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