1 /* logll.c
2 *
3 * Natural logarithm for 128-bit long double precision.
4 *
5 *
6 *
7 * SYNOPSIS:
8 *
9 * long double x, y, logl();
10 *
11 * y = logl( x );
12 *
13 *
14 *
15 * DESCRIPTION:
16 *
17 * Returns the base e (2.718...) logarithm of x.
18 *
19 * The argument is separated into its exponent and fractional
20 * parts. Use of a lookup table increases the speed of the routine.
21 * The program uses logarithms tabulated at intervals of 1/128 to
22 * cover the domain from approximately 0.7 to 1.4.
23 *
24 * On the interval [-1/128, +1/128] the logarithm of 1+x is approximated by
25 * log(1+x) = x - 0.5 x^2 + x^3 P(x) .
26 *
27 *
28 *
29 * ACCURACY:
30 *
31 * Relative error:
32 * arithmetic domain # trials peak rms
33 * IEEE 0.875, 1.125 100000 1.2e-34 4.1e-35
34 * IEEE 0.125, 8 100000 1.2e-34 4.1e-35
35 *
36 *
37 * WARNING:
38 *
39 * This program uses integer operations on bit fields of floating-point
40 * numbers. It does not work with data structures other than the
41 * structure assumed.
42 *
43 */
44
45 /* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
46
47 This library is free software; you can redistribute it and/or
48 modify it under the terms of the GNU Lesser General Public
49 License as published by the Free Software Foundation; either
50 version 2.1 of the License, or (at your option) any later version.
51
52 This library is distributed in the hope that it will be useful,
53 but WITHOUT ANY WARRANTY; without even the implied warranty of
54 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
55 Lesser General Public License for more details.
56
57 You should have received a copy of the GNU Lesser General Public
58 License along with this library; if not, see
59 <https://www.gnu.org/licenses/>. */
60
61 #include <math.h>
62 #include <math_private.h>
63 #include <libm-alias-finite.h>
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 _Float128
69 l3 = L(3.333333333333333333333333333333336096926E-1),
70 l4 = L(-2.499999999999999999999999999486853077002E-1),
71 l5 = L(1.999999999999999999999999998515277861905E-1),
72 l6 = L(-1.666666666666666666666798448356171665678E-1),
73 l7 = L(1.428571428571428571428808945895490721564E-1),
74 l8 = L(-1.249999999999999987884655626377588149000E-1),
75 l9 = L(1.111111111111111093947834982832456459186E-1),
76 l10 = L(-1.000000000000532974938900317952530453248E-1),
77 l11 = L(9.090909090915566247008015301349979892689E-2),
78 l12 = L(-8.333333211818065121250921925397567745734E-2),
79 l13 = L(7.692307559897661630807048686258659316091E-2),
80 l14 = L(-7.144242754190814657241902218399056829264E-2),
81 l15 = L(6.668057591071739754844678883223432347481E-2);
82
83 /* Lookup table of ln(t) - (t-1)
84 t = 0.5 + (k+26)/128)
85 k = 0, ..., 91 */
86 static const _Float128 logtbl[92] = {
87 L(-5.5345593589352099112142921677820359632418E-2),
88 L(-5.2108257402767124761784665198737642086148E-2),
89 L(-4.8991686870576856279407775480686721935120E-2),
90 L(-4.5993270766361228596215288742353061431071E-2),
91 L(-4.3110481649613269682442058976885699556950E-2),
92 L(-4.0340872319076331310838085093194799765520E-2),
93 L(-3.7682072451780927439219005993827431503510E-2),
94 L(-3.5131785416234343803903228503274262719586E-2),
95 L(-3.2687785249045246292687241862699949178831E-2),
96 L(-3.0347913785027239068190798397055267411813E-2),
97 L(-2.8110077931525797884641940838507561326298E-2),
98 L(-2.5972247078357715036426583294246819637618E-2),
99 L(-2.3932450635346084858612873953407168217307E-2),
100 L(-2.1988775689981395152022535153795155900240E-2),
101 L(-2.0139364778244501615441044267387667496733E-2),
102 L(-1.8382413762093794819267536615342902718324E-2),
103 L(-1.6716169807550022358923589720001638093023E-2),
104 L(-1.5138929457710992616226033183958974965355E-2),
105 L(-1.3649036795397472900424896523305726435029E-2),
106 L(-1.2244881690473465543308397998034325468152E-2),
107 L(-1.0924898127200937840689817557742469105693E-2),
108 L(-9.6875626072830301572839422532631079809328E-3),
109 L(-8.5313926245226231463436209313499745894157E-3),
110 L(-7.4549452072765973384933565912143044991706E-3),
111 L(-6.4568155251217050991200599386801665681310E-3),
112 L(-5.5356355563671005131126851708522185605193E-3),
113 L(-4.6900728132525199028885749289712348829878E-3),
114 L(-3.9188291218610470766469347968659624282519E-3),
115 L(-3.2206394539524058873423550293617843896540E-3),
116 L(-2.5942708080877805657374888909297113032132E-3),
117 L(-2.0385211375711716729239156839929281289086E-3),
118 L(-1.5522183228760777967376942769773768850872E-3),
119 L(-1.1342191863606077520036253234446621373191E-3),
120 L(-7.8340854719967065861624024730268350459991E-4),
121 L(-4.9869831458030115699628274852562992756174E-4),
122 L(-2.7902661731604211834685052867305795169688E-4),
123 L(-1.2335696813916860754951146082826952093496E-4),
124 L(-3.0677461025892873184042490943581654591817E-5),
125 #define ZERO logtbl[38]
126 L(0.0000000000000000000000000000000000000000E0),
127 L(-3.0359557945051052537099938863236321874198E-5),
128 L(-1.2081346403474584914595395755316412213151E-4),
129 L(-2.7044071846562177120083903771008342059094E-4),
130 L(-4.7834133324631162897179240322783590830326E-4),
131 L(-7.4363569786340080624467487620270965403695E-4),
132 L(-1.0654639687057968333207323853366578860679E-3),
133 L(-1.4429854811877171341298062134712230604279E-3),
134 L(-1.8753781835651574193938679595797367137975E-3),
135 L(-2.3618380914922506054347222273705859653658E-3),
136 L(-2.9015787624124743013946600163375853631299E-3),
137 L(-3.4938307889254087318399313316921940859043E-3),
138 L(-4.1378413103128673800485306215154712148146E-3),
139 L(-4.8328735414488877044289435125365629849599E-3),
140 L(-5.5782063183564351739381962360253116934243E-3),
141 L(-6.3731336597098858051938306767880719015261E-3),
142 L(-7.2169643436165454612058905294782949315193E-3),
143 L(-8.1090214990427641365934846191367315083867E-3),
144 L(-9.0486422112807274112838713105168375482480E-3),
145 L(-1.0035177140880864314674126398350812606841E-2),
146 L(-1.1067990155502102718064936259435676477423E-2),
147 L(-1.2146457974158024928196575103115488672416E-2),
148 L(-1.3269969823361415906628825374158424754308E-2),
149 L(-1.4437927104692837124388550722759686270765E-2),
150 L(-1.5649743073340777659901053944852735064621E-2),
151 L(-1.6904842527181702880599758489058031645317E-2),
152 L(-1.8202661505988007336096407340750378994209E-2),
153 L(-1.9542647000370545390701192438691126552961E-2),
154 L(-2.0924256670080119637427928803038530924742E-2),
155 L(-2.2346958571309108496179613803760727786257E-2),
156 L(-2.3810230892650362330447187267648486279460E-2),
157 L(-2.5313561699385640380910474255652501521033E-2),
158 L(-2.6856448685790244233704909690165496625399E-2),
159 L(-2.8438398935154170008519274953860128449036E-2),
160 L(-3.0058928687233090922411781058956589863039E-2),
161 L(-3.1717563112854831855692484086486099896614E-2),
162 L(-3.3413836095418743219397234253475252001090E-2),
163 L(-3.5147290019036555862676702093393332533702E-2),
164 L(-3.6917475563073933027920505457688955423688E-2),
165 L(-3.8723951502862058660874073462456610731178E-2),
166 L(-4.0566284516358241168330505467000838017425E-2),
167 L(-4.2444048996543693813649967076598766917965E-2),
168 L(-4.4356826869355401653098777649745233339196E-2),
169 L(-4.6304207416957323121106944474331029996141E-2),
170 L(-4.8285787106164123613318093945035804818364E-2),
171 L(-5.0301169421838218987124461766244507342648E-2),
172 L(-5.2349964705088137924875459464622098310997E-2),
173 L(-5.4431789996103111613753440311680967840214E-2),
174 L(-5.6546268881465384189752786409400404404794E-2),
175 L(-5.8693031345788023909329239565012647817664E-2),
176 L(-6.0871713627532018185577188079210189048340E-2),
177 L(-6.3081958078862169742820420185833800925568E-2),
178 L(-6.5323413029406789694910800219643791556918E-2),
179 L(-6.7595732653791419081537811574227049288168E-2)
180 };
181
182 /* ln(2) = ln2a + ln2b with extended precision. */
183 static const _Float128
184 ln2a = L(6.93145751953125e-1),
185 ln2b = L(1.4286068203094172321214581765680755001344E-6);
186
187 _Float128
188 __ieee754_logl(_Float128 x)
189 {
190 _Float128 z, y, w;
191 ieee854_long_double_shape_type u, t;
192 unsigned int m;
193 int k, e;
194
195 u.value = x;
196 m = u.parts32.w0;
197
198 /* Check for IEEE special cases. */
199 k = m & 0x7fffffff;
200 /* log(0) = -infinity. */
201 if ((k | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0)
202 {
203 return L(-0.5) / ZERO;
204 }
205 /* log ( x < 0 ) = NaN */
206 if (m & 0x80000000)
207 {
208 return (x - x) / ZERO;
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 u.value = __frexpl (x, &e);
218 m = u.parts32.w0 & 0xffff;
219 m |= 0x10000;
220 /* Find lookup table index k from high order bits of the significand. */
221 if (m < 0x16800)
222 {
223 k = (m - 0xff00) >> 9;
224 /* t is the argument 0.5 + (k+26)/128
225 of the nearest item to u in the lookup table. */
226 t.parts32.w0 = 0x3fff0000 + (k << 9);
227 t.parts32.w1 = 0;
228 t.parts32.w2 = 0;
229 t.parts32.w3 = 0;
230 u.parts32.w0 += 0x10000;
231 e -= 1;
232 k += 64;
233 }
234 else
235 {
236 k = (m - 0xfe00) >> 10;
237 t.parts32.w0 = 0x3ffe0000 + (k << 10);
238 t.parts32.w1 = 0;
239 t.parts32.w2 = 0;
240 t.parts32.w3 = 0;
241 }
242 /* On this interval the table is not used due to cancellation error. */
243 if ((x <= L(1.0078125)) && (x >= L(0.9921875)))
244 {
245 if (x == 1)
246 return 0;
247 z = x - 1;
248 k = 64;
249 t.value = 1;
250 e = 0;
251 }
252 else
253 {
254 /* log(u) = log( t u/t ) = log(t) + log(u/t)
255 log(t) is tabulated in the lookup table.
256 Express log(u/t) = log(1+z), where z = u/t - 1 = (u-t)/t.
257 cf. Cody & Waite. */
258 z = (u.value - t.value) / t.value;
259 }
260 /* Series expansion of log(1+z). */
261 w = z * z;
262 y = ((((((((((((l15 * z
263 + l14) * z
264 + l13) * z
265 + l12) * z
266 + l11) * z
267 + l10) * z
268 + l9) * z
269 + l8) * z
270 + l7) * z
271 + l6) * z
272 + l5) * z
273 + l4) * z
274 + l3) * z * w;
275 y -= 0.5 * w;
276 y += e * ln2b; /* Base 2 exponent offset times ln(2). */
277 y += z;
278 y += logtbl[k-26]; /* log(t) - (t-1) */
279 y += (t.value - 1);
280 y += e * ln2a;
281 return y;
282 }
283 libm_alias_finite (__ieee754_logl, __logl)