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Cygwin: mmap: allow remapping part of an existing anonymous mapping
[newlib-cygwin.git] / newlib / libm / ld80 / s_logl.c
blob01e541b497380a7bc157e6b4c2665bf097888de7
1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
3 *
4 * Copyright (c) 2007-2013 Bruce D. Evans
5 * 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 unmodified, this list of conditions, and the following
12 * disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in the
15 * documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 */
28
29 #include <sys/cdefs.h>
30 __FBSDID("$FreeBSD$");
31
32 /**
33 * Implementation of the natural logarithm of x for Intel 80-bit format.
34 *
35 * First decompose x into its base 2 representation:
36 *
37 * log(x) = log(X * 2**k), where X is in [1, 2)
38 * = log(X) + k * log(2).
39 *
40 * Let X = X_i + e, where X_i is the center of one of the intervals
41 * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256)
42 * and X is in this interval. Then
43 *
44 * log(X) = log(X_i + e)
45 * = log(X_i * (1 + e / X_i))
46 * = log(X_i) + log(1 + e / X_i).
47 *
48 * The values log(X_i) are tabulated below. Let d = e / X_i and use
49 *
50 * log(1 + d) = p(d)
51 *
52 * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of
53 * suitably high degree.
54 *
55 * To get sufficiently small roundoff errors, k * log(2), log(X_i), and
56 * sometimes (if |k| is not large) the first term in p(d) must be evaluated
57 * and added up in extra precision. Extra precision is not needed for the
58 * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final
59 * error is controlled mainly by the error in the second term in p(d). The
60 * error in this term itself is at most 0.5 ulps from the d*d operation in
61 * it. The error in this term relative to the first term is thus at most
62 * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of
63 * at most twice this at the point of the final rounding step. Thus the
64 * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive
65 * testing of a float variant of this function showed a maximum final error
66 * of 0.5008 ulps. Non-exhaustive testing of a double variant of this
67 * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256).
68 *
69 * We made the maximum of |d| (and thus the total relative error and the
70 * degree of p(d)) small by using a large number of intervals. Using
71 * centers of intervals instead of endpoints reduces this maximum by a
72 * factor of 2 for a given number of intervals. p(d) is special only
73 * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen
74 * naturally. The most accurate minimax polynomial of a given degree might
75 * be different, but then we wouldn't want it since we would have to do
76 * extra work to avoid roundoff error (especially for P0*d instead of d).
77 */
78
79 #ifdef DEBUG
80 #include <assert.h>
81 #include <fenv.h>
82 #endif
83
84 #ifdef __i386__
85 #include <ieeefp.h>
86 #endif
87
88 #include "../ld/fpmath.h"
89 #include "math.h"
90 #define i386_SSE_GOOD
91 #ifndef NO_STRUCT_RETURN
92 #define STRUCT_RETURN
93 #endif
94 #include "../ld/math_private.h"
95
96 #if !defined(NO_UTAB) && !defined(NO_UTABL)
97 #define USE_UTAB
98 #endif
99
100 /*
101 * Domain [-0.005280, 0.004838], range ~[-5.1736e-22, 5.1738e-22]:
102 * |log(1 + d)/d - p(d)| < 2**-70.7
103 */
104 static const double
105 P2 = -0.5,
106 P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */
107 P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */
108 P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */
109 P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */
110 P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */
111 P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */
112
113 static volatile const double zero = 0;
114
115 #define INTERVALS 128
116 #define LOG2_INTERVALS 7
117 #define TSIZE (INTERVALS + 1)
118 #define G(i) (T[(i)].G)
119 #define F_hi(i) (T[(i)].F_hi)
120 #define F_lo(i) (T[(i)].F_lo)
121 #define ln2_hi F_hi(TSIZE - 1)
122 #define ln2_lo F_lo(TSIZE - 1)
123 #define E(i) (U[(i)].E)
124 #define H(i) (U[(i)].H)
125
126 static const struct {
127 float G; /* 1/(1 + i/128) rounded to 8/9 bits */
128 float F_hi; /* log(1 / G_i) rounded (see below) */
129 double F_lo; /* next 53 bits for log(1 / G_i) */
130 } T[TSIZE] = {
131 /*
132 * ln2_hi and each F_hi(i) are rounded to a number of bits that
133 * makes F_hi(i) + dk*ln2_hi exact for all i and all dk.
134 *
135 * The last entry (for X just below 2) is used to define ln2_hi
136 * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly
137 * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1.
138 * This is needed for accuracy when x is just below 1. (To avoid
139 * special cases, such x are "reduced" strangely to X just below
140 * 2 and dk = -1, and then the exact cancellation is needed
141 * because any the error from any non-exactness would be too
142 * large).
143 *
144 * We want to share this table between double precision and ld80,
145 * so the relevant range of dk is the larger one of ld80
146 * ([-16445, 16383]) and the relevant exactness requirement is
147 * the stricter one of double precision. The maximum number of
148 * bits in F_hi(i) that works is very dependent on i but has
149 * a minimum of 33. We only need about 12 bits in F_hi(i) for
150 * it to provide enough extra precision in double precision (11
151 * more than that are required for ld80).
152 *
153 * We round F_hi(i) to 24 bits so that it can have type float,
154 * mainly to minimize the size of the table. Using all 24 bits
155 * in a float for it automatically satisfies the above constraints.
156 */
157 { 0x800000.0p-23, 0, 0 },
158 { 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6675.0p-84 },
159 { 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84 },
160 { 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83 },
161 { 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82 },
162 { 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82 },
163 { 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83 },
164 { 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82 },
165 { 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91 },
166 { 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81 },
167 { 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82 },
168 { 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83 },
169 { 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81 },
170 { 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83 },
171 { 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85 },
172 { 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84 },
173 { 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81 },
174 { 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82 },
175 { 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80 },
176 { 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83 },
177 { 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82 },
178 { 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80 },
179 { 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82 },
180 { 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81 },
181 { 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84 },
182 { 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80 },
183 { 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81 },
184 { 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80 },
185 { 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82 },
186 { 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81 },
187 { 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80 },
188 { 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81 },
189 { 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81 },
190 { 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85 },
191 { 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87 },
192 { 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81 },
193 { 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80 },
194 { 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79 },
195 { 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79 },
196 { 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81 },
197 { 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79 },
198 { 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81 },
199 { 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79 },
200 { 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80 },
201 { 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81 },
202 { 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79 },
203 { 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79 },
204 { 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81 },
205 { 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81 },
206 { 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82 },
207 { 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b2.0p-80 },
208 { 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a424234.0p-79 },
209 { 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a4.0p-83 },
210 { 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770634.0p-79 },
211 { 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b152.0p-82 },
212 { 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f09.0p-80 },
213 { 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad89.0p-79 },
214 { 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf.0p-79 },
215 { 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab90486409.0p-80 },
216 { 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333.0p-79 },
217 { 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fc0.0p-80 },
218 { 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c9.0p-80 },
219 { 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8.0p-79 },
220 { 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a87.0p-81 },
221 { 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3cb.0p-79 },
222 { 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d.0p-81 },
223 { 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac.0p-81 },
224 { 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549e.0p-79 },
225 { 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b61.0p-80 },
226 { 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a3.0p-80 },
227 { 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f1.0p-82 },
228 { 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b68.0p-80 },
229 { 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f57.0p-80 },
230 { 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d9.0p-80 },
231 { 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d4.0p-80 },
232 { 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd.0p-79 },
233 { 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f730190.0p-79 },
234 { 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cd.0p-80 },
235 { 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d.0p-81 },
236 { 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af2.0p-79 },
237 { 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e0.0p-84 },
238 { 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade0.0p-79 },
239 { 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1.0p-79 },
240 { 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c53.0p-79 },
241 { 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f.0p-78 },
242 { 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e5.0p-81 },
243 { 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79 },
244 { 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78 },
245 { 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78 },
246 { 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79 },
247 { 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80 },
248 { 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78 },
249 { 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78 },
250 { 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79 },
251 { 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79 },
252 { 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78 },
253 { 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80 },
254 { 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79 },
255 { 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79 },
256 { 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79 },
257 { 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79 },
258 { 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78 },
259 { 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79 },
260 { 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78 },
261 { 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79 },
262 { 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78 },
263 { 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78 },
264 { 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78 },
265 { 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78 },
266 { 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79 },
267 { 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79 },
268 { 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81 },
269 { 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79 },
270 { 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78 },
271 { 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79 },
272 { 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79 },
273 { 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78 },
274 { 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80 },
275 { 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80 },
276 { 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79 },
277 { 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79 },
278 { 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80 },
279 { 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79 },
280 { 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80 },
281 { 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80 },
282 { 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81 },
283 { 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78 },
284 { 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78 },
285 { 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81 },
286 };
287
288 #ifdef USE_UTAB
289 static const struct {
290 float H; /* 1 + i/INTERVALS (exact) */
291 float E; /* H(i) * G(i) - 1 (exact) */
292 } U[TSIZE] = {
293 { 0x800000.0p-23, 0 },
294 { 0x810000.0p-23, -0x800000.0p-37 },
295 { 0x820000.0p-23, -0x800000.0p-35 },
296 { 0x830000.0p-23, -0x900000.0p-34 },
297 { 0x840000.0p-23, -0x800000.0p-33 },
298 { 0x850000.0p-23, -0xc80000.0p-33 },
299 { 0x860000.0p-23, -0xa00000.0p-36 },
300 { 0x870000.0p-23, 0x940000.0p-33 },
301 { 0x880000.0p-23, 0x800000.0p-35 },
302 { 0x890000.0p-23, -0xc80000.0p-34 },
303 { 0x8a0000.0p-23, 0xe00000.0p-36 },
304 { 0x8b0000.0p-23, 0x900000.0p-33 },
305 { 0x8c0000.0p-23, -0x800000.0p-35 },
306 { 0x8d0000.0p-23, -0xe00000.0p-33 },
307 { 0x8e0000.0p-23, 0x880000.0p-33 },
308 { 0x8f0000.0p-23, -0xa80000.0p-34 },
309 { 0x900000.0p-23, -0x800000.0p-35 },
310 { 0x910000.0p-23, 0x800000.0p-37 },
311 { 0x920000.0p-23, 0x900000.0p-35 },
312 { 0x930000.0p-23, 0xd00000.0p-35 },
313 { 0x940000.0p-23, 0xe00000.0p-35 },
314 { 0x950000.0p-23, 0xc00000.0p-35 },
315 { 0x960000.0p-23, 0xe00000.0p-36 },
316 { 0x970000.0p-23, -0x800000.0p-38 },
317 { 0x980000.0p-23, -0xc00000.0p-35 },
318 { 0x990000.0p-23, -0xd00000.0p-34 },
319 { 0x9a0000.0p-23, 0x880000.0p-33 },
320 { 0x9b0000.0p-23, 0xe80000.0p-35 },
321 { 0x9c0000.0p-23, -0x800000.0p-35 },
322 { 0x9d0000.0p-23, 0xb40000.0p-33 },
323 { 0x9e0000.0p-23, 0x880000.0p-34 },
324 { 0x9f0000.0p-23, -0xe00000.0p-35 },
325 { 0xa00000.0p-23, 0x800000.0p-33 },
326 { 0xa10000.0p-23, -0x900000.0p-36 },
327 { 0xa20000.0p-23, -0xb00000.0p-33 },
328 { 0xa30000.0p-23, -0xa00000.0p-36 },
329 { 0xa40000.0p-23, 0x800000.0p-33 },
330 { 0xa50000.0p-23, -0xf80000.0p-35 },
331 { 0xa60000.0p-23, 0x880000.0p-34 },
332 { 0xa70000.0p-23, -0x900000.0p-33 },
333 { 0xa80000.0p-23, -0x800000.0p-35 },
334 { 0xa90000.0p-23, 0x900000.0p-34 },
335 { 0xaa0000.0p-23, 0xa80000.0p-33 },
336 { 0xab0000.0p-23, -0xac0000.0p-34 },
337 { 0xac0000.0p-23, -0x800000.0p-37 },
338 { 0xad0000.0p-23, 0xf80000.0p-35 },
339 { 0xae0000.0p-23, 0xf80000.0p-34 },
340 { 0xaf0000.0p-23, -0xac0000.0p-33 },
341 { 0xb00000.0p-23, -0x800000.0p-33 },
342 { 0xb10000.0p-23, -0xb80000.0p-34 },
343 { 0xb20000.0p-23, -0x800000.0p-34 },
344 { 0xb30000.0p-23, -0xb00000.0p-35 },
345 { 0xb40000.0p-23, -0x800000.0p-35 },
346 { 0xb50000.0p-23, -0xe00000.0p-36 },
347 { 0xb60000.0p-23, -0x800000.0p-35 },
348 { 0xb70000.0p-23, -0xb00000.0p-35 },
349 { 0xb80000.0p-23, -0x800000.0p-34 },
350 { 0xb90000.0p-23, -0xb80000.0p-34 },
351 { 0xba0000.0p-23, -0x800000.0p-33 },
352 { 0xbb0000.0p-23, -0xac0000.0p-33 },
353 { 0xbc0000.0p-23, 0x980000.0p-33 },
354 { 0xbd0000.0p-23, 0xbc0000.0p-34 },
355 { 0xbe0000.0p-23, 0xe00000.0p-36 },
356 { 0xbf0000.0p-23, -0xb80000.0p-35 },
357 { 0xc00000.0p-23, -0x800000.0p-33 },
358 { 0xc10000.0p-23, 0xa80000.0p-33 },
359 { 0xc20000.0p-23, 0x900000.0p-34 },
360 { 0xc30000.0p-23, -0x800000.0p-35 },
361 { 0xc40000.0p-23, -0x900000.0p-33 },
362 { 0xc50000.0p-23, 0x820000.0p-33 },
363 { 0xc60000.0p-23, 0x800000.0p-38 },
364 { 0xc70000.0p-23, -0x820000.0p-33 },
365 { 0xc80000.0p-23, 0x800000.0p-33 },
366 { 0xc90000.0p-23, -0xa00000.0p-36 },
367 { 0xca0000.0p-23, -0xb00000.0p-33 },
368 { 0xcb0000.0p-23, 0x840000.0p-34 },
369 { 0xcc0000.0p-23, -0xd00000.0p-34 },
370 { 0xcd0000.0p-23, 0x800000.0p-33 },
371 { 0xce0000.0p-23, -0xe00000.0p-35 },
372 { 0xcf0000.0p-23, 0xa60000.0p-33 },
373 { 0xd00000.0p-23, -0x800000.0p-35 },
374 { 0xd10000.0p-23, 0xb40000.0p-33 },
375 { 0xd20000.0p-23, -0x800000.0p-35 },
376 { 0xd30000.0p-23, 0xaa0000.0p-33 },
377 { 0xd40000.0p-23, -0xe00000.0p-35 },
378 { 0xd50000.0p-23, 0x880000.0p-33 },
379 { 0xd60000.0p-23, -0xd00000.0p-34 },
380 { 0xd70000.0p-23, 0x9c0000.0p-34 },
381 { 0xd80000.0p-23, -0xb00000.0p-33 },
382 { 0xd90000.0p-23, -0x800000.0p-38 },
383 { 0xda0000.0p-23, 0xa40000.0p-33 },
384 { 0xdb0000.0p-23, -0xdc0000.0p-34 },
385 { 0xdc0000.0p-23, 0xc00000.0p-35 },
386 { 0xdd0000.0p-23, 0xca0000.0p-33 },
387 { 0xde0000.0p-23, -0xb80000.0p-34 },
388 { 0xdf0000.0p-23, 0xd00000.0p-35 },
389 { 0xe00000.0p-23, 0xc00000.0p-33 },
390 { 0xe10000.0p-23, -0xf40000.0p-34 },
391 { 0xe20000.0p-23, 0x800000.0p-37 },
392 { 0xe30000.0p-23, 0x860000.0p-33 },
393 { 0xe40000.0p-23, -0xc80000.0p-33 },
394 { 0xe50000.0p-23, -0xa80000.0p-34 },
395 { 0xe60000.0p-23, 0xe00000.0p-36 },
396 { 0xe70000.0p-23, 0x880000.0p-33 },
397 { 0xe80000.0p-23, -0xe00000.0p-33 },
398 { 0xe90000.0p-23, -0xfc0000.0p-34 },
399 { 0xea0000.0p-23, -0x800000.0p-35 },
400 { 0xeb0000.0p-23, 0xe80000.0p-35 },
401 { 0xec0000.0p-23, 0x900000.0p-33 },
402 { 0xed0000.0p-23, 0xe20000.0p-33 },
403 { 0xee0000.0p-23, -0xac0000.0p-33 },
404 { 0xef0000.0p-23, -0xc80000.0p-34 },
405 { 0xf00000.0p-23, -0x800000.0p-35 },
406 { 0xf10000.0p-23, 0x800000.0p-35 },
407 { 0xf20000.0p-23, 0xb80000.0p-34 },
408 { 0xf30000.0p-23, 0x940000.0p-33 },
409 { 0xf40000.0p-23, 0xc80000.0p-33 },
410 { 0xf50000.0p-23, -0xf20000.0p-33 },
411 { 0xf60000.0p-23, -0xc80000.0p-33 },
412 { 0xf70000.0p-23, -0xa20000.0p-33 },
413 { 0xf80000.0p-23, -0x800000.0p-33 },
414 { 0xf90000.0p-23, -0xc40000.0p-34 },
415 { 0xfa0000.0p-23, -0x900000.0p-34 },
416 { 0xfb0000.0p-23, -0xc80000.0p-35 },
417 { 0xfc0000.0p-23, -0x800000.0p-35 },
418 { 0xfd0000.0p-23, -0x900000.0p-36 },
419 { 0xfe0000.0p-23, -0x800000.0p-37 },
420 { 0xff0000.0p-23, -0x800000.0p-39 },
421 { 0x800000.0p-22, 0 },
422 };
423 #endif /* USE_UTAB */
424
425 #ifdef STRUCT_RETURN
426 #define RETURN1(rp, v) do { \
427 (rp)->hi = (v); \
428 (rp)->lo_set = 0; \
429 return; \
430 } while (0)
431
432 #define RETURN2(rp, h, l) do { \
433 (rp)->hi = (h); \
434 (rp)->lo = (l); \
435 (rp)->lo_set = 1; \
436 return; \
437 } while (0)
438
439 struct ld {
440 long double hi;
441 long double lo;
442 int lo_set;
443 };
444 #else
445 #define RETURN1(rp, v) RETURNF(v)
446 #define RETURN2(rp, h, l) RETURNI((h) + (l))
447 #endif
448
449 #ifdef STRUCT_RETURN
450 static inline __always_inline void
451 k_logl(long double x, struct ld *rp)
452 #else
453 long double
454 logl(long double x)
455 #endif
456 {
457 long double d, dk, val_hi, val_lo, z;
458 uint64_t ix, lx;
459 int i, k;
460 uint16_t hx;
461
462 EXTRACT_LDBL80_WORDS(hx, lx, x);
463 k = -16383;
464 #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */
465 if (x == 1)
466 RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */
467 #endif
468 if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */
469 if (((hx & 0x7fff) | lx) == 0)
470 RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */
471 if (hx != 0)
472 /* log(neg or [pseudo-]NaN) = qNaN: */
473 RETURN1(rp, (x - x) / zero);
474 x *= 0x1.0p65; /* subnormal; scale up x */
475 /* including pseudo-subnormals */
476 EXTRACT_LDBL80_WORDS(hx, lx, x);
477 k = -16383 - 65;
478 } else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0)
479 RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */
480 /* log(pseudo-Inf) = qNaN */
481 /* log(pseudo-NaN) = qNaN */
482 /* log(unnormal) = qNaN */
483 #ifndef STRUCT_RETURN
484 ENTERI();
485 #endif
486 k += hx;
487 ix = lx & 0x7fffffffffffffffULL;
488 dk = k;
489
490 /* Scale x to be in [1, 2). */
491 SET_LDBL_EXPSIGN(x, 0x3fff);
492
493 /* 0 <= i <= INTERVALS: */
494 #define L2I (64 - LOG2_INTERVALS)
495 i = (ix + (1LL << (L2I - 2))) >> (L2I - 1);
496
497 /*
498 * -0.005280 < d < 0.004838. In particular, the infinite-
499 * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits
500 * ensures that d is representable without extra precision for
501 * this bound on |d| (since when this calculation is expressed
502 * as x*G(i)-1, the multiplication needs as many extra bits as
503 * G(i) has and the subtraction cancels 8 bits). But for
504 * most i (107 cases out of 129), the infinite-precision |d|
505 * is <= 2**-8. G(i) is rounded to 9 bits for such i to give
506 * better accuracy (this works by improving the bound on |d|,
507 * which in turn allows rounding to 9 bits in more cases).
508 * This is only important when the original x is near 1 -- it
509 * lets us avoid using a special method to give the desired
510 * accuracy for such x.
511 */
512 if (0)
513 d = x * G(i) - 1;
514 else {
515 #ifdef USE_UTAB
516 d = (x - H(i)) * G(i) + E(i);
517 #else
518 long double x_hi, x_lo;
519 float fx_hi;
520
521 /*
522 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly.
523 * G(i) has at most 9 bits, so the splitting point is not
524 * critical.
525 */
526 SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
527 x_hi = fx_hi;
528 x_lo = x - x_hi;
529 d = x_hi * G(i) - 1 + x_lo * G(i);
530 #endif
531 }
532
533 /*
534 * Our algorithm depends on exact cancellation of F_lo(i) and
535 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is
536 * at the end of the table. This and other technical complications
537 * make it difficult to avoid the double scaling in (dk*ln2) *
538 * log(base) for base != e without losing more accuracy and/or
539 * efficiency than is gained.
540 */
541 z = d * d;
542 val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
543 (F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * P2;
544 val_hi = d;
545 #ifdef DEBUG
546 if (fetestexcept(FE_UNDERFLOW))
547 breakpoint();
548 #endif
549
550 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
551 RETURN2(rp, val_hi, val_lo);
552 }
553
554 long double
555 log1pl(long double x)
556 {
557 long double d, d_hi, d_lo, dk, f_lo, val_hi, val_lo, z;
558 long double f_hi, twopminusk;
559 uint64_t ix, lx;
560 int i, k;
561 int16_t ax, hx;
562
563 DOPRINT_START(&x);
564 EXTRACT_LDBL80_WORDS(hx, lx, x);
565 if (hx < 0x3fff) { /* x < 1, or x neg NaN */
566 ax = hx & 0x7fff;
567 if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */
568 if (ax == 0x3fff && lx == 0x8000000000000000ULL)
569 RETURNP(-1 / zero); /* log1p(-1) = -Inf */
570 /* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */
571 RETURNP((x - x) / (x - x));
572 }
573 if (ax <= 0x3fbe) { /* |x| < 2**-64 */
574 if ((int)x == 0)
575 RETURNP(x); /* x with inexact if x != 0 */
576 }
577 f_hi = 1;
578 f_lo = x;
579 } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */
580 RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */
581 /* log1p(pseudo-Inf) = qNaN */
582 /* log1p(pseudo-NaN) = qNaN */
583 /* log1p(unnormal) = qNaN */
584 } else if (hx < 0x407f) { /* 1 <= x < 2**128 */
585 f_hi = x;
586 f_lo = 1;
587 } else { /* 2**128 <= x < +Inf */
588 f_hi = x;
589 f_lo = 0; /* avoid underflow of the P5 term */
590 }
591 ENTERI();
592 x = f_hi + f_lo;
593 f_lo = (f_hi - x) + f_lo;
594
595 EXTRACT_LDBL80_WORDS(hx, lx, x);
596 k = -16383;
597
598 k += hx;
599 ix = lx & 0x7fffffffffffffffULL;
600 dk = k;
601
602 SET_LDBL_EXPSIGN(x, 0x3fff);
603 twopminusk = 1;
604 SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
605 f_lo *= twopminusk;
606
607 i = (ix + (1LL << (L2I - 2))) >> (L2I - 1);
608
609 /*
610 * x*G(i)-1 (with a reduced x) can be represented exactly, as
611 * above, but now we need to evaluate the polynomial on d =
612 * (x+f_lo)*G(i)-1 and extra precision is needed for that.
613 * Since x+x_lo is a hi+lo decomposition and subtracting 1
614 * doesn't lose too many bits, an inexact calculation for
615 * f_lo*G(i) is good enough.
616 */
617 if (0)
618 d_hi = x * G(i) - 1;
619 else {
620 #ifdef USE_UTAB
621 d_hi = (x - H(i)) * G(i) + E(i);
622 #else
623 long double x_hi, x_lo;
624 float fx_hi;
625
626 SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000);
627 x_hi = fx_hi;
628 x_lo = x - x_hi;
629 d_hi = x_hi * G(i) - 1 + x_lo * G(i);
630 #endif
631 }
632 d_lo = f_lo * G(i);
633
634 /*
635 * This is _2sumF(d_hi, d_lo) inlined. The condition
636 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not
637 * always satisifed, so it is not clear that this works, but
638 * it works in practice. It works even if it gives a wrong
639 * normalized d_lo, since |d_lo| > |d_hi| implies that i is
640 * nonzero and d is tiny, so the F(i) term dominates d_lo.
641 * In float precision:
642 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25.
643 * And if d is only a little tinier than that, we would have
644 * another underflow problem for the P3 term; this is also ruled
645 * out by exhaustive testing.)
646 */
647 d = d_hi + d_lo;
648 d_lo = d_hi - d + d_lo;
649 d_hi = d;
650
651 z = d * d;
652 val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) +
653 (F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * P2;
654 val_hi = d_hi;
655 #ifdef DEBUG
656 if (fetestexcept(FE_UNDERFLOW))
657 breakpoint();
658 #endif
659
660 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
661 RETURN2PI(val_hi, val_lo);
662 }
663
664 #ifdef STRUCT_RETURN
665
666 long double
667 logl(long double x)
668 {
669 struct ld r;
670
671 ENTERI();
672 DOPRINT_START(&x);
673 k_logl(x, &r);
674 RETURNSPI(&r);
675 }
676
677 /* Use macros since GCC < 8 rejects static const expressions in initializers. */
678 #define invln10_hi 4.3429448190317999e-1 /* 0x1bcb7b1526e000.0p-54 */
679 #define invln10_lo 7.1842412889749798e-14 /* 0x1438ca9aadd558.0p-96 */
680 #define invln2_hi 1.4426950408887933e0 /* 0x171547652b8000.0p-52 */
681 #define invln2_lo 1.7010652264631490e-13 /* 0x17f0bbbe87fed0.0p-95 */
682 /* Let the compiler pre-calculate this sum to avoid FE_INEXACT at run time. */
683 static const double invln10_lo_plus_hi = invln10_lo + invln10_hi;
684 static const double invln2_lo_plus_hi = invln2_lo + invln2_hi;
685
686 long double
687 log10l(long double x)
688 {
689 struct ld r;
690 long double hi, lo;
691
692 ENTERI();
693 DOPRINT_START(&x);
694 k_logl(x, &r);
695 if (!r.lo_set)
696 RETURNPI(r.hi);
697 _2sumF(r.hi, r.lo);
698 hi = (float)r.hi;
699 lo = r.lo + (r.hi - hi);
700 RETURN2PI(invln10_hi * hi,
701 invln10_lo_plus_hi * lo + invln10_lo * hi);
702 }
703
704 long double
705 log2l(long double x)
706 {
707 struct ld r;
708 long double hi, lo;
709
710 ENTERI();
711 DOPRINT_START(&x);
712 k_logl(x, &r);
713 if (!r.lo_set)
714 RETURNPI(r.hi);
715 _2sumF(r.hi, r.lo);
716 hi = (float)r.hi;
717 lo = r.lo + (r.hi - hi);
718 RETURN2PI(invln2_hi * hi,
719 invln2_lo_plus_hi * lo + invln2_lo * hi);
720 }
721
722 #endif /* STRUCT_RETURN */