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