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: src/lib/msun/bsdsrc/b_log.c,v 1.8 2005/09/19 11:28:19 bde Exp $"); */ 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 N 128 81 82 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 83 * Used for generation of extend precision logarithms. 84 * The constant 35184372088832 is 2^45, so the divide is exact. 85 * It ensures correct reading of logF_head, even for inaccurate 86 * decimal-to-binary conversion routines. (Everybody gets the 87 * right answer for integers less than 2^53.) 88 * Values for log(F) were generated using error < 10^-57 absolute 89 * with the bc -l package. 90 */ 91 static double A1 = .08333333333333178827; 92 static double A2 = .01250000000377174923; 93 static double A3 = .002232139987919447809; 94 static double A4 = .0004348877777076145742; 95 96 static double logF_head[N+1] = { 97 0., 98 .007782140442060381246, 99 .015504186535963526694, 100 .023167059281547608406, 101 .030771658666765233647, 102 .038318864302141264488, 103 .045809536031242714670, 104 .053244514518837604555, 105 .060624621816486978786, 106 .067950661908525944454, 107 .075223421237524235039, 108 .082443669210988446138, 109 .089612158689760690322, 110 .096729626458454731618, 111 .103796793681567578460, 112 .110814366340264314203, 113 .117783035656430001836, 114 .124703478501032805070, 115 .131576357788617315236, 116 .138402322859292326029, 117 .145182009844575077295, 118 .151916042025732167530, 119 .158605030176659056451, 120 .165249572895390883786, 121 .171850256926518341060, 122 .178407657472689606947, 123 .184922338493834104156, 124 .191394852999565046047, 125 .197825743329758552135, 126 .204215541428766300668, 127 .210564769107350002741, 128 .216873938300523150246, 129 .223143551314024080056, 130 .229374101064877322642, 131 .235566071312860003672, 132 .241719936886966024758, 133 .247836163904594286577, 134 .253915209980732470285, 135 .259957524436686071567, 136 .265963548496984003577, 137 .271933715484010463114, 138 .277868451003087102435, 139 .283768173130738432519, 140 .289633292582948342896, 141 .295464212893421063199, 142 .301261330578199704177, 143 .307025035294827830512, 144 .312755710004239517729, 145 .318453731118097493890, 146 .324119468654316733591, 147 .329753286372579168528, 148 .335355541920762334484, 149 .340926586970454081892, 150 .346466767346100823488, 151 .351976423156884266063, 152 .357455888922231679316, 153 .362905493689140712376, 154 .368325561158599157352, 155 .373716409793814818840, 156 .379078352934811846353, 157 .384411698910298582632, 158 .389716751140440464951, 159 .394993808240542421117, 160 .400243164127459749579, 161 .405465108107819105498, 162 .410659924985338875558, 163 .415827895143593195825, 164 .420969294644237379543, 165 .426084395310681429691, 166 .431173464818130014464, 167 .436236766774527495726, 168 .441274560805140936281, 169 .446287102628048160113, 170 .451274644139630254358, 171 .456237433481874177232, 172 .461175715122408291790, 173 .466089729924533457960, 174 .470979715219073113985, 175 .475845904869856894947, 176 .480688529345570714212, 177 .485507815781602403149, 178 .490303988045525329653, 179 .495077266798034543171, 180 .499827869556611403822, 181 .504556010751912253908, 182 .509261901790523552335, 183 .513945751101346104405, 184 .518607764208354637958, 185 .523248143765158602036, 186 .527867089620485785417, 187 .532464798869114019908, 188 .537041465897345915436, 189 .541597282432121573947, 190 .546132437597407260909, 191 .550647117952394182793, 192 .555141507540611200965, 193 .559615787935399566777, 194 .564070138285387656651, 195 .568504735352689749561, 196 .572919753562018740922, 197 .577315365035246941260, 198 .581691739635061821900, 199 .586049045003164792433, 200 .590387446602107957005, 201 .594707107746216934174, 202 .599008189645246602594, 203 .603290851438941899687, 204 .607555250224322662688, 205 .611801541106615331955, 206 .616029877215623855590, 207 .620240409751204424537, 208 .624433288012369303032, 209 .628608659422752680256, 210 .632766669570628437213, 211 .636907462236194987781, 212 .641031179420679109171, 213 .645137961373620782978, 214 .649227946625615004450, 215 .653301272011958644725, 216 .657358072709030238911, 217 .661398482245203922502, 218 .665422632544505177065, 219 .669430653942981734871, 220 .673422675212350441142, 221 .677398823590920073911, 222 .681359224807238206267, 223 .685304003098281100392, 224 .689233281238557538017, 225 .693147180560117703862 226 }; 227 228 static double logF_tail[N+1] = { 229 0., 230 -.00000000000000543229938420049, 231 .00000000000000172745674997061, 232 -.00000000000001323017818229233, 233 -.00000000000001154527628289872, 234 -.00000000000000466529469958300, 235 .00000000000005148849572685810, 236 -.00000000000002532168943117445, 237 -.00000000000005213620639136504, 238 -.00000000000001819506003016881, 239 .00000000000006329065958724544, 240 .00000000000008614512936087814, 241 -.00000000000007355770219435028, 242 .00000000000009638067658552277, 243 .00000000000007598636597194141, 244 .00000000000002579999128306990, 245 -.00000000000004654729747598444, 246 -.00000000000007556920687451336, 247 .00000000000010195735223708472, 248 -.00000000000017319034406422306, 249 -.00000000000007718001336828098, 250 .00000000000010980754099855238, 251 -.00000000000002047235780046195, 252 -.00000000000008372091099235912, 253 .00000000000014088127937111135, 254 .00000000000012869017157588257, 255 .00000000000017788850778198106, 256 .00000000000006440856150696891, 257 .00000000000016132822667240822, 258 -.00000000000007540916511956188, 259 -.00000000000000036507188831790, 260 .00000000000009120937249914984, 261 .00000000000018567570959796010, 262 -.00000000000003149265065191483, 263 -.00000000000009309459495196889, 264 .00000000000017914338601329117, 265 -.00000000000001302979717330866, 266 .00000000000023097385217586939, 267 .00000000000023999540484211737, 268 .00000000000015393776174455408, 269 -.00000000000036870428315837678, 270 .00000000000036920375082080089, 271 -.00000000000009383417223663699, 272 .00000000000009433398189512690, 273 .00000000000041481318704258568, 274 -.00000000000003792316480209314, 275 .00000000000008403156304792424, 276 -.00000000000034262934348285429, 277 .00000000000043712191957429145, 278 -.00000000000010475750058776541, 279 -.00000000000011118671389559323, 280 .00000000000037549577257259853, 281 .00000000000013912841212197565, 282 .00000000000010775743037572640, 283 .00000000000029391859187648000, 284 -.00000000000042790509060060774, 285 .00000000000022774076114039555, 286 .00000000000010849569622967912, 287 -.00000000000023073801945705758, 288 .00000000000015761203773969435, 289 .00000000000003345710269544082, 290 -.00000000000041525158063436123, 291 .00000000000032655698896907146, 292 -.00000000000044704265010452446, 293 .00000000000034527647952039772, 294 -.00000000000007048962392109746, 295 .00000000000011776978751369214, 296 -.00000000000010774341461609578, 297 .00000000000021863343293215910, 298 .00000000000024132639491333131, 299 .00000000000039057462209830700, 300 -.00000000000026570679203560751, 301 .00000000000037135141919592021, 302 -.00000000000017166921336082431, 303 -.00000000000028658285157914353, 304 -.00000000000023812542263446809, 305 .00000000000006576659768580062, 306 -.00000000000028210143846181267, 307 .00000000000010701931762114254, 308 .00000000000018119346366441110, 309 .00000000000009840465278232627, 310 -.00000000000033149150282752542, 311 -.00000000000018302857356041668, 312 -.00000000000016207400156744949, 313 .00000000000048303314949553201, 314 -.00000000000071560553172382115, 315 .00000000000088821239518571855, 316 -.00000000000030900580513238244, 317 -.00000000000061076551972851496, 318 .00000000000035659969663347830, 319 .00000000000035782396591276383, 320 -.00000000000046226087001544578, 321 .00000000000062279762917225156, 322 .00000000000072838947272065741, 323 .00000000000026809646615211673, 324 -.00000000000010960825046059278, 325 .00000000000002311949383800537, 326 -.00000000000058469058005299247, 327 -.00000000000002103748251144494, 328 -.00000000000023323182945587408, 329 -.00000000000042333694288141916, 330 -.00000000000043933937969737844, 331 .00000000000041341647073835565, 332 .00000000000006841763641591466, 333 .00000000000047585534004430641, 334 .00000000000083679678674757695, 335 -.00000000000085763734646658640, 336 .00000000000021913281229340092, 337 -.00000000000062242842536431148, 338 -.00000000000010983594325438430, 339 .00000000000065310431377633651, 340 -.00000000000047580199021710769, 341 -.00000000000037854251265457040, 342 .00000000000040939233218678664, 343 .00000000000087424383914858291, 344 .00000000000025218188456842882, 345 -.00000000000003608131360422557, 346 -.00000000000050518555924280902, 347 .00000000000078699403323355317, 348 -.00000000000067020876961949060, 349 .00000000000016108575753932458, 350 .00000000000058527188436251509, 351 -.00000000000035246757297904791, 352 -.00000000000018372084495629058, 353 .00000000000088606689813494916, 354 .00000000000066486268071468700, 355 .00000000000063831615170646519, 356 .00000000000025144230728376072, 357 -.00000000000017239444525614834 358 }; 359 360 #if 0 361 double 362 #ifdef _ANSI_SOURCE 363 log(double x) 364 #else 365 log(x) double x; 366 #endif 367 { 368 int m, j; 369 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 370 volatile double u1; 371 372 /* Catch special cases */ 373 if (x <= 0) 374 if (x == zero) /* log(0) = -Inf */ 375 return (-one/zero); 376 else /* log(neg) = NaN */ 377 return (zero/zero); 378 else if (!finite(x)) 379 return (x+x); /* x = NaN, Inf */ 380 381 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 382 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 383 384 m = logb(x); 385 g = ldexp(x, -m); 386 if (m == -1022) { 387 j = logb(g), m += j; 388 g = ldexp(g, -j); 389 } 390 j = N*(g-1) + .5; 391 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 392 f = g - F; 393 394 /* Approximate expansion for log(1+f/F) ~= u + q */ 395 g = 1/(2*F+f); 396 u = 2*f*g; 397 v = u*u; 398 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 399 400 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 401 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 402 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 403 */ 404 if (m | j) 405 u1 = u + 513, u1 -= 513; 406 407 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 408 * u1 = u to 24 bits. 409 */ 410 else 411 u1 = u, TRUNC(u1); 412 u2 = (2.0*(f - F*u1) - u1*f) * g; 413 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 414 415 /* log(x) = log(2^m*F*(1+f/F)) = */ 416 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 417 /* (exact) + (tiny) */ 418 419 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 420 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 421 u2 += logF_tail[N]*m; 422 return (u1 + u2); 423 } 424 #endif 425 426 /* 427 * Extra precision variant, returning struct {double a, b;}; 428 * log(x) = a+b to 63 bits, with a rounded to 26 bits. 429 */ 430 struct Double 431 #ifdef _ANSI_SOURCE 432 __log__D(double x) 433 #else 434 __log__D(x) double x; 435 #endif 436 { 437 int m, j; 438 double F, f, g, q, u, v, u2; 439 volatile double u1; 440 struct Double r; 441 442 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 443 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 444 445 m = logb(x); 446 g = ldexp(x, -m); 447 if (m == -1022) { 448 j = logb(g), m += j; 449 g = ldexp(g, -j); 450 } 451 j = N*(g-1) + .5; 452 F = (1.0/N) * j + 1; 453 f = g - F; 454 455 g = 1/(2*F+f); 456 u = 2*f*g; 457 v = u*u; 458 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 459 if (m | j) 460 u1 = u + 513, u1 -= 513; 461 else 462 u1 = u, TRUNC(u1); 463 u2 = (2.0*(f - F*u1) - u1*f) * g; 464 465 u1 += m*logF_head[N] + logF_head[j]; 466 467 u2 += logF_tail[j]; u2 += q; 468 u2 += logF_tail[N]*m; 469 r.a = u1 + u2; /* Only difference is here */ 470 TRUNC(r.a); 471 r.b = (u1 - r.a) + u2; 472 return (r); 473 } 474
