1/*-
2 * SPDX-License-Identifier: BSD-3-Clause
3 *
4 * Copyright (c) 1992, 1993
5 *	The Regents of the University of California.  All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 *    notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 *    notice, this list of conditions and the following disclaimer in the
14 *    documentation and/or other materials provided with the distribution.
15 * 3. Neither the name of the University nor the names of its contributors
16 *    may be used to endorse or promote products derived from this software
17 *    without specific prior written permission.
18 *
19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
22 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
29 * SUCH DAMAGE.
30 */
31
32/*
33 * See bsdsrc/b_log.c for implementation details.
34 *
35 * bsdrc/b_log.c converted to long double by Steven G. Kargl.
36 */
37
38#define N 128
39
40/*
41 * Coefficients in the polynomial approximation of log(1+f/F).
42 * Domain of x is [0,1./256] with 2**(-84.48) precision.
43 */
44static const union ieee_ext_u
45    a1u = LD80C(0xaaaaaaaaaaaaaaab,    -4,  8.33333333333333333356e-02L),
46    a2u = LD80C(0xcccccccccccccd29,    -7,  1.25000000000000000781e-02L),
47    a3u = LD80C(0x9249249241ed3764,    -9,  2.23214285711721994134e-03L),
48    a4u = LD80C(0xe38e959e1e7e01cf,   -12,  4.34030476540000360640e-04L);
49#define	A1	(a1u.extu_ld)
50#define	A2	(a2u.extu_ld)
51#define	A3	(a3u.extu_ld)
52#define	A4	(a4u.extu_ld)
53
54/*
55 * Table of log(Fj) = logF_head[j] + logF_tail[j], for Fj = 1+j/128.
56 * Used for generation of extend precision logarithms.
57 * The constant 35184372088832 is 2^45, so the divide is exact.
58 * It ensures correct reading of logF_head, even for inaccurate
59 * decimal-to-binary conversion routines.  (Everybody gets the
60 * right answer for integers less than 2^53.)
61 * Values for log(F) were generated using error < 10^-57 absolute
62 * with the bc -l package.
63 */
64
65static double logF_head[N+1] = {
66	0.,
67	.007782140442060381246,
68	.015504186535963526694,
69	.023167059281547608406,
70	.030771658666765233647,
71	.038318864302141264488,
72	.045809536031242714670,
73	.053244514518837604555,
74	.060624621816486978786,
75	.067950661908525944454,
76	.075223421237524235039,
77	.082443669210988446138,
78	.089612158689760690322,
79	.096729626458454731618,
80	.103796793681567578460,
81	.110814366340264314203,
82	.117783035656430001836,
83	.124703478501032805070,
84	.131576357788617315236,
85	.138402322859292326029,
86	.145182009844575077295,
87	.151916042025732167530,
88	.158605030176659056451,
89	.165249572895390883786,
90	.171850256926518341060,
91	.178407657472689606947,
92	.184922338493834104156,
93	.191394852999565046047,
94	.197825743329758552135,
95	.204215541428766300668,
96	.210564769107350002741,
97	.216873938300523150246,
98	.223143551314024080056,
99	.229374101064877322642,
100	.235566071312860003672,
101	.241719936886966024758,
102	.247836163904594286577,
103	.253915209980732470285,
104	.259957524436686071567,
105	.265963548496984003577,
106	.271933715484010463114,
107	.277868451003087102435,
108	.283768173130738432519,
109	.289633292582948342896,
110	.295464212893421063199,
111	.301261330578199704177,
112	.307025035294827830512,
113	.312755710004239517729,
114	.318453731118097493890,
115	.324119468654316733591,
116	.329753286372579168528,
117	.335355541920762334484,
118	.340926586970454081892,
119	.346466767346100823488,
120	.351976423156884266063,
121	.357455888922231679316,
122	.362905493689140712376,
123	.368325561158599157352,
124	.373716409793814818840,
125	.379078352934811846353,
126	.384411698910298582632,
127	.389716751140440464951,
128	.394993808240542421117,
129	.400243164127459749579,
130	.405465108107819105498,
131	.410659924985338875558,
132	.415827895143593195825,
133	.420969294644237379543,
134	.426084395310681429691,
135	.431173464818130014464,
136	.436236766774527495726,
137	.441274560805140936281,
138	.446287102628048160113,
139	.451274644139630254358,
140	.456237433481874177232,
141	.461175715122408291790,
142	.466089729924533457960,
143	.470979715219073113985,
144	.475845904869856894947,
145	.480688529345570714212,
146	.485507815781602403149,
147	.490303988045525329653,
148	.495077266798034543171,
149	.499827869556611403822,
150	.504556010751912253908,
151	.509261901790523552335,
152	.513945751101346104405,
153	.518607764208354637958,
154	.523248143765158602036,
155	.527867089620485785417,
156	.532464798869114019908,
157	.537041465897345915436,
158	.541597282432121573947,
159	.546132437597407260909,
160	.550647117952394182793,
161	.555141507540611200965,
162	.559615787935399566777,
163	.564070138285387656651,
164	.568504735352689749561,
165	.572919753562018740922,
166	.577315365035246941260,
167	.581691739635061821900,
168	.586049045003164792433,
169	.590387446602107957005,
170	.594707107746216934174,
171	.599008189645246602594,
172	.603290851438941899687,
173	.607555250224322662688,
174	.611801541106615331955,
175	.616029877215623855590,
176	.620240409751204424537,
177	.624433288012369303032,
178	.628608659422752680256,
179	.632766669570628437213,
180	.636907462236194987781,
181	.641031179420679109171,
182	.645137961373620782978,
183	.649227946625615004450,
184	.653301272011958644725,
185	.657358072709030238911,
186	.661398482245203922502,
187	.665422632544505177065,
188	.669430653942981734871,
189	.673422675212350441142,
190	.677398823590920073911,
191	.681359224807238206267,
192	.685304003098281100392,
193	.689233281238557538017,
194	.693147180560117703862
195};
196
197static double logF_tail[N+1] = {
198	0.,
199	-.00000000000000543229938420049,
200	 .00000000000000172745674997061,
201	-.00000000000001323017818229233,
202	-.00000000000001154527628289872,
203	-.00000000000000466529469958300,
204	 .00000000000005148849572685810,
205	-.00000000000002532168943117445,
206	-.00000000000005213620639136504,
207	-.00000000000001819506003016881,
208	 .00000000000006329065958724544,
209	 .00000000000008614512936087814,
210	-.00000000000007355770219435028,
211	 .00000000000009638067658552277,
212	 .00000000000007598636597194141,
213	 .00000000000002579999128306990,
214	-.00000000000004654729747598444,
215	-.00000000000007556920687451336,
216	 .00000000000010195735223708472,
217	-.00000000000017319034406422306,
218	-.00000000000007718001336828098,
219	 .00000000000010980754099855238,
220	-.00000000000002047235780046195,
221	-.00000000000008372091099235912,
222	 .00000000000014088127937111135,
223	 .00000000000012869017157588257,
224	 .00000000000017788850778198106,
225	 .00000000000006440856150696891,
226	 .00000000000016132822667240822,
227	-.00000000000007540916511956188,
228	-.00000000000000036507188831790,
229	 .00000000000009120937249914984,
230	 .00000000000018567570959796010,
231	-.00000000000003149265065191483,
232	-.00000000000009309459495196889,
233	 .00000000000017914338601329117,
234	-.00000000000001302979717330866,
235	 .00000000000023097385217586939,
236	 .00000000000023999540484211737,
237	 .00000000000015393776174455408,
238	-.00000000000036870428315837678,
239	 .00000000000036920375082080089,
240	-.00000000000009383417223663699,
241	 .00000000000009433398189512690,
242	 .00000000000041481318704258568,
243	-.00000000000003792316480209314,
244	 .00000000000008403156304792424,
245	-.00000000000034262934348285429,
246	 .00000000000043712191957429145,
247	-.00000000000010475750058776541,
248	-.00000000000011118671389559323,
249	 .00000000000037549577257259853,
250	 .00000000000013912841212197565,
251	 .00000000000010775743037572640,
252	 .00000000000029391859187648000,
253	-.00000000000042790509060060774,
254	 .00000000000022774076114039555,
255	 .00000000000010849569622967912,
256	-.00000000000023073801945705758,
257	 .00000000000015761203773969435,
258	 .00000000000003345710269544082,
259	-.00000000000041525158063436123,
260	 .00000000000032655698896907146,
261	-.00000000000044704265010452446,
262	 .00000000000034527647952039772,
263	-.00000000000007048962392109746,
264	 .00000000000011776978751369214,
265	-.00000000000010774341461609578,
266	 .00000000000021863343293215910,
267	 .00000000000024132639491333131,
268	 .00000000000039057462209830700,
269	-.00000000000026570679203560751,
270	 .00000000000037135141919592021,
271	-.00000000000017166921336082431,
272	-.00000000000028658285157914353,
273	-.00000000000023812542263446809,
274	 .00000000000006576659768580062,
275	-.00000000000028210143846181267,
276	 .00000000000010701931762114254,
277	 .00000000000018119346366441110,
278	 .00000000000009840465278232627,
279	-.00000000000033149150282752542,
280	-.00000000000018302857356041668,
281	-.00000000000016207400156744949,
282	 .00000000000048303314949553201,
283	-.00000000000071560553172382115,
284	 .00000000000088821239518571855,
285	-.00000000000030900580513238244,
286	-.00000000000061076551972851496,
287	 .00000000000035659969663347830,
288	 .00000000000035782396591276383,
289	-.00000000000046226087001544578,
290	 .00000000000062279762917225156,
291	 .00000000000072838947272065741,
292	 .00000000000026809646615211673,
293	-.00000000000010960825046059278,
294	 .00000000000002311949383800537,
295	-.00000000000058469058005299247,
296	-.00000000000002103748251144494,
297	-.00000000000023323182945587408,
298	-.00000000000042333694288141916,
299	-.00000000000043933937969737844,
300	 .00000000000041341647073835565,
301	 .00000000000006841763641591466,
302	 .00000000000047585534004430641,
303	 .00000000000083679678674757695,
304	-.00000000000085763734646658640,
305	 .00000000000021913281229340092,
306	-.00000000000062242842536431148,
307	-.00000000000010983594325438430,
308	 .00000000000065310431377633651,
309	-.00000000000047580199021710769,
310	-.00000000000037854251265457040,
311	 .00000000000040939233218678664,
312	 .00000000000087424383914858291,
313	 .00000000000025218188456842882,
314	-.00000000000003608131360422557,
315	-.00000000000050518555924280902,
316	 .00000000000078699403323355317,
317	-.00000000000067020876961949060,
318	 .00000000000016108575753932458,
319	 .00000000000058527188436251509,
320	-.00000000000035246757297904791,
321	-.00000000000018372084495629058,
322	 .00000000000088606689813494916,
323	 .00000000000066486268071468700,
324	 .00000000000063831615170646519,
325	 .00000000000025144230728376072,
326	-.00000000000017239444525614834
327};
328/*
329 * Extra precision variant, returning struct {double a, b;};
330 * log(x) = a + b to 63 bits, with 'a' rounded to 24 bits.
331 */
332static struct Double
333__log__D(long double x)
334{
335	int m, j;
336	long double F, f, g, q, u, v, u1, u2;
337	struct Double r;
338
339	/*
340	 * Argument reduction: 1 <= g < 2; x/2^m = g;
341	 * y = F*(1 + f/F) for |f| <= 2^-8
342	 */
343	g = frexpl(x, &m);
344	g *= 2;
345	m--;
346	if (m == DBL_MIN_EXP - 1) {
347		j = ilogbl(g);
348		m += j;
349		g = ldexpl(g, -j);
350	}
351	j = N * (g - 1) + 0.5L;
352	F = (1.L / N) * j + 1;
353	f = g - F;
354
355	g = 1 / (2 * F + f);
356	u = 2 * f * g;
357	v = u * u;
358	q = u * v * (A1 + v * (A2 + v * (A3 + v * A4)));
359	if (m | j) {
360		u1 = u + 513;
361		u1 -= 513;
362	} else {
363		u1 = (float)u;
364	}
365	u2 = (2 * (f - F * u1) - u1 * f) * g;
366
367	u1 += m * (long double)logF_head[N] + logF_head[j];
368
369	u2 += logF_tail[j];
370	u2 += q;
371	u2 += logF_tail[N] * m;
372	r.a = (float)(u1 + u2);		/* Only difference is here. */
373	r.b = (u1 - r.a) + u2;
374	return (r);
375}
376