1 /* $NetBSD: n_log.c,v 1.1 1995/10/10 23:36:57 ragge Exp $ */ 2 /* 3 * Copyright (c) 1992, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 static char sccsid[] = "@(#)log.c 8.2 (Berkeley) 11/30/93"; 37 #endif /* not lint */ 38 39 #include <math.h> 40 #include <errno.h> 41 42 #include "mathimpl.h" 43 44 /* Table-driven natural logarithm. 45 * 46 * This code was derived, with minor modifications, from: 47 * Peter Tang, "Table-Driven Implementation of the 48 * Logarithm in IEEE Floating-Point arithmetic." ACM Trans. 49 * Math Software, vol 16. no 4, pp 378-400, Dec 1990). 50 * 51 * Calculates log(2^m*F*(1+f/F)), |f/j| <= 1/256, 52 * where F = j/128 for j an integer in [0, 128]. 53 * 54 * log(2^m) = log2_hi*m + log2_tail*m 55 * since m is an integer, the dominant term is exact. 56 * m has at most 10 digits (for subnormal numbers), 57 * and log2_hi has 11 trailing zero bits. 58 * 59 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h 60 * logF_hi[] + 512 is exact. 61 * 62 * log(1+f/F) = 2*f/(2*F + f) + 1/12 * (2*f/(2*F + f))**3 + ... 63 * the leading term is calculated to extra precision in two 64 * parts, the larger of which adds exactly to the dominant 65 * m and F terms. 66 * There are two cases: 67 * 1. when m, j are non-zero (m | j), use absolute 68 * precision for the leading term. 69 * 2. when m = j = 0, |1-x| < 1/256, and log(x) ~= (x-1). 70 * In this case, use a relative precision of 24 bits. 71 * (This is done differently in the original paper) 72 * 73 * Special cases: 74 * 0 return signalling -Inf 75 * neg return signalling NaN 76 * +Inf return +Inf 77 */ 78 79 #if defined(vax) || defined(tahoe) 80 #define _IEEE 0 81 #define TRUNC(x) x = (double) (float) (x) 82 #else 83 #define _IEEE 1 84 #define endian (((*(int *) &one)) ? 1 : 0) 85 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000 86 #define infnan(x) 0.0 87 #endif 88 89 #define N 128 90 91 /* Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128. 92 * Used for generation of extend precision logarithms. 93 * The constant 35184372088832 is 2^45, so the divide is exact. 94 * It ensures correct reading of logF_head, even for inaccurate 95 * decimal-to-binary conversion routines. (Everybody gets the 96 * right answer for integers less than 2^53.) 97 * Values for log(F) were generated using error < 10^-57 absolute 98 * with the bc -l package. 99 */ 100 static double A1 = .08333333333333178827; 101 static double A2 = .01250000000377174923; 102 static double A3 = .002232139987919447809; 103 static double A4 = .0004348877777076145742; 104 105 static double logF_head[N+1] = { 106 0., 107 .007782140442060381246, 108 .015504186535963526694, 109 .023167059281547608406, 110 .030771658666765233647, 111 .038318864302141264488, 112 .045809536031242714670, 113 .053244514518837604555, 114 .060624621816486978786, 115 .067950661908525944454, 116 .075223421237524235039, 117 .082443669210988446138, 118 .089612158689760690322, 119 .096729626458454731618, 120 .103796793681567578460, 121 .110814366340264314203, 122 .117783035656430001836, 123 .124703478501032805070, 124 .131576357788617315236, 125 .138402322859292326029, 126 .145182009844575077295, 127 .151916042025732167530, 128 .158605030176659056451, 129 .165249572895390883786, 130 .171850256926518341060, 131 .178407657472689606947, 132 .184922338493834104156, 133 .191394852999565046047, 134 .197825743329758552135, 135 .204215541428766300668, 136 .210564769107350002741, 137 .216873938300523150246, 138 .223143551314024080056, 139 .229374101064877322642, 140 .235566071312860003672, 141 .241719936886966024758, 142 .247836163904594286577, 143 .253915209980732470285, 144 .259957524436686071567, 145 .265963548496984003577, 146 .271933715484010463114, 147 .277868451003087102435, 148 .283768173130738432519, 149 .289633292582948342896, 150 .295464212893421063199, 151 .301261330578199704177, 152 .307025035294827830512, 153 .312755710004239517729, 154 .318453731118097493890, 155 .324119468654316733591, 156 .329753286372579168528, 157 .335355541920762334484, 158 .340926586970454081892, 159 .346466767346100823488, 160 .351976423156884266063, 161 .357455888922231679316, 162 .362905493689140712376, 163 .368325561158599157352, 164 .373716409793814818840, 165 .379078352934811846353, 166 .384411698910298582632, 167 .389716751140440464951, 168 .394993808240542421117, 169 .400243164127459749579, 170 .405465108107819105498, 171 .410659924985338875558, 172 .415827895143593195825, 173 .420969294644237379543, 174 .426084395310681429691, 175 .431173464818130014464, 176 .436236766774527495726, 177 .441274560805140936281, 178 .446287102628048160113, 179 .451274644139630254358, 180 .456237433481874177232, 181 .461175715122408291790, 182 .466089729924533457960, 183 .470979715219073113985, 184 .475845904869856894947, 185 .480688529345570714212, 186 .485507815781602403149, 187 .490303988045525329653, 188 .495077266798034543171, 189 .499827869556611403822, 190 .504556010751912253908, 191 .509261901790523552335, 192 .513945751101346104405, 193 .518607764208354637958, 194 .523248143765158602036, 195 .527867089620485785417, 196 .532464798869114019908, 197 .537041465897345915436, 198 .541597282432121573947, 199 .546132437597407260909, 200 .550647117952394182793, 201 .555141507540611200965, 202 .559615787935399566777, 203 .564070138285387656651, 204 .568504735352689749561, 205 .572919753562018740922, 206 .577315365035246941260, 207 .581691739635061821900, 208 .586049045003164792433, 209 .590387446602107957005, 210 .594707107746216934174, 211 .599008189645246602594, 212 .603290851438941899687, 213 .607555250224322662688, 214 .611801541106615331955, 215 .616029877215623855590, 216 .620240409751204424537, 217 .624433288012369303032, 218 .628608659422752680256, 219 .632766669570628437213, 220 .636907462236194987781, 221 .641031179420679109171, 222 .645137961373620782978, 223 .649227946625615004450, 224 .653301272011958644725, 225 .657358072709030238911, 226 .661398482245203922502, 227 .665422632544505177065, 228 .669430653942981734871, 229 .673422675212350441142, 230 .677398823590920073911, 231 .681359224807238206267, 232 .685304003098281100392, 233 .689233281238557538017, 234 .693147180560117703862 235 }; 236 237 static double logF_tail[N+1] = { 238 0., 239 -.00000000000000543229938420049, 240 .00000000000000172745674997061, 241 -.00000000000001323017818229233, 242 -.00000000000001154527628289872, 243 -.00000000000000466529469958300, 244 .00000000000005148849572685810, 245 -.00000000000002532168943117445, 246 -.00000000000005213620639136504, 247 -.00000000000001819506003016881, 248 .00000000000006329065958724544, 249 .00000000000008614512936087814, 250 -.00000000000007355770219435028, 251 .00000000000009638067658552277, 252 .00000000000007598636597194141, 253 .00000000000002579999128306990, 254 -.00000000000004654729747598444, 255 -.00000000000007556920687451336, 256 .00000000000010195735223708472, 257 -.00000000000017319034406422306, 258 -.00000000000007718001336828098, 259 .00000000000010980754099855238, 260 -.00000000000002047235780046195, 261 -.00000000000008372091099235912, 262 .00000000000014088127937111135, 263 .00000000000012869017157588257, 264 .00000000000017788850778198106, 265 .00000000000006440856150696891, 266 .00000000000016132822667240822, 267 -.00000000000007540916511956188, 268 -.00000000000000036507188831790, 269 .00000000000009120937249914984, 270 .00000000000018567570959796010, 271 -.00000000000003149265065191483, 272 -.00000000000009309459495196889, 273 .00000000000017914338601329117, 274 -.00000000000001302979717330866, 275 .00000000000023097385217586939, 276 .00000000000023999540484211737, 277 .00000000000015393776174455408, 278 -.00000000000036870428315837678, 279 .00000000000036920375082080089, 280 -.00000000000009383417223663699, 281 .00000000000009433398189512690, 282 .00000000000041481318704258568, 283 -.00000000000003792316480209314, 284 .00000000000008403156304792424, 285 -.00000000000034262934348285429, 286 .00000000000043712191957429145, 287 -.00000000000010475750058776541, 288 -.00000000000011118671389559323, 289 .00000000000037549577257259853, 290 .00000000000013912841212197565, 291 .00000000000010775743037572640, 292 .00000000000029391859187648000, 293 -.00000000000042790509060060774, 294 .00000000000022774076114039555, 295 .00000000000010849569622967912, 296 -.00000000000023073801945705758, 297 .00000000000015761203773969435, 298 .00000000000003345710269544082, 299 -.00000000000041525158063436123, 300 .00000000000032655698896907146, 301 -.00000000000044704265010452446, 302 .00000000000034527647952039772, 303 -.00000000000007048962392109746, 304 .00000000000011776978751369214, 305 -.00000000000010774341461609578, 306 .00000000000021863343293215910, 307 .00000000000024132639491333131, 308 .00000000000039057462209830700, 309 -.00000000000026570679203560751, 310 .00000000000037135141919592021, 311 -.00000000000017166921336082431, 312 -.00000000000028658285157914353, 313 -.00000000000023812542263446809, 314 .00000000000006576659768580062, 315 -.00000000000028210143846181267, 316 .00000000000010701931762114254, 317 .00000000000018119346366441110, 318 .00000000000009840465278232627, 319 -.00000000000033149150282752542, 320 -.00000000000018302857356041668, 321 -.00000000000016207400156744949, 322 .00000000000048303314949553201, 323 -.00000000000071560553172382115, 324 .00000000000088821239518571855, 325 -.00000000000030900580513238244, 326 -.00000000000061076551972851496, 327 .00000000000035659969663347830, 328 .00000000000035782396591276383, 329 -.00000000000046226087001544578, 330 .00000000000062279762917225156, 331 .00000000000072838947272065741, 332 .00000000000026809646615211673, 333 -.00000000000010960825046059278, 334 .00000000000002311949383800537, 335 -.00000000000058469058005299247, 336 -.00000000000002103748251144494, 337 -.00000000000023323182945587408, 338 -.00000000000042333694288141916, 339 -.00000000000043933937969737844, 340 .00000000000041341647073835565, 341 .00000000000006841763641591466, 342 .00000000000047585534004430641, 343 .00000000000083679678674757695, 344 -.00000000000085763734646658640, 345 .00000000000021913281229340092, 346 -.00000000000062242842536431148, 347 -.00000000000010983594325438430, 348 .00000000000065310431377633651, 349 -.00000000000047580199021710769, 350 -.00000000000037854251265457040, 351 .00000000000040939233218678664, 352 .00000000000087424383914858291, 353 .00000000000025218188456842882, 354 -.00000000000003608131360422557, 355 -.00000000000050518555924280902, 356 .00000000000078699403323355317, 357 -.00000000000067020876961949060, 358 .00000000000016108575753932458, 359 .00000000000058527188436251509, 360 -.00000000000035246757297904791, 361 -.00000000000018372084495629058, 362 .00000000000088606689813494916, 363 .00000000000066486268071468700, 364 .00000000000063831615170646519, 365 .00000000000025144230728376072, 366 -.00000000000017239444525614834 367 }; 368 369 double 370 #ifdef _ANSI_SOURCE 371 log(double x) 372 #else 373 log(x) double x; 374 #endif 375 { 376 int m, j; 377 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0; 378 volatile double u1; 379 380 /* Catch special cases */ 381 if (x <= 0) 382 if (_IEEE && x == zero) /* log(0) = -Inf */ 383 return (-one/zero); 384 else if (_IEEE) /* log(neg) = NaN */ 385 return (zero/zero); 386 else if (x == zero) /* NOT REACHED IF _IEEE */ 387 return (infnan(-ERANGE)); 388 else 389 return (infnan(EDOM)); 390 else if (!finite(x)) 391 if (_IEEE) /* x = NaN, Inf */ 392 return (x+x); 393 else 394 return (infnan(ERANGE)); 395 396 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 397 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 398 399 m = logb(x); 400 g = ldexp(x, -m); 401 if (_IEEE && m == -1022) { 402 j = logb(g), m += j; 403 g = ldexp(g, -j); 404 } 405 j = N*(g-1) + .5; 406 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */ 407 f = g - F; 408 409 /* Approximate expansion for log(1+f/F) ~= u + q */ 410 g = 1/(2*F+f); 411 u = 2*f*g; 412 v = u*u; 413 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 414 415 /* case 1: u1 = u rounded to 2^-43 absolute. Since u < 2^-8, 416 * u1 has at most 35 bits, and F*u1 is exact, as F has < 8 bits. 417 * It also adds exactly to |m*log2_hi + log_F_head[j] | < 750 418 */ 419 if (m | j) 420 u1 = u + 513, u1 -= 513; 421 422 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero; 423 * u1 = u to 24 bits. 424 */ 425 else 426 u1 = u, TRUNC(u1); 427 u2 = (2.0*(f - F*u1) - u1*f) * g; 428 /* u1 + u2 = 2f/(2F+f) to extra precision. */ 429 430 /* log(x) = log(2^m*F*(1+f/F)) = */ 431 /* (m*log2_hi+logF_head[j]+u1) + (m*log2_lo+logF_tail[j]+q); */ 432 /* (exact) + (tiny) */ 433 434 u1 += m*logF_head[N] + logF_head[j]; /* exact */ 435 u2 = (u2 + logF_tail[j]) + q; /* tiny */ 436 u2 += logF_tail[N]*m; 437 return (u1 + u2); 438 } 439 440 /* 441 * Extra precision variant, returning struct {double a, b;}; 442 * log(x) = a+b to 63 bits, with a is rounded to 26 bits. 443 */ 444 struct Double 445 #ifdef _ANSI_SOURCE 446 __log__D(double x) 447 #else 448 __log__D(x) double x; 449 #endif 450 { 451 int m, j; 452 double F, f, g, q, u, v, u2, one = 1.0; 453 volatile double u1; 454 struct Double r; 455 456 /* Argument reduction: 1 <= g < 2; x/2^m = g; */ 457 /* y = F*(1 + f/F) for |f| <= 2^-8 */ 458 459 m = logb(x); 460 g = ldexp(x, -m); 461 if (_IEEE && m == -1022) { 462 j = logb(g), m += j; 463 g = ldexp(g, -j); 464 } 465 j = N*(g-1) + .5; 466 F = (1.0/N) * j + 1; 467 f = g - F; 468 469 g = 1/(2*F+f); 470 u = 2*f*g; 471 v = u*u; 472 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4))); 473 if (m | j) 474 u1 = u + 513, u1 -= 513; 475 else 476 u1 = u, TRUNC(u1); 477 u2 = (2.0*(f - F*u1) - u1*f) * g; 478 479 u1 += m*logF_head[N] + logF_head[j]; 480 481 u2 += logF_tail[j]; u2 += q; 482 u2 += logF_tail[N]*m; 483 r.a = u1 + u2; /* Only difference is here */ 484 TRUNC(r.a); 485 r.b = (u1 - r.a) + u2; 486 return (r); 487 } 488