2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
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6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
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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.
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
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30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
34 /* @(#)log.c 8.2 (Berkeley) 11/30/93 */
35 #include <sys/cdefs.h>
36 __FBSDID("$FreeBSD$");
43 /* Table-driven natural logarithm.
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).
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].
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.
58 * log(F) = logF_hi[j] + logF_lo[j] is in tabular form in log_table.h
59 * logF_hi[] + 512 is exact.
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
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)
73 * 0 return signalling -Inf
74 * neg return signalling NaN
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.
89 static double A1 = .08333333333333178827;
90 static double A2 = .01250000000377174923;
91 static double A3 = .002232139987919447809;
92 static double A4 = .0004348877777076145742;
94 static double logF_head[N+1] = {
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
226 static double logF_tail[N+1] = {
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
367 double F, f, g, q, u, u2, v, zero = 0.0, one = 1.0;
370 /* Catch special cases */
372 if (x == zero) /* log(0) = -Inf */
374 else /* log(neg) = NaN */
377 return (x+x); /* x = NaN, Inf */
379 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
380 /* y = F*(1 + f/F) for |f| <= 2^-8 */
389 F = (1.0/N) * j + 1; /* F*128 is an integer in [128, 512] */
392 /* Approximate expansion for log(1+f/F) ~= u + q */
396 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
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
403 u1 = u + 513, u1 -= 513;
405 /* case 2: |1-x| < 1/256. The m- and j- dependent terms are zero;
410 u2 = (2.0*(f - F*u1) - u1*f) * g;
411 /* u1 + u2 = 2f/(2F+f) to extra precision. */
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) */
417 u1 += m*logF_head[N] + logF_head[j]; /* exact */
418 u2 = (u2 + logF_tail[j]) + q; /* tiny */
419 u2 += logF_tail[N]*m;
425 * Extra precision variant, returning struct {double a, b;};
426 * log(x) = a+b to 63 bits, with a rounded to 26 bits.
432 __log__D(x) double x;
436 double F, f, g, q, u, v, u2;
440 /* Argument reduction: 1 <= g < 2; x/2^m = g; */
441 /* y = F*(1 + f/F) for |f| <= 2^-8 */
456 q = u*v*(A1 + v*(A2 + v*(A3 + v*A4)));
458 u1 = u + 513, u1 -= 513;
461 u2 = (2.0*(f - F*u1) - u1*f) * g;
463 u1 += m*logF_head[N] + logF_head[j];
465 u2 += logF_tail[j]; u2 += q;
466 u2 += logF_tail[N]*m;
467 r.a = u1 + u2; /* Only difference is here */
469 r.b = (u1 - r.a) + u2;