1 /* Data for the log part of pow.
2 Copyright (C) 2018-2023 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19 #include "math_config.h"
20
21 #define N (1 << POW_LOG_TABLE_BITS)
22
23 const struct pow_log_data __pow_log_data = {
24 .ln2hi = 0x1.62e42fefa3800p-1,
25 .ln2lo = 0x1.ef35793c76730p-45,
26 .poly = {
27 #if N == 128 && POW_LOG_POLY_ORDER == 8
28 // relative error: 0x1.11922ap-70
29 // in -0x1.6bp-8 0x1.6bp-8
30 // Coefficients are scaled to match the scaling during evaluation.
31 -0x1p-1,
32 0x1.555555555556p-2 * -2,
33 -0x1.0000000000006p-2 * -2,
34 0x1.999999959554ep-3 * 4,
35 -0x1.555555529a47ap-3 * 4,
36 0x1.2495b9b4845e9p-3 * -8,
37 -0x1.0002b8b263fc3p-3 * -8,
38 #endif
39 },
40 /* Algorithm:
41
42 x = 2^k z
43 log(x) = k ln2 + log(c) + log(z/c)
44 log(z/c) = poly(z/c - 1)
45
46 where z is in [0x1.69555p-1; 0x1.69555p0] which is split into N subintervals
47 and z falls into the ith one, then table entries are computed as
48
49 tab[i].invc = 1/c
50 tab[i].logc = round(0x1p43*log(c))/0x1p43
51 tab[i].logctail = (double)(log(c) - logc)
52
53 where c is chosen near the center of the subinterval such that 1/c has only a
54 few precision bits so z/c - 1 is exactly representible as double:
55
56 1/c = center < 1 ? round(N/center)/N : round(2*N/center)/N/2
57
58 Note: |z/c - 1| < 1/N for the chosen c, |log(c) - logc - logctail| < 0x1p-97,
59 the last few bits of logc are rounded away so k*ln2hi + logc has no rounding
60 error and the interval for z is selected such that near x == 1, where log(x)
61 is tiny, large cancellation error is avoided in logc + poly(z/c - 1). */
62 .tab = {
63 #if N == 128
64 #define A(a,b,c) {a,0,b,c},
65 A(0x1.6a00000000000p+0, -0x1.62c82f2b9c800p-2, 0x1.ab42428375680p-48)
66 A(0x1.6800000000000p+0, -0x1.5d1bdbf580800p-2, -0x1.ca508d8e0f720p-46)
67 A(0x1.6600000000000p+0, -0x1.5767717455800p-2, -0x1.362a4d5b6506dp-45)
68 A(0x1.6400000000000p+0, -0x1.51aad872df800p-2, -0x1.684e49eb067d5p-49)
69 A(0x1.6200000000000p+0, -0x1.4be5f95777800p-2, -0x1.41b6993293ee0p-47)
70 A(0x1.6000000000000p+0, -0x1.4618bc21c6000p-2, 0x1.3d82f484c84ccp-46)
71 A(0x1.5e00000000000p+0, -0x1.404308686a800p-2, 0x1.c42f3ed820b3ap-50)
72 A(0x1.5c00000000000p+0, -0x1.3a64c55694800p-2, 0x1.0b1c686519460p-45)
73 A(0x1.5a00000000000p+0, -0x1.347dd9a988000p-2, 0x1.5594dd4c58092p-45)
74 A(0x1.5800000000000p+0, -0x1.2e8e2bae12000p-2, 0x1.67b1e99b72bd8p-45)
75 A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46)
76 A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46)
77 A(0x1.5400000000000p+0, -0x1.22941fbcf7800p-2, -0x1.65a242853da76p-46)
78 A(0x1.5200000000000p+0, -0x1.1c898c1699800p-2, -0x1.fafbc68e75404p-46)
79 A(0x1.5000000000000p+0, -0x1.1675cababa800p-2, 0x1.f1fc63382a8f0p-46)
80 A(0x1.4e00000000000p+0, -0x1.1058bf9ae4800p-2, -0x1.6a8c4fd055a66p-45)
81 A(0x1.4c00000000000p+0, -0x1.0a324e2739000p-2, -0x1.c6bee7ef4030ep-47)
82 A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48)
83 A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48)
84 A(0x1.4800000000000p+0, -0x1.fb9186d5e4000p-3, 0x1.d572aab993c87p-47)
85 A(0x1.4600000000000p+0, -0x1.ef0adcbdc6000p-3, 0x1.b26b79c86af24p-45)
86 A(0x1.4400000000000p+0, -0x1.e27076e2af000p-3, -0x1.72f4f543fff10p-46)
87 A(0x1.4200000000000p+0, -0x1.d5c216b4fc000p-3, 0x1.1ba91bbca681bp-45)
88 A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45)
89 A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45)
90 A(0x1.3e00000000000p+0, -0x1.bc286742d9000p-3, 0x1.94eb0318bb78fp-46)
91 A(0x1.3c00000000000p+0, -0x1.af3c94e80c000p-3, 0x1.a4e633fcd9066p-52)
92 A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45)
93 A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45)
94 A(0x1.3800000000000p+0, -0x1.9525a9cf45000p-3, -0x1.ad1d904c1d4e3p-45)
95 A(0x1.3600000000000p+0, -0x1.87fa06520d000p-3, 0x1.bbdbf7fdbfa09p-45)
96 A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45)
97 A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45)
98 A(0x1.3200000000000p+0, -0x1.6d60fe719d000p-3, -0x1.0e46aa3b2e266p-46)
99 A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46)
100 A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46)
101 A(0x1.2e00000000000p+0, -0x1.526e5e3a1b000p-3, -0x1.0de8b90075b8fp-45)
102 A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46)
103 A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46)
104 A(0x1.2a00000000000p+0, -0x1.371fc201e9000p-3, 0x1.178864d27543ap-48)
105 A(0x1.2800000000000p+0, -0x1.29552f81ff000p-3, -0x1.48d301771c408p-45)
106 A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45)
107 A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45)
108 A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47)
109 A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47)
110 A(0x1.2200000000000p+0, -0x1.fec9131dbe000p-4, -0x1.575545ca333f2p-45)
111 A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45)
112 A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45)
113 A(0x1.1e00000000000p+0, -0x1.c5e548f5bc000p-4, -0x1.d0c57585fbe06p-46)
114 A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45)
115 A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45)
116 A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46)
117 A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46)
118 A(0x1.1800000000000p+0, -0x1.6f0d28ae56000p-4, -0x1.69737c93373dap-45)
119 A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46)
120 A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46)
121 A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45)
122 A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45)
123 A(0x1.1200000000000p+0, -0x1.16536eea38000p-4, 0x1.47c5e768fa309p-46)
124 A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45)
125 A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45)
126 A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46)
127 A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46)
128 A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45)
129 A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45)
130 A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48)
131 A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48)
132 A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45)
133 A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45)
134 A(0x1.0600000000000p+0, -0x1.7b91b07d58000p-6, -0x1.88d5493faa639p-45)
135 A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50)
136 A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50)
137 A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46)
138 A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46)
139 A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0)
140 A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0)
141 A(0x1.fc00000000000p-1, 0x1.0101575890000p-7, -0x1.0c76b999d2be8p-46)
142 A(0x1.f800000000000p-1, 0x1.0205658938000p-6, -0x1.3dc5b06e2f7d2p-45)
143 A(0x1.f400000000000p-1, 0x1.8492528c90000p-6, -0x1.aa0ba325a0c34p-45)
144 A(0x1.f000000000000p-1, 0x1.0415d89e74000p-5, 0x1.111c05cf1d753p-47)
145 A(0x1.ec00000000000p-1, 0x1.466aed42e0000p-5, -0x1.c167375bdfd28p-45)
146 A(0x1.e800000000000p-1, 0x1.894aa149fc000p-5, -0x1.97995d05a267dp-46)
147 A(0x1.e400000000000p-1, 0x1.ccb73cdddc000p-5, -0x1.a68f247d82807p-46)
148 A(0x1.e200000000000p-1, 0x1.eea31c006c000p-5, -0x1.e113e4fc93b7bp-47)
149 A(0x1.de00000000000p-1, 0x1.1973bd1466000p-4, -0x1.5325d560d9e9bp-45)
150 A(0x1.da00000000000p-1, 0x1.3bdf5a7d1e000p-4, 0x1.cc85ea5db4ed7p-45)
151 A(0x1.d600000000000p-1, 0x1.5e95a4d97a000p-4, -0x1.c69063c5d1d1ep-45)
152 A(0x1.d400000000000p-1, 0x1.700d30aeac000p-4, 0x1.c1e8da99ded32p-49)
153 A(0x1.d000000000000p-1, 0x1.9335e5d594000p-4, 0x1.3115c3abd47dap-45)
154 A(0x1.cc00000000000p-1, 0x1.b6ac88dad6000p-4, -0x1.390802bf768e5p-46)
155 A(0x1.ca00000000000p-1, 0x1.c885801bc4000p-4, 0x1.646d1c65aacd3p-45)
156 A(0x1.c600000000000p-1, 0x1.ec739830a2000p-4, -0x1.dc068afe645e0p-45)
157 A(0x1.c400000000000p-1, 0x1.fe89139dbe000p-4, -0x1.534d64fa10afdp-45)
158 A(0x1.c000000000000p-1, 0x1.1178e8227e000p-3, 0x1.1ef78ce2d07f2p-45)
159 A(0x1.be00000000000p-1, 0x1.1aa2b7e23f000p-3, 0x1.ca78e44389934p-45)
160 A(0x1.ba00000000000p-1, 0x1.2d1610c868000p-3, 0x1.39d6ccb81b4a1p-47)
161 A(0x1.b800000000000p-1, 0x1.365fcb0159000p-3, 0x1.62fa8234b7289p-51)
162 A(0x1.b400000000000p-1, 0x1.4913d8333b000p-3, 0x1.5837954fdb678p-45)
163 A(0x1.b200000000000p-1, 0x1.527e5e4a1b000p-3, 0x1.633e8e5697dc7p-45)
164 A(0x1.ae00000000000p-1, 0x1.6574ebe8c1000p-3, 0x1.9cf8b2c3c2e78p-46)
165 A(0x1.ac00000000000p-1, 0x1.6f0128b757000p-3, -0x1.5118de59c21e1p-45)
166 A(0x1.aa00000000000p-1, 0x1.7898d85445000p-3, -0x1.c661070914305p-46)
167 A(0x1.a600000000000p-1, 0x1.8beafeb390000p-3, -0x1.73d54aae92cd1p-47)
168 A(0x1.a400000000000p-1, 0x1.95a5adcf70000p-3, 0x1.7f22858a0ff6fp-47)
169 A(0x1.a000000000000p-1, 0x1.a93ed3c8ae000p-3, -0x1.8724350562169p-45)
170 A(0x1.9e00000000000p-1, 0x1.b31d8575bd000p-3, -0x1.c358d4eace1aap-47)
171 A(0x1.9c00000000000p-1, 0x1.bd087383be000p-3, -0x1.d4bc4595412b6p-45)
172 A(0x1.9a00000000000p-1, 0x1.c6ffbc6f01000p-3, -0x1.1ec72c5962bd2p-48)
173 A(0x1.9600000000000p-1, 0x1.db13db0d49000p-3, -0x1.aff2af715b035p-45)
174 A(0x1.9400000000000p-1, 0x1.e530effe71000p-3, 0x1.212276041f430p-51)
175 A(0x1.9200000000000p-1, 0x1.ef5ade4dd0000p-3, -0x1.a211565bb8e11p-51)
176 A(0x1.9000000000000p-1, 0x1.f991c6cb3b000p-3, 0x1.bcbecca0cdf30p-46)
177 A(0x1.8c00000000000p-1, 0x1.07138604d5800p-2, 0x1.89cdb16ed4e91p-48)
178 A(0x1.8a00000000000p-1, 0x1.0c42d67616000p-2, 0x1.7188b163ceae9p-45)
179 A(0x1.8800000000000p-1, 0x1.1178e8227e800p-2, -0x1.c210e63a5f01cp-45)
180 A(0x1.8600000000000p-1, 0x1.16b5ccbacf800p-2, 0x1.b9acdf7a51681p-45)
181 A(0x1.8400000000000p-1, 0x1.1bf99635a6800p-2, 0x1.ca6ed5147bdb7p-45)
182 A(0x1.8200000000000p-1, 0x1.214456d0eb800p-2, 0x1.a87deba46baeap-47)
183 A(0x1.7e00000000000p-1, 0x1.2bef07cdc9000p-2, 0x1.a9cfa4a5004f4p-45)
184 A(0x1.7c00000000000p-1, 0x1.314f1e1d36000p-2, -0x1.8e27ad3213cb8p-45)
185 A(0x1.7a00000000000p-1, 0x1.36b6776be1000p-2, 0x1.16ecdb0f177c8p-46)
186 A(0x1.7800000000000p-1, 0x1.3c25277333000p-2, 0x1.83b54b606bd5cp-46)
187 A(0x1.7600000000000p-1, 0x1.419b423d5e800p-2, 0x1.8e436ec90e09dp-47)
188 A(0x1.7400000000000p-1, 0x1.4718dc271c800p-2, -0x1.f27ce0967d675p-45)
189 A(0x1.7200000000000p-1, 0x1.4c9e09e173000p-2, -0x1.e20891b0ad8a4p-45)
190 A(0x1.7000000000000p-1, 0x1.522ae0738a000p-2, 0x1.ebe708164c759p-45)
191 A(0x1.6e00000000000p-1, 0x1.57bf753c8d000p-2, 0x1.fadedee5d40efp-46)
192 A(0x1.6c00000000000p-1, 0x1.5d5bddf596000p-2, -0x1.a0b2a08a465dcp-47)
193 #endif
194 },
195 };