1 /* $OpenBSD: e_logl.c,v 1.1 2011/07/06 00:02:42 martynas Exp $ */
2
3 /*
4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5 *
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
9 *
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17 */
18
19 /* logl.c
20 *
21 * Natural logarithm for 128-bit long double precision.
22 *
23 *
24 *
25 * SYNOPSIS:
26 *
27 * long double x, y, logl();
28 *
29 * y = logl( x );
30 *
31 *
32 *
33 * DESCRIPTION:
34 *
35 * Returns the base e (2.718...) logarithm of x.
36 *
37 * The argument is separated into its exponent and fractional
38 * parts. Use of a lookup table increases the speed of the routine.
39 * The program uses logarithms tabulated at intervals of 1/128 to
40 * cover the domain from approximately 0.7 to 1.4.
41 *
42 * On the interval [-1/128, +1/128] the logarithm of 1+x is approximated by
43 * log(1+x) = x - 0.5 x^2 + x^3 P(x) .
44 *
45 *
46 *
47 * ACCURACY:
48 *
49 * Relative error:
50 * arithmetic domain # trials peak rms
51 * IEEE 0.875, 1.125 100000 1.2e-34 4.1e-35
52 * IEEE 0.125, 8 100000 1.2e-34 4.1e-35
53 *
54 *
55 * WARNING:
56 *
57 * This program uses integer operations on bit fields of floating-point
58 * numbers. It does not work with data structures other than the
59 * structure assumed.
60 *
61 */
62
63
64
65 /* log(1+x) = x - .5 x^2 + x^3 l(x)
66 -.0078125 <= x <= +.0078125
67 peak relative error 1.2e-37 */
68 static const long double
69 l3 = 3.333333333333333333333333333333336096926E-1L,
70 l4 = -2.499999999999999999999999999486853077002E-1L,
71 l5 = 1.999999999999999999999999998515277861905E-1L,
72 l6 = -1.666666666666666666666798448356171665678E-1L,
73 l7 = 1.428571428571428571428808945895490721564E-1L,
74 l8 = -1.249999999999999987884655626377588149000E-1L,
75 l9 = 1.111111111111111093947834982832456459186E-1L,
76 l10 = -1.000000000000532974938900317952530453248E-1L,
77 l11 = 9.090909090915566247008015301349979892689E-2L,
78 l12 = -8.333333211818065121250921925397567745734E-2L,
79 l13 = 7.692307559897661630807048686258659316091E-2L,
80 l14 = -7.144242754190814657241902218399056829264E-2L,
81 l15 = 6.668057591071739754844678883223432347481E-2L;
82
83 /* Lookup table of ln(t) - (t-1)
84 t = 0.5 + (k+26)/128)
85 k = 0, ..., 91 */
86 static const long double logtbl[92] = {
87 -5.5345593589352099112142921677820359632418E-2L,
88 -5.2108257402767124761784665198737642086148E-2L,
89 -4.8991686870576856279407775480686721935120E-2L,
90 -4.5993270766361228596215288742353061431071E-2L,
91 -4.3110481649613269682442058976885699556950E-2L,
92 -4.0340872319076331310838085093194799765520E-2L,
93 -3.7682072451780927439219005993827431503510E-2L,
94 -3.5131785416234343803903228503274262719586E-2L,
95 -3.2687785249045246292687241862699949178831E-2L,
96 -3.0347913785027239068190798397055267411813E-2L,
97 -2.8110077931525797884641940838507561326298E-2L,
98 -2.5972247078357715036426583294246819637618E-2L,
99 -2.3932450635346084858612873953407168217307E-2L,
100 -2.1988775689981395152022535153795155900240E-2L,
101 -2.0139364778244501615441044267387667496733E-2L,
102 -1.8382413762093794819267536615342902718324E-2L,
103 -1.6716169807550022358923589720001638093023E-2L,
104 -1.5138929457710992616226033183958974965355E-2L,
105 -1.3649036795397472900424896523305726435029E-2L,
106 -1.2244881690473465543308397998034325468152E-2L,
107 -1.0924898127200937840689817557742469105693E-2L,
108 -9.6875626072830301572839422532631079809328E-3L,
109 -8.5313926245226231463436209313499745894157E-3L,
110 -7.4549452072765973384933565912143044991706E-3L,
111 -6.4568155251217050991200599386801665681310E-3L,
112 -5.5356355563671005131126851708522185605193E-3L,
113 -4.6900728132525199028885749289712348829878E-3L,
114 -3.9188291218610470766469347968659624282519E-3L,
115 -3.2206394539524058873423550293617843896540E-3L,
116 -2.5942708080877805657374888909297113032132E-3L,
117 -2.0385211375711716729239156839929281289086E-3L,
118 -1.5522183228760777967376942769773768850872E-3L,
119 -1.1342191863606077520036253234446621373191E-3L,
120 -7.8340854719967065861624024730268350459991E-4L,
121 -4.9869831458030115699628274852562992756174E-4L,
122 -2.7902661731604211834685052867305795169688E-4L,
123 -1.2335696813916860754951146082826952093496E-4L,
124 -3.0677461025892873184042490943581654591817E-5L,
125 #define ZERO logtbl[38]
126 0.0000000000000000000000000000000000000000E0L,
127 -3.0359557945051052537099938863236321874198E-5L,
128 -1.2081346403474584914595395755316412213151E-4L,
129 -2.7044071846562177120083903771008342059094E-4L,
130 -4.7834133324631162897179240322783590830326E-4L,
131 -7.4363569786340080624467487620270965403695E-4L,
132 -1.0654639687057968333207323853366578860679E-3L,
133 -1.4429854811877171341298062134712230604279E-3L,
134 -1.8753781835651574193938679595797367137975E-3L,
135 -2.3618380914922506054347222273705859653658E-3L,
136 -2.9015787624124743013946600163375853631299E-3L,
137 -3.4938307889254087318399313316921940859043E-3L,
138 -4.1378413103128673800485306215154712148146E-3L,
139 -4.8328735414488877044289435125365629849599E-3L,
140 -5.5782063183564351739381962360253116934243E-3L,
141 -6.3731336597098858051938306767880719015261E-3L,
142 -7.2169643436165454612058905294782949315193E-3L,
143 -8.1090214990427641365934846191367315083867E-3L,
144 -9.0486422112807274112838713105168375482480E-3L,
145 -1.0035177140880864314674126398350812606841E-2L,
146 -1.1067990155502102718064936259435676477423E-2L,
147 -1.2146457974158024928196575103115488672416E-2L,
148 -1.3269969823361415906628825374158424754308E-2L,
149 -1.4437927104692837124388550722759686270765E-2L,
150 -1.5649743073340777659901053944852735064621E-2L,
151 -1.6904842527181702880599758489058031645317E-2L,
152 -1.8202661505988007336096407340750378994209E-2L,
153 -1.9542647000370545390701192438691126552961E-2L,
154 -2.0924256670080119637427928803038530924742E-2L,
155 -2.2346958571309108496179613803760727786257E-2L,
156 -2.3810230892650362330447187267648486279460E-2L,
157 -2.5313561699385640380910474255652501521033E-2L,
158 -2.6856448685790244233704909690165496625399E-2L,
159 -2.8438398935154170008519274953860128449036E-2L,
160 -3.0058928687233090922411781058956589863039E-2L,
161 -3.1717563112854831855692484086486099896614E-2L,
162 -3.3413836095418743219397234253475252001090E-2L,
163 -3.5147290019036555862676702093393332533702E-2L,
164 -3.6917475563073933027920505457688955423688E-2L,
165 -3.8723951502862058660874073462456610731178E-2L,
166 -4.0566284516358241168330505467000838017425E-2L,
167 -4.2444048996543693813649967076598766917965E-2L,
168 -4.4356826869355401653098777649745233339196E-2L,
169 -4.6304207416957323121106944474331029996141E-2L,
170 -4.8285787106164123613318093945035804818364E-2L,
171 -5.0301169421838218987124461766244507342648E-2L,
172 -5.2349964705088137924875459464622098310997E-2L,
173 -5.4431789996103111613753440311680967840214E-2L,
174 -5.6546268881465384189752786409400404404794E-2L,
175 -5.8693031345788023909329239565012647817664E-2L,
176 -6.0871713627532018185577188079210189048340E-2L,
177 -6.3081958078862169742820420185833800925568E-2L,
178 -6.5323413029406789694910800219643791556918E-2L,
179 -6.7595732653791419081537811574227049288168E-2L
180 };
181
182 /* ln(2) = ln2a + ln2b with extended precision. */
183 static const long double
184 ln2a = 6.93145751953125e-1L,
185 ln2b = 1.4286068203094172321214581765680755001344E-6L;
186
187 long double
logl(long double x)188 logl(long double x)
189 {
190 long double z, y, w;
191 ieee_quad_shape_type u, t;
192 unsigned int m;
193 int k, e;
194
195 u.value = x;
196 m = u.parts32.mswhi;
197
198 /* Check for IEEE special cases. */
199 k = m & 0x7fffffff;
200 /* log(0) = -infinity. */
201 if ((k | u.parts32.mswlo | u.parts32.lswhi | u.parts32.lswlo) == 0)
202 {
203 return __math_divzerol(1);
204 }
205 /* log ( x < 0 ) = NaN */
206 if (m & 0x80000000)
207 {
208 return __math_invalidl(x);
209 }
210 /* log (infinity or NaN) */
211 if (k >= 0x7fff0000)
212 {
213 return x + x;
214 }
215
216 /* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */
217 e = (int) (m >> 16);
218
219 if (e == 0) {
220 /* Handle denorm */
221 e = 1;
222 while (!(m & 0x10000)) {
223 m = (m << 1) | ((int32_t) u.parts32.mswlo < 0);
224 u.parts32.mswlo = (u.parts32.mswlo << 1) | ((int32_t) u.parts32.lswhi < 0);
225 u.parts32.lswhi = (u.parts32.lswhi << 1) | ((int32_t) u.parts32.lswlo < 0);
226 u.parts32.lswlo = (u.parts32.lswlo << 1);
227 e--;
228 }
229 }
230 m &= 0xffff;
231 e -= (int) 0x3ffe;
232 u.parts32.mswhi = m | 0x3ffe0000;
233 m |= 0x10000;
234 /* Find lookup table index k from high order bits of the significand. */
235 if (m < 0x16800)
236 {
237 k = (m - 0xff00) >> 9;
238 /* t is the argument 0.5 + (k+26)/128
239 of the nearest item to u in the lookup table. */
240 t.parts32.mswhi = 0x3fff0000 + (k << 9);
241 t.parts32.mswlo = 0;
242 t.parts32.lswhi = 0;
243 t.parts32.lswlo = 0;
244 u.parts32.mswhi += 0x10000;
245 e -= 1;
246 k += 64;
247 }
248 else
249 {
250 k = (m - 0xfe00) >> 10;
251 t.parts32.mswhi = 0x3ffe0000 + (k << 10);
252 t.parts32.mswlo = 0;
253 t.parts32.lswhi = 0;
254 t.parts32.lswlo = 0;
255 }
256 /* On this interval the table is not used due to cancellation error. */
257 if ((x <= 1.0078125L) && (x >= 0.9921875L))
258 {
259 z = x - 1.0L;
260 k = 64;
261 t.value = 1.0L;
262 e = 0;
263 }
264 else
265 {
266 /* log(u) = log( t u/t ) = log(t) + log(u/t)
267 log(t) is tabulated in the lookup table.
268 Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t.
269 cf. Cody & Waite. */
270 z = (u.value - t.value) / t.value;
271 }
272 /* Series expansion of log(1+z). */
273 w = z * z;
274 y = ((((((((((((l15 * z
275 + l14) * z
276 + l13) * z
277 + l12) * z
278 + l11) * z
279 + l10) * z
280 + l9) * z
281 + l8) * z
282 + l7) * z
283 + l6) * z
284 + l5) * z
285 + l4) * z
286 + l3) * z * w;
287 y -= 0.5L * w;
288 y += e * ln2b; /* Base 2 exponent offset times ln(2). */
289 y += z;
290 y += logtbl[k-26]; /* log(t) - (t-1) */
291 y += (t.value - 1.0L);
292 y += e * ln2a;
293 return y;
294 }
295
