 419 */ 420 if (m | j) 421 u1 = u + 513, u1 -= 513; 422 423 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 424 * u1 = u to 24 bits. 425 */ 426 else 427 u1 = u, TRUNC(u1); 428 u2 = (2.0*(f - F*u1) - u1*f) * g; 429 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 430 431 /* log(x) = log(2^m*F*(1+f/F)) = */ 432 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 433 /* (exact) + (tiny) */ 434 435 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 436 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 437 u2 += logF_tail[N]*m; 438 return (u1 + u2); 439 } 440 441 /* 442 * Extra precision variant, returning struct {double a, b;}; 443 * log(x) = a+b to 63 bits, with a is rounded to 26 bits. 444 */ 445 struct Double 446 #ifdef _ANSI_SOURCE 447 __log__D(double x) 448 #else 449 __log__D(x) double x; 450 #endif 451 { 452 int m, j; 453 double F, f, g, q, u, v, u2, one = 1.0; 454 volatile double u1; 455 struct Double r; 456 457 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 458 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 459 460 m = logb(x); 461 g = ldexp(x, -m); 462 if (_IEEE && m == -1022) { 463 j = logb(g), m += j; 464 g = ldexp(g, -j); 465 } 466 j = N*(g-1) + .5; 467 F = (1.0/N) * j + 1; 468 f = g - F; 469 470 g = 1/(2*F+f); 471 u = 2*f*g; 472 v = u*u; 473 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 474 if (m | j) 475 u1 = u + 513, u1 -= 513; 476 else 477 u1 = u, TRUNC(u1); 478 u2 = (2.0*(f - F*u1) - u1*f) * g; 479 480 u1 += m*logF_head[N] + logF_head[j]; 481 482 u2 += logF_tail[j]; u2 += q; 483 u2 += logF_tail[N]*m; 484 r.a = u1 + u2; /* Only difference is here */ 485 TRUNC(r.a); 486 r.b = (u1 - r.a) + u2; 487 return (r); 488 } 489