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 128-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 #include "fpmath.h"
85 #include "math.h"
86 #ifndef NO_STRUCT_RETURN
87 #define STRUCT_RETURN
88 #endif
89 #include "math_private.h"
90
91 #if !defined(NO_UTAB) && !defined(NO_UTABL)
92 #define USE_UTAB
93 #endif
94
95 /*
96 * Domain [-0.005280, 0.004838], range ~[-1.1577e-37, 1.1582e-37]:
97 * |log(1 + d)/d - p(d)| < 2**-122.7
98 */
99 static const long double
100 P2 = -0.5L,
101 P3 = 3.33333333333333333333333333333233795e-1L, /* 0x15555555555555555555555554d42.0p-114L */
102 P4 = -2.49999999999999999999999999941139296e-1L, /* -0x1ffffffffffffffffffffffdab14e.0p-115L */
103 P5 = 2.00000000000000000000000085468039943e-1L, /* 0x19999999999999999999a6d3567f4.0p-115L */
104 P6 = -1.66666666666666666666696142372698408e-1L, /* -0x15555555555555555567267a58e13.0p-115L */
105 P7 = 1.42857142857142857119522943477166120e-1L, /* 0x1249249249249248ed79a0ae434de.0p-115L */
106 P8 = -1.24999999999999994863289015033581301e-1L; /* -0x1fffffffffffffa13e91765e46140.0p-116L */
107 /* Double precision gives ~ 53 + log2(P9 * max(|d|)**8) ~= 120 bits. */
108 static const double
109 P9 = 1.1111111111111401e-1, /* 0x1c71c71c71c7ed.0p-56 */
110 P10 = -1.0000000000040135e-1, /* -0x199999999a0a92.0p-56 */
111 P11 = 9.0909090728136258e-2, /* 0x1745d173962111.0p-56 */
112 P12 = -8.3333318851855284e-2, /* -0x1555551722c7a3.0p-56 */
113 P13 = 7.6928634666404178e-2, /* 0x13b1985204a4ae.0p-56 */
114 P14 = -7.1626810078462499e-2; /* -0x12562276cdc5d0.0p-56 */
115
116 static volatile const double zero = 0;
117
118 #define INTERVALS 128
119 #define LOG2_INTERVALS 7
120 #define TSIZE (INTERVALS + 1)
121 #define G(i) (T[(i)].G)
122 #define F_hi(i) (T[(i)].F_hi)
123 #define F_lo(i) (T[(i)].F_lo)
124 #define ln2_hi F_hi(TSIZE - 1)
125 #define ln2_lo F_lo(TSIZE - 1)
126 #define E(i) (U[(i)].E)
127 #define H(i) (U[(i)].H)
128
129 static const struct {
130 float G; /* 1/(1 + i/128) rounded to 8/9 bits */
131 float F_hi; /* log(1 / G_i) rounded (see below) */
132 /* The compiler will insert 8 bytes of padding here. */
133 long double F_lo; /* next 113 bits for log(1 / G_i) */
134 } T[TSIZE] = {
135 /*
136 * ln2_hi and each F_hi(i) are rounded to a number of bits that
137 * makes F_hi(i) + dk*ln2_hi exact for all i and all dk.
138 *
139 * The last entry (for X just below 2) is used to define ln2_hi
140 * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly
141 * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1.
142 * This is needed for accuracy when x is just below 1. (To avoid
143 * special cases, such x are "reduced" strangely to X just below
144 * 2 and dk = -1, and then the exact cancellation is needed
145 * because any the error from any non-exactness would be too
146 * large).
147 *
148 * The relevant range of dk is [-16445, 16383]. The maximum number
149 * of bits in F_hi(i) that works is very dependent on i but has
150 * a minimum of 93. We only need about 12 bits in F_hi(i) for
151 * it to provide enough extra precision.
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, -0x14ee431dae6674afa0c4bfe16e8fd.0p-144L,
159 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83717be918e1119642ab.0p-144L,
160 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173697cf302cc9476f561.0p-143L,
161 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e78eba9b1113bc1c18.0p-142L,
162 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7bfa509bec8da5f22.0p-142L,
163 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a107674986dcdca6709c.0p-143L,
164 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb958897a3ea46e84abb3.0p-142L,
165 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c484993c549c4bf40.0p-151L,
166 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560d9e9ab3d6ebab580.0p-141L,
167 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d5037108f4ec21e48cd.0p-142L,
168 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a70f7a684c596b12.0p-143L,
169 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da99ded322fb08b8462.0p-141L,
170 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150ae09996d7bb4653e.0p-143L,
171 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251aefe0ded34c8318f52.0p-145L,
172 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d56699c1799a244d4.0p-144L,
173 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e6766abceccab1d7174.0p-141L,
174 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f4215f93936b709efb.0p-142L,
175 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6affd511b534b72a28e.0p-140L,
176 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1d9b2ef7e68680598.0p-143L,
177 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b3f46662238a9425a.0p-142L,
178 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9691aed4d5e3df94.0p-140L,
179 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c46c186384993e1c93.0p-142L,
180 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e5697dc6a402a56fce1.0p-141L,
181 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba9367707ebfa540e45350c.0p-144L,
182 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d31ef0f4c9d43f79b2.0p-140L,
183 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b75e7d900b521c48d.0p-141L,
184 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06508aeb00d2ae3e9.0p-140L,
185 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3af80485c2f409633c.0p-142L,
186 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d686581799fbce0b5f19.0p-141L,
187 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae54f550444ecf8b995.0p-140L,
188 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc4595412b5d2517aaac13.0p-141L,
189 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d3a85b5b43c0e727.0p-141L,
190 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df841a71b79dd5645b46.0p-145L,
191 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe0bcfbe6d6db9f66.0p-147L,
192 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa6911c7bafcb4d84fb.0p-141L,
193 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb328337cc050c6d83b22.0p-140L,
194 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e5fcf1a212e2a91e.0p-139L,
195 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f465e5ecab5f2a6f81.0p-139L,
196 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a0fe396f40f1dda9.0p-141L,
197 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de945a049a962e66c6.0p-139L,
198 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5147bdb6ddcaf59c425.0p-141L,
199 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba46bae9827221dc98.0p-139L,
200 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b730e28aba001a9b5e0.0p-140L,
201 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed72e23e13431adc4a5.0p-141L,
202 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7148e8d80caa10b7.0p-139L,
203 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c5663663d15faed7771.0p-139L,
204 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb2438273918db7df5c.0p-141L,
205 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698298adddd7f32686.0p-141L,
206 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123615b147a5d47bc208f.0p-142L,
207 0xb80000.0p-24, 0xa91570.0p-25, 0x1ce28f5f3840b263acb4351104631.0p-140L,
208 0xb70000.0p-24, 0xabdfbb.0p-25, -0x186e5c0a42423457e22d8c650b355.0p-139L,
209 0xb60000.0p-24, 0xaeadef.0p-25, -0x14d41a0b2a08a465dc513b13f567d.0p-143L,
210 0xb50000.0p-24, 0xb18018.0p-25, 0x16755892770633947ffe651e7352f.0p-139L,
211 0xb40000.0p-24, 0xb45642.0p-25, -0x16395ebe59b15228bfe8798d10ff0.0p-142L,
212 0xb30000.0p-24, 0xb73077.0p-25, 0x1abc65c8595f088b61a335f5b688c.0p-140L,
213 0xb20000.0p-24, 0xba0ec4.0p-25, -0x1273089d3dad88e7d353e9967d548.0p-139L,
214 0xb10000.0p-24, 0xbcf133.0p-25, 0x10f9f67b1f4bbf45de06ecebfaf6d.0p-139L,
215 0xb00000.0p-24, 0xbfd7d2.0p-25, -0x109fab904864092b34edda19a831e.0p-140L,
216 0xaf0000.0p-24, 0xc2c2ac.0p-25, -0x1124680aa43333221d8a9b475a6ba.0p-139L,
217 0xae8000.0p-24, 0xc439b3.0p-25, -0x1f360cc4710fbfe24b633f4e8d84d.0p-140L,
218 0xad8000.0p-24, 0xc72afd.0p-25, -0x132d91f21d89c89c45003fc5d7807.0p-140L,
219 0xac8000.0p-24, 0xca20a2.0p-25, -0x16bf9b4d1f8da8002f2449e174504.0p-139L,
220 0xab8000.0p-24, 0xcd1aae.0p-25, 0x19deb5ce6a6a8717d5626e16acc7d.0p-141L,
221 0xaa8000.0p-24, 0xd0192f.0p-25, 0x1a29fb48f7d3ca87dabf351aa41f4.0p-139L,
222 0xaa0000.0p-24, 0xd19a20.0p-25, 0x1127d3c6457f9d79f51dcc73014c9.0p-141L,
223 0xa90000.0p-24, 0xd49f6a.0p-25, -0x1ba930e486a0ac42d1bf9199188e7.0p-141L,
224 0xa80000.0p-24, 0xd7a94b.0p-25, -0x1b6e645f31549dd1160bcc45c7e2c.0p-139L,
225 0xa70000.0p-24, 0xdab7d0.0p-25, 0x1118a425494b610665377f15625b6.0p-140L,
226 0xa68000.0p-24, 0xdc40d5.0p-25, 0x1966f24d29d3a2d1b2176010478be.0p-140L,
227 0xa58000.0p-24, 0xdf566d.0p-25, -0x1d8e52eb2248f0c95dd83626d7333.0p-142L,
228 0xa48000.0p-24, 0xe270ce.0p-25, -0x1ee370f96e6b67ccb006a5b9890ea.0p-140L,
229 0xa40000.0p-24, 0xe3ffce.0p-25, 0x1d155324911f56db28da4d629d00a.0p-140L,
230 0xa30000.0p-24, 0xe72179.0p-25, -0x1fe6e2f2f867d8f4d60c713346641.0p-140L,
231 0xa20000.0p-24, 0xea4812.0p-25, 0x1b7be9add7f4d3b3d406b6cbf3ce5.0p-140L,
232 0xa18000.0p-24, 0xebdd3d.0p-25, 0x1b3cfb3f7511dd73692609040ccc2.0p-139L,
233 0xa08000.0p-24, 0xef0b5b.0p-25, -0x1220de1f7301901b8ad85c25afd09.0p-139L,
234 0xa00000.0p-24, 0xf0a451.0p-25, -0x176364c9ac81cc8a4dfb804de6867.0p-140L,
235 0x9f0000.0p-24, 0xf3da16.0p-25, 0x1eed6b9aafac8d42f78d3e65d3727.0p-141L,
236 0x9e8000.0p-24, 0xf576e9.0p-25, 0x1d593218675af269647b783d88999.0p-139L,
237 0x9d8000.0p-24, 0xf8b47c.0p-25, -0x13e8eb7da053e063714615f7cc91d.0p-144L,
238 0x9d0000.0p-24, 0xfa553f.0p-25, 0x1c063259bcade02951686d5373aec.0p-139L,
239 0x9c0000.0p-24, 0xfd9ac5.0p-25, 0x1ef491085fa3c1649349630531502.0p-139L,
240 0x9b8000.0p-24, 0xff3f8c.0p-25, 0x1d607a7c2b8c5320619fb9433d841.0p-139L,
241 0x9a8000.0p-24, 0x814697.0p-24, -0x12ad3817004f3f0bdff99f932b273.0p-138L,
242 0x9a0000.0p-24, 0x821b06.0p-24, -0x189fc53117f9e54e78103a2bc1767.0p-141L,
243 0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b7d7f47ddb45c5a3.0p-139L,
244 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb82873b04a9af1dd692c.0p-138L,
245 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9b9770d8cb6573540.0p-138L,
246 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f002e836dfd47bd41.0p-139L,
247 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd5cd7cc94306fb3ff.0p-140L,
248 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de2b41eeebd550702.0p-138L,
249 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f45275c917a30df4ac9.0p-138L,
250 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af71a89df4e6df2e93b.0p-139L,
251 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfdec367828734cae5.0p-139L,
252 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f76b87333891e0dec4.0p-138L,
253 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a263e3bf446c6e3f69.0p-140L,
254 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d723a4c7380a448d8.0p-139L,
255 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3236f330972da2a7a87.0p-139L,
256 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d21148c6002becd3.0p-139L,
257 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c336af90e00533323ba.0p-139L,
258 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf2f105a89060046aa.0p-138L,
259 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf162409d8ea99d4c0.0p-139L,
260 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507b9dc10dac743ad7a.0p-138L,
261 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e97c4990b23d9ac1f5.0p-139L,
262 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea74540bdd2aa99a42.0p-138L,
263 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f9527e6aba8f2d783c1.0p-138L,
264 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe7ba81c664c107e0.0p-138L,
265 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576aad348ae79867223.0p-138L,
266 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a246726b304ccae56.0p-139L,
267 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c2f9b47466d6123fe.0p-139L,
268 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f38b4619a2483399.0p-141L,
269 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3597043be78eaf49.0p-139L,
270 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d204147dc69a07a649.0p-138L,
271 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c008d3602a7b41c6e8.0p-139L,
272 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541aca7d5844606b2421.0p-139L,
273 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4571acbcfb03f16daf4.0p-138L,
274 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c0a345ad743ae1ae.0p-140L,
275 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d749362382a7688479e24.0p-140L,
276 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce532661ea9643a3a2d378.0p-139L,
277 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d257530a682b80490.0p-139L,
278 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b35ba3e1f868fd0b5e.0p-140L,
279 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3303dd481779df69.0p-139L,
280 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900345a85d2d86161742e.0p-140L,
281 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f79b026b64b42caa1.0p-140L,
282 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a82ab19f77652d977a.0p-141L,
283 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b48d7b98c1cf7234.0p-138L,
284 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a197e9c7359dd94152f.0p-138L,
285 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c3898cff81a12a17e2.0p-141L,
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
k_logl(long double x,struct ld * rp)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, val_hi, val_lo;
458 double dd, dk;
459 uint64_t lx, llx;
460 int i, k;
461 uint16_t hx;
462
463 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
464 k = -16383;
465 #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */
466 if (x == 1)
467 RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */
468 #endif
469 if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */
470 if (((hx & 0x7fff) | lx | llx) == 0)
471 RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */
472 if (hx != 0)
473 /* log(neg or NaN) = qNaN: */
474 RETURN1(rp, (x - x) / zero);
475 x *= 0x1.0p113; /* subnormal; scale up x */
476 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
477 k = -16383 - 113;
478 } else if (hx >= 0x7fff)
479 RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */
480 #ifndef STRUCT_RETURN
481 ENTERI();
482 #endif
483 k += hx;
484 dk = k;
485
486 /* Scale x to be in [1, 2). */
487 SET_LDBL_EXPSIGN(x, 0x3fff);
488
489 /* 0 <= i <= INTERVALS: */
490 #define L2I (49 - LOG2_INTERVALS)
491 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
492
493 /*
494 * -0.005280 < d < 0.004838. In particular, the infinite-
495 * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits
496 * ensures that d is representable without extra precision for
497 * this bound on |d| (since when this calculation is expressed
498 * as x*G(i)-1, the multiplication needs as many extra bits as
499 * G(i) has and the subtraction cancels 8 bits). But for
500 * most i (107 cases out of 129), the infinite-precision |d|
501 * is <= 2**-8. G(i) is rounded to 9 bits for such i to give
502 * better accuracy (this works by improving the bound on |d|,
503 * which in turn allows rounding to 9 bits in more cases).
504 * This is only important when the original x is near 1 -- it
505 * lets us avoid using a special method to give the desired
506 * accuracy for such x.
507 */
508 if (0)
509 d = x * G(i) - 1;
510 else {
511 #ifdef USE_UTAB
512 d = (x - H(i)) * G(i) + E(i);
513 #else
514 long double x_hi;
515 double x_lo;
516
517 /*
518 * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly.
519 * G(i) has at most 9 bits, so the splitting point is not
520 * critical.
521 */
522 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
523 llx & 0xffffffffff000000ULL);
524 x_lo = x - x_hi;
525 d = x_hi * G(i) - 1 + x_lo * G(i);
526 #endif
527 }
528
529 /*
530 * Our algorithm depends on exact cancellation of F_lo(i) and
531 * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is
532 * at the end of the table. This and other technical complications
533 * make it difficult to avoid the double scaling in (dk*ln2) *
534 * log(base) for base != e without losing more accuracy and/or
535 * efficiency than is gained.
536 */
537 /*
538 * Use double precision operations wherever possible, since
539 * long double operations are emulated and were very slow on
540 * the old sparc64 and unknown on the newer aarch64 and riscv
541 * machines. Also, don't try to improve parallelism by
542 * increasing the number of operations, since any parallelism
543 * on such machines is needed for the emulation. Horner's
544 * method is good for this, and is also good for accuracy.
545 * Horner's method doesn't handle the `lo' term well, either
546 * for efficiency or accuracy. However, for accuracy we
547 * evaluate d * d * P2 separately to take advantage of by P2
548 * being exact, and this gives a good place to sum the 'lo'
549 * term too.
550 */
551 dd = (double)d;
552 val_lo = d * d * d * (P3 +
553 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
554 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
555 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo) + d * d * P2;
556 val_hi = d;
557 #ifdef DEBUG
558 if (fetestexcept(FE_UNDERFLOW))
559 breakpoint();
560 #endif
561
562 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
563 RETURN2(rp, val_hi, val_lo);
564 }
565
566 long double
log1pl(long double x)567 log1pl(long double x)
568 {
569 long double d, d_hi, f_lo, val_hi, val_lo;
570 long double f_hi, twopminusk;
571 double d_lo, dd, dk;
572 uint64_t lx, llx;
573 int i, k;
574 int16_t ax, hx;
575
576 DOPRINT_START(&x);
577 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
578 if (hx < 0x3fff) { /* x < 1, or x neg NaN */
579 ax = hx & 0x7fff;
580 if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */
581 if (ax == 0x3fff && (lx | llx) == 0)
582 RETURNP(-1 / zero); /* log1p(-1) = -Inf */
583 /* log1p(x < 1, or x NaN) = qNaN: */
584 RETURNP((x - x) / (x - x));
585 }
586 if (ax <= 0x3f8d) { /* |x| < 2**-113 */
587 if ((int)x == 0)
588 RETURNP(x); /* x with inexact if x != 0 */
589 }
590 f_hi = 1;
591 f_lo = x;
592 } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */
593 RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */
594 } else if (hx < 0x40e1) { /* 1 <= x < 2**226 */
595 f_hi = x;
596 f_lo = 1;
597 } else { /* 2**226 <= x < +Inf */
598 f_hi = x;
599 f_lo = 0; /* avoid underflow of the P3 term */
600 }
601 ENTERI();
602 x = f_hi + f_lo;
603 f_lo = (f_hi - x) + f_lo;
604
605 EXTRACT_LDBL128_WORDS(hx, lx, llx, x);
606 k = -16383;
607
608 k += hx;
609 dk = k;
610
611 SET_LDBL_EXPSIGN(x, 0x3fff);
612 twopminusk = 1;
613 SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff));
614 f_lo *= twopminusk;
615
616 i = (lx + (1LL << (L2I - 2))) >> (L2I - 1);
617
618 /*
619 * x*G(i)-1 (with a reduced x) can be represented exactly, as
620 * above, but now we need to evaluate the polynomial on d =
621 * (x+f_lo)*G(i)-1 and extra precision is needed for that.
622 * Since x+x_lo is a hi+lo decomposition and subtracting 1
623 * doesn't lose too many bits, an inexact calculation for
624 * f_lo*G(i) is good enough.
625 */
626 if (0)
627 d_hi = x * G(i) - 1;
628 else {
629 #ifdef USE_UTAB
630 d_hi = (x - H(i)) * G(i) + E(i);
631 #else
632 long double x_hi;
633 double x_lo;
634
635 INSERT_LDBL128_WORDS(x_hi, 0x3fff, lx,
636 llx & 0xffffffffff000000ULL);
637 x_lo = x - x_hi;
638 d_hi = x_hi * G(i) - 1 + x_lo * G(i);
639 #endif
640 }
641 d_lo = f_lo * G(i);
642
643 /*
644 * This is _2sumF(d_hi, d_lo) inlined. The condition
645 * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not
646 * always satisifed, so it is not clear that this works, but
647 * it works in practice. It works even if it gives a wrong
648 * normalized d_lo, since |d_lo| > |d_hi| implies that i is
649 * nonzero and d is tiny, so the F(i) term dominates d_lo.
650 * In float precision:
651 * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25.
652 * And if d is only a little tinier than that, we would have
653 * another underflow problem for the P3 term; this is also ruled
654 * out by exhaustive testing.)
655 */
656 d = d_hi + d_lo;
657 d_lo = d_hi - d + d_lo;
658 d_hi = d;
659
660 dd = (double)d;
661 val_lo = d * d * d * (P3 +
662 d * (P4 + d * (P5 + d * (P6 + d * (P7 + d * (P8 +
663 dd * (P9 + dd * (P10 + dd * (P11 + dd * (P12 + dd * (P13 +
664 dd * P14))))))))))) + (F_lo(i) + dk * ln2_lo + d_lo) + d * d * P2;
665 val_hi = d_hi;
666 #ifdef DEBUG
667 if (fetestexcept(FE_UNDERFLOW))
668 breakpoint();
669 #endif
670
671 _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi);
672 RETURN2PI(val_hi, val_lo);
673 }
674
675 #ifdef STRUCT_RETURN
676
677 long double
logl(long double x)678 logl(long double x)
679 {
680 struct ld r;
681
682 ENTERI();
683 DOPRINT_START(&x);
684 k_logl(x, &r);
685 RETURNSPI(&r);
686 }
687
688 /*
689 * 29+113 bit decompositions. The bits are distributed so that the products
690 * of the hi terms are exact in double precision. The types are chosen so
691 * that the products of the hi terms are done in at least double precision,
692 * without any explicit conversions. More natural choices would require a
693 * slow long double precision multiplication.
694 */
695 static const double
696 invln10_hi = 4.3429448176175356e-1, /* 0x1bcb7b15000000.0p-54 */
697 invln2_hi = 1.4426950402557850e0; /* 0x17154765000000.0p-52 */
698 static const long double
699 invln10_lo = 1.41498268538580090791605082294397000e-10L, /* 0x137287195355baaafad33dc323ee3.0p-145L */
700 invln2_lo = 6.33178418956604368501892137426645911e-10L; /* 0x15c17f0bbbe87fed0691d3e88eb57.0p-143L */
701
702 long double
log10l(long double x)703 log10l(long double x)
704 {
705 struct ld r;
706 long double lo;
707 float hi;
708
709 ENTERI();
710 DOPRINT_START(&x);
711 k_logl(x, &r);
712 if (!r.lo_set)
713 RETURNPI(r.hi);
714 _2sumF(r.hi, r.lo);
715 hi = r.hi;
716 lo = r.lo + (r.hi - hi);
717 RETURN2PI(invln10_hi * hi,
718 (invln10_lo + invln10_hi) * lo + invln10_lo * hi);
719 }
720
721 long double
log2l(long double x)722 log2l(long double x)
723 {
724 struct ld r;
725 long double lo;
726 float hi;
727
728 ENTERI();
729 DOPRINT_START(&x);
730 k_logl(x, &r);
731 if (!r.lo_set)
732 RETURNPI(r.hi);
733 _2sumF(r.hi, r.lo);
734 hi = r.hi;
735 lo = r.lo + (r.hi - hi);
736 RETURN2PI(invln2_hi * hi,
737 (invln2_lo + invln2_hi) * lo + invln2_lo * hi);
738 }
739
740 #endif /* STRUCT_RETURN */
741