1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause
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 /**
31 * Implementation of the natural logarithm of x for 128-bit format.
32 *
33 * First decompose x into its base 2 representation:
34 *
35 * log(x) = log(X * 2**k), where X is in [1, 2)
36 * = log(X) + k * log(2).
37 *
38 * Let X = X_i + e, where X_i is the center of one of the intervals
39 * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256)
40 * and X is in this interval. Then
41 *
42 * log(X) = log(X_i + e)
43 * = log(X_i * (1 + e / X_i))
44 * = log(X_i) + log(1 + e / X_i).
45 *
46 * The values log(X_i) are tabulated below. Let d = e / X_i and use
47 *
48 * log(1 + d) = p(d)
49 *
50 * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of
51 * suitably high degree.
52 *
53 * To get sufficiently small roundoff errors, k * log(2), log(X_i), and
54 * sometimes (if |k| is not large) the first term in p(d) must be evaluated
55 * and added up in extra precision. Extra precision is not needed for the
56 * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final
57 * error is controlled mainly by the error in the second term in p(d). The
58 * error in this term itself is at most 0.5 ulps from the d*d operation in
59 * it. The error in this term relative to the first term is thus at most
60 * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of
61 * at most twice this at the point of the final rounding step. Thus the
62 * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive
63 * testing of a float variant of this function showed a maximum final error
64 * of 0.5008 ulps. Non-exhaustive testing of a double variant of this
65 * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256).
66 *
67 * We made the maximum of |d| (and thus the total relative error and the
68 * degree of p(d)) small by using a large number of intervals. Using
69 * centers of intervals instead of endpoints reduces this maximum by a
70 * factor of 2 for a given number of intervals. p(d) is special only
71 * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen
72 * naturally. The most accurate minimax polynomial of a given degree might
73 * be different, but then we wouldn't want it since we would have to do
74 * extra work to avoid roundoff error (especially for P0*d instead of d).
75 */
76
77 #ifdef DEBUG
78 #include <fenv.h>
79 #endif
80
81 #include "math.h"
82 #ifndef NO_STRUCT_RETURN
83 #define STRUCT_RETURN
84 #endif
85 #include "math_private.h"
86
87 #if !defined(NO_UTAB) && !defined(NO_UTABL)
88 #define USE_UTAB
89 #endif
90
91 /*
92 * Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]:
93 * |log(1 + d)/d - p(d)| < 2**-122.7
94 */
95 static const long double
96 P2 = -0.5L,
97 P3 = 3.33333333333333333333333333333233795e-1L, /* 0x15555555555555555555555554d42.0p-114L */
98 P4 = -2.49999999999999999999999999941139296e-1L, /* -0x1ffffffffffffffffffffffdab14e.0p-115L */
99 P5 = 2.00000000000000000000000085468039943e-1L, /* 0x19999999999999999999a6d3567f4.0p-115L */
100 P6 = -1.66666666666666666666696142372698408e-1L, /* -0x15555555555555555567267a58e13.0p-115L */
101 P7 = 1.42857142857142857119522943477166120e-1L, /* 0x1249249249249248ed79a0ae434de.0p-115L */
102 P8 = -1.24999999999999994863289015033581301e-1L; /* -0x1fffffffffffffa13e91765e46140.0p-116L */
103 /* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */
104 static const double
105 P9 = 1.1111111111111401e-1, /* 0x1c71c71c71c7ed.0p-56 */
106 P10 = -1.0000000000040135e-1, /* -0x199999999a0a92.0p-56 */
107 P11 = 9.0909090728136258e-2, /* 0x1745d173962111.0p-56 */
108 P12 = -8.3333318851855284e-2, /* -0x1555551722c7a3.0p-56 */
109 P13 = 7.6928634666404178e-2, /* 0x13b1985204a4ae.0p-56 */
110 P14 = -7.1626810078462499e-2; /* -0x12562276cdc5d0.0p-56 */
111
112 static volatile const double zero = 0;
113
114 #define INTERVALS 128
115 #define LOG2_INTERVALS 7
116 #define TSIZE (INTERVALS + 1)
117 #define G(i) (T[(i)].G)
118 #define F_hi(i) (T[(i)].F_hi)
119 #define F_lo(i) (T[(i)].F_lo)
120 #define ln2_hi F_hi(TSIZE - 1)
121 #define ln2_lo F_lo(TSIZE - 1)
122 #define E(i) (U[(i)].E)
123 #define H(i) (U[(i)].H)
124
125 static const struct {
126 float G; /* 1/(1 + i/128) rounded to 8/9 bits */
127 float F_hi; /* log(1 / G_i) rounded (see below) */
128 /* The compiler will insert 8 bytes of padding here. */
129 long double F_lo; /* next 113 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 * The relevant range of dk is [-16445, 16383]. The maximum number
145 * of bits in F_hi(i) that works is very dependent on i but has
146 * a minimum of 93. We only need about 12 bits in F_hi(i) for
147 * it to provide enough extra precision.
148 *
149 * We round F_hi(i) to 24 bits so that it can have type float,
150 * mainly to minimize the size of the table. Using all 24 bits
151 * in a float for it automatically satisfies the above constraints.
152 */
153 { 0x800000.0p-23, 0, 0 },
154 { 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L },
155 { 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L },
156 { 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173697cf302cc9476f561.0p-143L },
157 { 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e78eba9b1113bc1c18.0p-142L },
158 { 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L },
159 { 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L },
160 { 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L },
161 { 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c484993c549c4bf40.0p-151L },
162 { 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L },
163 { 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L },
164 { 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L },
165 { 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da99ded322fb08b8462.0p-141L },
166 { 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L },
167 { 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251aefe0ded34c8318f52.0p-145L },
168 { 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d56699c1799a244d4.0p-144L },
169 { 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e6766abceccab1d7174.0p-141L },
170 { 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L },
171 { 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6affd511b534b72a28e.0p-140L },
172 { 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L },
173 { 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L },
174 { 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L },
175 { 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L },
176 { 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e5697dc6a402a56fce1.0p-141L },
177 { 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba9367707ebfa540e45350c.0p-144L },
178 { 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d31ef0f4c9d43f79b2.0p-140L },
179 { 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b75e7d900b521c48d.0p-141L },
180 { 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L },
181 { 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L },
182 { 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d686581799fbce0b5f19.0p-141L },
183 { 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae54f550444ecf8b995.0p-140L },
184 { 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L },
185 { 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d3a85b5b43c0e727.0p-141L },
186 { 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L },
187 { 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L },
188 { 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa6911c7bafcb4d84fb.0p-141L },
189 { 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb328337cc050c6d83b22.0p-140L },
190 { 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e5fcf1a212e2a91e.0p-139L },
191 { 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L },
192 { 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a0fe396f40f1dda9.0p-141L },
193 { 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de945a049a962e66c6.0p-139L },
194 { 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L },
195 { 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba46bae9827221dc98.0p-139L },
196 { 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L },
197 { 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L },
198 { 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L },
199 { 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L },
200 { 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb2438273918db7df5c.0p-141L },
201 { 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698298adddd7f32686.0p-141L },
202 { 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L },
203 { 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b263acb4351104631.0p-140L },
204 { 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L },
205 { 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L },
206 { 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770633947ffe651e7352f.0p-139L },
207 { 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L },
208 { 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f088b61a335f5b688c.0p-140L },
209 { 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L },
210 { 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L },
211 { 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L },
212 { 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333221d8a9b475a6ba.0p-139L },
213 { 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fbfe24b633f4e8d84d.0p-140L },
214 { 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c89c45003fc5d7807.0p-140L },
215 { 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8002f2449e174504.0p-139L },
216 { 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a8717d5626e16acc7d.0p-141L },
217 { 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3ca87dabf351aa41f4.0p-139L },
218 { 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d79f51dcc73014c9.0p-141L },
219 { 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac42d1bf9199188e7.0p-141L },
220 { 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549dd1160bcc45c7e2c.0p-139L },
221 { 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b610665377f15625b6.0p-140L },
222 { 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a2d1b2176010478be.0p-140L },
223 { 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L },
224 { 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L },
225 { 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f56db28da4d629d00a.0p-140L },
226 { 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L },
227 { 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L },
228 { 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd73692609040ccc2.0p-139L },
229 { 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L },
230 { 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L },
231 { 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d42f78d3e65d3727.0p-141L },
232 { 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af269647b783d88999.0p-139L },
233 { 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L },
234 { 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade02951686d5373aec.0p-139L },
235 { 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1649349630531502.0p-139L },
236 { 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c5320619fb9433d841.0p-139L },
237 { 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f0bdff99f932b273.0p-138L },
238 { 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e54e78103a2bc1767.0p-141L },
239 { 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b7d7f47ddb45c5a3.0p-139L },
240 { 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb82873b04a9af1dd692c.0p-138L },
241 { 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9b9770d8cb6573540.0p-138L },
242 { 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f002e836dfd47bd41.0p-139L },
243 { 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd5cd7cc94306fb3ff.0p-140L },
244 { 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de2b41eeebd550702.0p-138L },
245 { 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f45275c917a30df4ac9.0p-138L },
246 { 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af71a89df4e6df2e93b.0p-139L },
247 { 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfdec367828734cae5.0p-139L },
248 { 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f76b87333891e0dec4.0p-138L },
249 { 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a263e3bf446c6e3f69.0p-140L },
250 { 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d723a4c7380a448d8.0p-139L },
251 { 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3236f330972da2a7a87.0p-139L },
252 { 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d21148c6002becd3.0p-139L },
253 { 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c336af90e00533323ba.0p-139L },
254 { 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf2f105a89060046aa.0p-138L },
255 { 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf162409d8ea99d4c0.0p-139L },
256 { 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507b9dc10dac743ad7a.0p-138L },
257 { 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e97c4990b23d9ac1f5.0p-139L },
258 { 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea74540bdd2aa99a42.0p-138L },
259 { 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f9527e6aba8f2d783c1.0p-138L },
260 { 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe7ba81c664c107e0.0p-138L },
261 { 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576aad348ae79867223.0p-138L },
262 { 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a246726b304ccae56.0p-139L },
263 { 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c2f9b47466d6123fe.0p-139L },
264 { 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f38b4619a2483399.0p-141L },
265 { 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3597043be78eaf49.0p-139L },
266 { 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d204147dc69a07a649.0p-138L },
267 { 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c008d3602a7b41c6e8.0p-139L },
268 { 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541aca7d5844606b2421.0p-139L },
269 { 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4571acbcfb03f16daf4.0p-138L },
270 { 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c0a345ad743ae1ae.0p-140L },
271 { 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d749362382a7688479e24.0p-140L },
272 { 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce532661ea9643a3a2d378.0p-139L },
273 { 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d257530a682b80490.0p-139L },
274 { 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b35ba3e1f868fd0b5e.0p-140L },
275 { 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3303dd481779df69.0p-139L },
276 { 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900345a85d2d86161742e.0p-140L },
277 { 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f79b026b64b42caa1.0p-140L },
278 { 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a82ab19f77652d977a.0p-141L },
279 { 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b48d7b98c1cf7234.0p-138L },
280 { 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a197e9c7359dd94152f.0p-138L },
281 { 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c3898cff81a12a17e2.0p-141L },
282 };
283
284 #ifdef USE_UTAB
285 static const struct {
286 float H; /* 1 + i/INTERVALS (exact) */
287 float E; /* H(i) * G(i) - 1 (exact) */
288 } U[TSIZE] = {
289 { 0x800000.0p-23, 0 },
290 { 0x810000.0p-23, -0x800000.0p-37 },
291 { 0x820000.0p-23, -0x800000.0p-35 },
292 { 0x830000.0p-23, -0x900000.0p-34 },
293 { 0x840000.0p-23, -0x800000.0p-33 },
294 { 0x850000.0p-23, -0xc80000.0p-33 },
295 { 0x860000.0p-23, -0xa00000.0p-36 },
296 { 0x870000.0p-23, 0x940000.0p-33 },
297 { 0x880000.0p-23, 0x800000.0p-35 },
298 { 0x890000.0p-23, -0xc80000.0p-34 },
299 { 0x8a0000.0p-23, 0xe00000.0p-36 },
300 { 0x8b0000.0p-23, 0x900000.0p-33 },
301 { 0x8c0000.0p-23, -0x800000.0p-35 },
302 { 0x8d0000.0p-23, -0xe00000.0p-33 },
303 { 0x8e0000.0p-23, 0x880000.0p-33 },
304 { 0x8f0000.0p-23, -0xa80000.0p-34 },
305 { 0x900000.0p-23, -0x800000.0p-35 },
306 { 0x910000.0p-23, 0x800000.0p-37 },
307 { 0x920000.0p-23, 0x900000.0p-35 },
308 { 0x930000.0p-23, 0xd00000.0p-35 },
309 { 0x940000.0p-23, 0xe00000.0p-35 },
310 { 0x950000.0p-23, 0xc00000.0p-35 },
311 { 0x960000.0p-23, 0xe00000.0p-36 },
312 { 0x970000.0p-23, -0x800000.0p-38 },
313 { 0x980000.0p-23, -0xc00000.0p-35 },
314 { 0x990000.0p-23, -0xd00000.0p-34 },
315 { 0x9a0000.0p-23, 0x880000.0p-33 },
316 { 0x9b0000.0p-23, 0xe80000.0p-35 },
317 { 0x9c0000.0p-23, -0x800000.0p-35 },
318 { 0x9d0000.0p-23, 0xb40000.0p-33 },
319 { 0x9e0000.0p-23, 0x880000.0p-34 },
320 { 0x9f0000.0p-23, -0xe00000.0p-35 },
321 { 0xa00000.0p-23, 0x800000.0p-33 },
322 { 0xa10000.0p-23, -0x900000.0p-36 },
323 { 0xa20000.0p-23, -0xb00000.0p-33 },
324 { 0xa30000.0p-23, -0xa00000.0p-36 },
325 { 0xa40000.0p-23, 0x800000.0p-33 },
326 { 0xa50000.0p-23, -0xf80000.0p-35 },
327 { 0xa60000.0p-23, 0x880000.0p-34 },
328 { 0xa70000.0p-23, -0x900000.0p-33 },
329 { 0xa80000.0p-23, -0x800000.0p-35 },
330 { 0xa90000.0p-23, 0x900000.0p-34 },
331 { 0xaa0000.0p-23, 0xa80000.0p-33 },
332 { 0xab0000.0p-23, -0xac0000.0p-34 },
333 { 0xac0000.0p-23, -0x800000.0p-37 },
334 { 0xad0000.0p-23, 0xf80000.0p-35 },
335 { 0xae0000.0p-23, 0xf80000.0p-34 },
336 { 0xaf0000.0p-23, -0xac0000.0p-33 },
337 { 0xb00000.0p-23, -0x800000.0p-33 },
338 { 0xb10000.0p-23, -0xb80000.0p-34 },
339 { 0xb20000.0p-23, -0x800000.0p-34 },
340 { 0xb30000.0p-23, -0xb00000.0p-35 },
341 { 0xb40000.0p-23, -0x800000.0p-35 },
342 { 0xb50000.0p-23, -0xe00000.0p-36 },
343 { 0xb60000.0p-23, -0x800000.0p-35 },
344 { 0xb70000.0p-23, -0xb00000.0p-35 },
345 { 0xb80000.0p-23, -0x800000.0p-34 },
346 { 0xb90000.0p-23, -0xb80000.0p-34 },
347 { 0xba0000.0p-23, -0x800000.0p-33 },
348 { 0xbb0000.0p-23, -0xac0000.0p-33 },
349 { 0xbc0000.0p-23, 0x980000.0p-33 },
350 { 0xbd0000.0p-23, 0xbc0000.0p-34 },
351 { 0xbe0000.0p-23, 0xe00000.0p-36 },
352 { 0xbf0000.0p-23, -0xb80000.0p-35 },
353 { 0xc00000.0p-23, -0x800000.0p-33 },
354 { 0xc10000.0p-23, 0xa80000.0p-33 },
355 { 0xc20000.0p-23, 0x900000.0p-34 },
356 { 0xc30000.0p-23, -0x800000.0p-35 },
357 { 0xc40000.0p-23, -0x900000.0p-33 },
358 { 0xc50000.0p-23, 0x820000.0p-33 },
359 { 0xc60000.0p-23, 0x800000.0p-38 },
360 { 0xc70000.0p-23, -0x820000.0p-33 },
361 { 0xc80000.0p-23, 0x800000.0p-33 },
362 { 0xc90000.0p-23, -0xa00000.0p-36 },
363 { 0xca0000.0p-23, -0xb00000.0p-33 },
364 { 0xcb0000.0p-23, 0x840000.0p-34 },
365 { 0xcc0000.0p-23, -0xd00000.0p-34 },
366 { 0xcd0000.0p-23, 0x800000.0p-33 },
367 { 0xce0000.0p-23, -0xe00000.0p-35 },
368 { 0xcf0000.0p-23, 0xa60000.0p-33 },
369 { 0xd00000.0p-23, -0x800000.0p-35 },
370 { 0xd10000.0p-23, 0xb40000.0p-33 },
371 { 0xd20000.0p-23, -0x800000.0p-35 },
372 { 0xd30000.0p-23, 0xaa0000.0p-33 },
373 { 0xd40000.0p-23, -0xe00000.0p-35 },
374 { 0xd50000.0p-23, 0x880000.0p-33 },
375 { 0xd60000.0p-23, -0xd00000.0p-34 },
376 { 0xd70000.0p-23, 0x9c0000.0p-34 },
377 { 0xd80000.0p-23, -0xb00000.0p-33 },
378 { 0xd90000.0p-23, -0x800000.0p-38 },
379 { 0xda0000.0p-23, 0xa40000.0p-33 },
380 { 0xdb0000.0p-23, -0xdc0000.0p-34 },
381 { 0xdc0000.0p-23, 0xc00000.0p-35 },
382 { 0xdd0000.0p-23, 0xca0000.0p-33 },
383 { 0xde0000.0p-23, -0xb80000.0p-34 },
384 { 0xdf0000.0p-23, 0xd00000.0p-35 },
385 { 0xe00000.0p-23, 0xc00000.0p-33 },
386 { 0xe10000.0p-23, -0xf40000.0p-34 },
387 { 0xe20000.0p-23, 0x800000.0p-37 },
388 { 0xe30000.0p-23, 0x860000.0p-33 },
389 { 0xe40000.0p-23, -0xc80000.0p-33 },
390 { 0xe50000.0p-23, -0xa80000.0p-34 },
391 { 0xe60000.0p-23, 0xe00000.0p-36 },
392 { 0xe70000.0p-23, 0x880000.0p-33 },
393 { 0xe80000.0p-23, -0xe00000.0p-33 },
394 { 0xe90000.0p-23, -0xfc0000.0p-34 },
395 { 0xea0000.0p-23, -0x800000.0p-35 },
396 { 0xeb0000.0p-23, 0xe80000.0p-35 },
397 { 0xec0000.0p-23, 0x900000.0p-33 },
398 { 0xed0000.0p-23, 0xe20000.0p-33 },
399 { 0xee0000.0p-23, -0xac0000.0p-33 },
400 { 0xef0000.0p-23, -0xc80000.0p-34 },
401 { 0xf00000.0p-23, -0x800000.0p-35 },
402 { 0xf10000.0p-23, 0x800000.0p-35 },
403 { 0xf20000.0p-23, 0xb80000.0p-34 },
404 { 0xf30000.0p-23, 0x940000.0p-33 },
405 { 0xf40000.0p-23, 0xc80000.0p-33 },
406 { 0xf50000.0p-23, -0xf20000.0p-33 },
407 { 0xf60000.0p-23, -0xc80000.0p-33 },
408 { 0xf70000.0p-23, -0xa20000.0p-33 },
409 { 0xf80000.0p-23, -0x800000.0p-33 },
410 { 0xf90000.0p-23, -0xc40000.0p-34 },
411 { 0xfa0000.0p-23, -0x900000.0p-34 },
412 { 0xfb0000.0p-23, -0xc80000.0p-35 },
413 { 0xfc0000.0p-23, -0x800000.0p-35 },
414 { 0xfd0000.0p-23, -0x900000.0p-36 },
415 { 0xfe0000.0p-23, -0x800000.0p-37 },
416 { 0xff0000.0p-23, -0x800000.0p-39 },
417 { 0x800000.0p-22, 0 },
418 };
419 #endif /* USE_UTAB */
420
421 #ifdef STRUCT_RETURN
422 #define RETURN1(rp, v) do { \
423 (rp)->hi = (v); \
424 (rp)->lo_set = 0; \
425 return; \
426 } while (0)
427
428 #define RETURN2(rp, h, l) do { \
429 (rp)->hi = (h); \
430 (rp)->lo = (l); \
431 (rp)->lo_set = 1; \
432 return; \
433 } while (0)
434
435 struct ld {
436 long double hi;
437 long double lo;
438 int lo_set;
439 };
440 #else
441 #define RETURN1(rp, v) RETURNF(v)
442 #define RETURN2(rp, h, l) RETURNI((h) + (l))
443 #endif
444
445 #ifdef STRUCT_RETURN
446 static inline __always_inline void
k_logl(long double x,struct ld * rp)447 k_logl(long double x, struct ld *rp)
448 #else
449 long double
450 logl(long double x)
451 #endif
452 {
453 long double d, val_hi, val_lo;
454 double dd, dk;
455 uint64_t lx, llx;
456 int i, k;
457 uint16_t hx;
458
459 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
460 k = -16383;
461 #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */
462 if (x == 1)
463 RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */
464 #endif
465 if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */
466 if (((hx & 0x7fff) | lx | llx) == 0)
467 RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */
468 if (hx != 0)
469 /* log(neg or NaN) = qNaN: */
470 RETURN1(rp, (x - x) / zero);
471 x *= 0x1.0p113; /* subnormal; scale up x */
472 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
473 k = -16383 - 113;
474 } else if (hx >= 0x7fff)
475 RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */
476 #ifndef STRUCT_RETURN
477 ENTERI();
478 #endif
479 k += hx;
480 dk = k;
481
482 /* Scale x to be in [1, 2). */
483 SET_LDBL_EXPSIGN(x, 0x3fff);
484
485 /* 0 <= i <= INTERVALS: */
486 #define L2I (49 - LOG2_INTERVALS)
487 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
488
489 /*
490 * -0.005280 < d < 0.004838. In particular, the infinite-
491 * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits
492 * ensures that d is representable without extra precision for
493 * this bound on |d| (since when this calculation is expressed
494 * as x*G(i)-1, the multiplication needs as many extra bits as
495 * G(i) has and the subtraction cancels 8 bits). But for
496 * most i (107 cases out of 129), the infinite-precision |d|
497 * is <= 2**-8. G(i) is rounded to 9 bits for such i to give
498 * better accuracy (this works by improving the bound on |d|,
499 * which in turn allows rounding to 9 bits in more cases).
500 * This is only important when the original x is near 1 -- it
501 * lets us avoid using a special method to give the desired
502 * accuracy for such x.
503 */
504 if (0)
505 d = x * G(i) - 1;
506 else {
507 #ifdef USE_UTAB
508 d = (x - H(i)) * G(i) + E(i);
509 #else
510 long double x_hi;
511 double x_lo;
512
513 /*
514 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly.
515 * G(i) has at most 9 bits, so the splitting point is not
516 * critical.
517 */
518 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
519 llx & 0xffffffffff000000ULL);
520 x_lo = x - x_hi;
521 d = x_hi * G(i) - 1 + x_lo * G(i);
522 #endif
523 }
524
525 /*
526 * Our algorithm depends on exact cancellation of F_lo(i) and
527 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is
528 * at the end of the table. This and other technical complications
529 * make it difficult to avoid the double scaling in (dk*ln2) *
530 * log(base) for base != e without losing more accuracy and/or
531 * efficiency than is gained.
532 */
533 /*
534 * Use double precision operations wherever possible, since
535 * long double operations are emulated and were very slow on
536 * the old sparc64 and unknown on the newer aarch64 and riscv
537 * machines. Also, don't try to improve parallelism by
538 * increasing the number of operations, since any parallelism
539 * on such machines is needed for the emulation. Horner's
540 * method is good for this, and is also good for accuracy.
541 * Horner's method doesn't handle the `lo' term well, either
542 * for efficiency or accuracy. However, for accuracy we
543 * evaluate d * d * P2 separately to take advantage of by P2
544 * being exact, and this gives a good place to sum the 'lo'
545 * term too.
546 */
547 dd = (double)d;
548 val_lo = d * d * d * (P3 +
549 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
550 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
551 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo) + d * d * P2;
552 val_hi = d;
553 #ifdef DEBUG
554 if (fetestexcept(FE_UNDERFLOW))
555 breakpoint();
556 #endif
557
558 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
559 RETURN2(rp, val_hi, val_lo);
560 }
561
562 long double
log1pl(long double x)563 log1pl(long double x)
564 {
565 long double d, d_hi, f_lo, val_hi, val_lo;
566 long double f_hi, twopminusk;
567 double d_lo, dd, dk;
568 uint64_t lx, llx;
569 int i, k;
570 int16_t ax, hx;
571
572 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
573 if (hx < 0x3fff) { /* x < 1, or x neg NaN */
574 ax = hx & 0x7fff;
575 if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */
576 if (ax == 0x3fff && (lx | llx) == 0)
577 RETURNF(-1 / zero); /* log1p(-1) = -Inf */
578 /* log1p(x < 1, or x NaN) = qNaN: */
579 RETURNF((x - x) / (x - x));
580 }
581 if (ax <= 0x3f8d) { /* |x| < 2**-113 */
582 if ((int)x == 0)
583 RETURNF(x); /* x with inexact if x != 0 */
584 }
585 f_hi = 1;
586 f_lo = x;
587 } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */
588 RETURNF(x + x); /* log1p(Inf or NaN) = Inf or qNaN */
589 } else if (hx < 0x40e1) { /* 1 <= x < 2**226 */
590 f_hi = x;
591 f_lo = 1;
592 } else { /* 2**226 <= x < +Inf */
593 f_hi = x;
594 f_lo = 0; /* avoid underflow of the P3 term */
595 }
596 ENTERI();
597 x = f_hi + f_lo;
598 f_lo = (f_hi - x) + f_lo;
599
600 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
601 k = -16383;
602
603 k += hx;
604 dk = k;
605
606 SET_LDBL_EXPSIGN(x, 0x3fff);
607 twopminusk = 1;
608 SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
609 f_lo *= twopminusk;
610
611 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
612
613 /*
614 * x*G(i)-1 (with a reduced x) can be represented exactly, as
615 * above, but now we need to evaluate the polynomial on d =
616 * (x+f_lo)*G(i)-1 and extra precision is needed for that.
617 * Since x+x_lo is a hi+lo decomposition and subtracting 1
618 * doesn't lose too many bits, an inexact calculation for
619 * f_lo*G(i) is good enough.
620 */
621 if (0)
622 d_hi = x * G(i) - 1;
623 else {
624 #ifdef USE_UTAB
625 d_hi = (x - H(i)) * G(i) + E(i);
626 #else
627 long double x_hi;
628 double x_lo;
629
630 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
631 llx & 0xffffffffff000000ULL);
632 x_lo = x - x_hi;
633 d_hi = x_hi * G(i) - 1 + x_lo * G(i);
634 #endif
635 }
636 d_lo = f_lo * G(i);
637
638 /*
639 * This is _2sumF(d_hi, d_lo) inlined. The condition
640 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not
641 * always satisifed, so it is not clear that this works, but
642 * it works in practice. It works even if it gives a wrong
643 * normalized d_lo, since |d_lo| > |d_hi| implies that i is
644 * nonzero and d is tiny, so the F(i) term dominates d_lo.
645 * In float precision:
646 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25.
647 * And if d is only a little tinier than that, we would have
648 * another underflow problem for the P3 term; this is also ruled
649 * out by exhaustive testing.)
650 */
651 d = d_hi + d_lo;
652 d_lo = d_hi - d + d_lo;
653 d_hi = d;
654
655 dd = (double)d;
656 val_lo = d * d * d * (P3 +
657 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
658 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
659 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo + d_lo) + d * d * P2;
660 val_hi = d_hi;
661 #ifdef DEBUG
662 if (fetestexcept(FE_UNDERFLOW))
663 breakpoint();
664 #endif
665
666 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
667 RETURNI(val_hi + val_lo);
668 }
669
670 #ifdef STRUCT_RETURN
671
672 long double
logl(long double x)673 logl(long double x)
674 {
675 struct ld r;
676
677 ENTERI();
678 k_logl(x, &r);
679 RETURNSPI(&r);
680 }
681
682 /*
683 * 29+113 bit decompositions. The bits are distributed so that the products
684 * of the hi terms are exact in double precision. The types are chosen so
685 * that the products of the hi terms are done in at least double precision,
686 * without any explicit conversions. More natural choices would require a
687 * slow long double precision multiplication.
688 */
689 static const double
690 invln10_hi = 4.3429448176175356e-1, /* 0x1bcb7b15000000.0p-54 */
691 invln2_hi = 1.4426950402557850e0; /* 0x17154765000000.0p-52 */
692 static const long double
693 invln10_lo = 1.41498268538580090791605082294397000e-10L, /* 0x137287195355baaafad33dc323ee3.0p-145L */
694 invln2_lo = 6.33178418956604368501892137426645911e-10L, /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */
695 invln10_lo_plus_hi = invln10_lo + invln10_hi,
696 invln2_lo_plus_hi = invln2_lo + invln2_hi;
697
698 long double
log10l(long double x)699 log10l(long double x)
700 {
701 struct ld r;
702 long double hi, lo;
703
704 ENTERI();
705 k_logl(x, &r);
706 if (!r.lo_set)
707 RETURNI(r.hi);
708 _2sumF(r.hi, r.lo);
709 hi = (float)r.hi;
710 lo = r.lo + (r.hi - hi);
711 RETURNI(invln10_hi * hi + (invln10_lo_plus_hi * lo + invln10_lo * hi));
712 }
713
714 long double
log2l(long double x)715 log2l(long double x)
716 {
717 struct ld r;
718 long double hi, lo;
719
720 ENTERI();
721 k_logl(x, &r);
722 if (!r.lo_set)
723 RETURNI(r.hi);
724 _2sumF(r.hi, r.lo);
725 hi = (float)r.hi;
726 lo = r.lo + (r.hi - hi);
727 RETURNI(invln2_hi * hi + (invln2_lo_plus_hi * lo + invln2_lo * hi));
728 }
729
730 #endif /* STRUCT_RETURN */
731