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1
2/*
3 * logf.s
4 *
5 * by Ian Ollmann
6 *
7 * Copyright (c) 2007, Apple Inc. All rights reserved
8 *
9 * Single precision implementation of C99 math library function logf and log1pf.
10 * Accurate to within 0.51 ulps over entire number range.
11 *
12 */
13
14
15#include <machine/asm.h>
16#include "abi.h"
17
18.const
19.align 4
20
21// 511 entry of Lookup table of values used for log and log1p calculation, generated as:
22//
23/*
24 #include <stdint.h>
25 #include <stdio.h>
26 #include <math.h>
27
28 int main (void)
29 {
30 int i;
31
32 for( i = -255; i < 256; i++ )
33 {
34 double d = (double) i / 256.0;
35 union
36 {
37 double d;
38 uint64_t u;
39 }u, u2;
40
41 u.d = (double) log1pl( (long double) d );
42 u2.d = 1.0 / (d+1.0);
43
44 printf( ".quad\t0x%llx,\t0x%llx\t// log1p(%7.5f), 1/%7.5f\n", u.u, u2.u, d, d+1.0 );
45 }
46
47 return 0;
48 }
49*/
50// logf only uses the entries starting at log_m_table. log1pf uses the full range.
51
52 .quad 0xbfe62e42fefa39ef, 0x4000000000000000 // log1p(-0.50000), 1/0.50000
53 .quad 0xbfe5ee82aa241920, 0x3fffc07f01fc07f0 // log1p(-0.49609), 1/0.50391
54 .quad 0xbfe5af405c3649e0, 0x3fff81f81f81f820 // log1p(-0.49219), 1/0.50781
55 .quad 0xbfe5707a26bb8c66, 0x3fff44659e4a4271 // log1p(-0.48828), 1/0.51172
56 .quad 0xbfe5322e26867857, 0x3fff07c1f07c1f08 // log1p(-0.48438), 1/0.51562
57 .quad 0xbfe4f45a835a4e19, 0x3ffecc07b301ecc0 // log1p(-0.48047), 1/0.51953
58 .quad 0xbfe4b6fd6f970c1f, 0x3ffe9131abf0b767 // log1p(-0.47656), 1/0.52344
59 .quad 0xbfe47a1527e8a2d3, 0x3ffe573ac901e574 // log1p(-0.47266), 1/0.52734
60 .quad 0xbfe43d9ff2f923c5, 0x3ffe1e1e1e1e1e1e // log1p(-0.46875), 1/0.53125
61 .quad 0xbfe4019c2125ca93, 0x3ffde5d6e3f8868a // log1p(-0.46484), 1/0.53516
62 .quad 0xbfe3c6080c36bfb5, 0x3ffdae6076b981db // log1p(-0.46094), 1/0.53906
63 .quad 0xbfe38ae2171976e7, 0x3ffd77b654b82c34 // log1p(-0.45703), 1/0.54297
64 .quad 0xbfe35028ad9d8c86, 0x3ffd41d41d41d41d // log1p(-0.45312), 1/0.54688
65 .quad 0xbfe315da4434068b, 0x3ffd0cb58f6ec074 // log1p(-0.44922), 1/0.55078
66 .quad 0xbfe2dbf557b0df43, 0x3ffcd85689039b0b // log1p(-0.44531), 1/0.55469
67 .quad 0xbfe2a2786d0ec107, 0x3ffca4b3055ee191 // log1p(-0.44141), 1/0.55859
68 .quad 0xbfe269621134db92, 0x3ffc71c71c71c71c // log1p(-0.43750), 1/0.56250
69 .quad 0xbfe230b0d8bebc98, 0x3ffc3f8f01c3f8f0 // log1p(-0.43359), 1/0.56641
70 .quad 0xbfe1f8635fc61659, 0x3ffc0e070381c0e0 // log1p(-0.42969), 1/0.57031
71 .quad 0xbfe1c07849ae6007, 0x3ffbdd2b899406f7 // log1p(-0.42578), 1/0.57422
72 .quad 0xbfe188ee40f23ca6, 0x3ffbacf914c1bad0 // log1p(-0.42188), 1/0.57812
73 .quad 0xbfe151c3f6f29612, 0x3ffb7d6c3dda338b // log1p(-0.41797), 1/0.58203
74 .quad 0xbfe11af823c75aa8, 0x3ffb4e81b4e81b4f // log1p(-0.41406), 1/0.58594
75 .quad 0xbfe0e4898611cce1, 0x3ffb2036406c80d9 // log1p(-0.41016), 1/0.58984
76 .quad 0xbfe0ae76e2d054fa, 0x3ffaf286bca1af28 // log1p(-0.40625), 1/0.59375
77 .quad 0xbfe078bf0533c568, 0x3ffac5701ac5701b // log1p(-0.40234), 1/0.59766
78 .quad 0xbfe04360be7603ad, 0x3ffa98ef606a63be // log1p(-0.39844), 1/0.60156
79 .quad 0xbfe00e5ae5b207ab, 0x3ffa6d01a6d01a6d // log1p(-0.39453), 1/0.60547
80 .quad 0xbfdfb358af7a4884, 0x3ffa41a41a41a41a // log1p(-0.39062), 1/0.60938
81 .quad 0xbfdf4aa7ee03192d, 0x3ffa16d3f97a4b02 // log1p(-0.38672), 1/0.61328
82 .quad 0xbfdee2a156b413e5, 0x3ff9ec8e951033d9 // log1p(-0.38281), 1/0.61719
83 .quad 0xbfde7b42c3ddad73, 0x3ff9c2d14ee4a102 // log1p(-0.37891), 1/0.62109
84 .quad 0xbfde148a1a2726ce, 0x3ff999999999999a // log1p(-0.37500), 1/0.62500
85 .quad 0xbfddae75484c9616, 0x3ff970e4f80cb872 // log1p(-0.37109), 1/0.62891
86 .quad 0xbfdd490246defa6b, 0x3ff948b0fcd6e9e0 // log1p(-0.36719), 1/0.63281
87 .quad 0xbfdce42f18064743, 0x3ff920fb49d0e229 // log1p(-0.36328), 1/0.63672
88 .quad 0xbfdc7ff9c74554c9, 0x3ff8f9c18f9c18fa // log1p(-0.35938), 1/0.64062
89 .quad 0xbfdc1c60693fa39e, 0x3ff8d3018d3018d3 // log1p(-0.35547), 1/0.64453
90 .quad 0xbfdbb9611b80e2fb, 0x3ff8acb90f6bf3aa // log1p(-0.35156), 1/0.64844
91 .quad 0xbfdb56fa04462909, 0x3ff886e5f0abb04a // log1p(-0.34766), 1/0.65234
92 .quad 0xbfdaf5295248cdd0, 0x3ff8618618618618 // log1p(-0.34375), 1/0.65625
93 .quad 0xbfda93ed3c8ad9e3, 0x3ff83c977ab2bedd // log1p(-0.33984), 1/0.66016
94 .quad 0xbfda33440224fa79, 0x3ff8181818181818 // log1p(-0.33594), 1/0.66406
95 .quad 0xbfd9d32bea15ed3b, 0x3ff7f405fd017f40 // log1p(-0.33203), 1/0.66797
96 .quad 0xbfd973a3431356ae, 0x3ff7d05f417d05f4 // log1p(-0.32812), 1/0.67188
97 .quad 0xbfd914a8635bf68a, 0x3ff7ad2208e0ecc3 // log1p(-0.32422), 1/0.67578
98 .quad 0xbfd8b639a88b2df5, 0x3ff78a4c8178a4c8 // log1p(-0.32031), 1/0.67969
99 .quad 0xbfd85855776dcbfb, 0x3ff767dce434a9b1 // log1p(-0.31641), 1/0.68359
100 .quad 0xbfd7fafa3bd8151c, 0x3ff745d1745d1746 // log1p(-0.31250), 1/0.68750
101 .quad 0xbfd79e26687cfb3e, 0x3ff724287f46debc // log1p(-0.30859), 1/0.69141
102 .quad 0xbfd741d876c67bb1, 0x3ff702e05c0b8170 // log1p(-0.30469), 1/0.69531
103 .quad 0xbfd6e60ee6af1972, 0x3ff6e1f76b4337c7 // log1p(-0.30078), 1/0.69922
104 .quad 0xbfd68ac83e9c6a14, 0x3ff6c16c16c16c17 // log1p(-0.29688), 1/0.70312
105 .quad 0xbfd630030b3aac49, 0x3ff6a13cd1537290 // log1p(-0.29297), 1/0.70703
106 .quad 0xbfd5d5bddf595f30, 0x3ff6816816816817 // log1p(-0.28906), 1/0.71094
107 .quad 0xbfd57bf753c8d1fb, 0x3ff661ec6a5122f9 // log1p(-0.28516), 1/0.71484
108 .quad 0xbfd522ae0738a3d8, 0x3ff642c8590b2164 // log1p(-0.28125), 1/0.71875
109 .quad 0xbfd4c9e09e172c3c, 0x3ff623fa77016240 // log1p(-0.27734), 1/0.72266
110 .quad 0xbfd4718dc271c41b, 0x3ff6058160581606 // log1p(-0.27344), 1/0.72656
111 .quad 0xbfd419b423d5e8c7, 0x3ff5e75bb8d015e7 // log1p(-0.26953), 1/0.73047
112 .quad 0xbfd3c25277333184, 0x3ff5c9882b931057 // log1p(-0.26562), 1/0.73438
113 .quad 0xbfd36b6776be1117, 0x3ff5ac056b015ac0 // log1p(-0.26172), 1/0.73828
114 .quad 0xbfd314f1e1d35ce4, 0x3ff58ed2308158ed // log1p(-0.25781), 1/0.74219
115 .quad 0xbfd2bef07cdc9354, 0x3ff571ed3c506b3a // log1p(-0.25391), 1/0.74609
116 .quad 0xbfd269621134db92, 0x3ff5555555555555 // log1p(-0.25000), 1/0.75000
117 .quad 0xbfd214456d0eb8d4, 0x3ff5390948f40feb // log1p(-0.24609), 1/0.75391
118 .quad 0xbfd1bf99635a6b95, 0x3ff51d07eae2f815 // log1p(-0.24219), 1/0.75781
119 .quad 0xbfd16b5ccbacfb73, 0x3ff5015015015015 // log1p(-0.23828), 1/0.76172
120 .quad 0xbfd1178e8227e47c, 0x3ff4e5e0a72f0539 // log1p(-0.23438), 1/0.76562
121 .quad 0xbfd0c42d676162e3, 0x3ff4cab88725af6e // log1p(-0.23047), 1/0.76953
122 .quad 0xbfd07138604d5862, 0x3ff4afd6a052bf5b // log1p(-0.22656), 1/0.77344
123 .quad 0xbfd01eae5626c691, 0x3ff49539e3b2d067 // log1p(-0.22266), 1/0.77734
124 .quad 0xbfcf991c6cb3b379, 0x3ff47ae147ae147b // log1p(-0.21875), 1/0.78125
125 .quad 0xbfcef5ade4dcffe6, 0x3ff460cbc7f5cf9a // log1p(-0.21484), 1/0.78516
126 .quad 0xbfce530effe71012, 0x3ff446f86562d9fb // log1p(-0.21094), 1/0.78906
127 .quad 0xbfcdb13db0d48940, 0x3ff42d6625d51f87 // log1p(-0.20703), 1/0.79297
128 .quad 0xbfcd1037f2655e7b, 0x3ff4141414141414 // log1p(-0.20312), 1/0.79688
129 .quad 0xbfcc6ffbc6f00f71, 0x3ff3fb013fb013fb // log1p(-0.19922), 1/0.80078
130 .quad 0xbfcbd087383bd8ad, 0x3ff3e22cbce4a902 // log1p(-0.19531), 1/0.80469
131 .quad 0xbfcb31d8575bce3d, 0x3ff3c995a47babe7 // log1p(-0.19141), 1/0.80859
132 .quad 0xbfca93ed3c8ad9e3, 0x3ff3b13b13b13b14 // log1p(-0.18750), 1/0.81250
133 .quad 0xbfc9f6c407089664, 0x3ff3991c2c187f63 // log1p(-0.18359), 1/0.81641
134 .quad 0xbfc95a5adcf7017f, 0x3ff3813813813814 // log1p(-0.17969), 1/0.82031
135 .quad 0xbfc8beafeb38fe8c, 0x3ff3698df3de0748 // log1p(-0.17578), 1/0.82422
136 .quad 0xbfc823c16551a3c2, 0x3ff3521cfb2b78c1 // log1p(-0.17188), 1/0.82812
137 .quad 0xbfc7898d85444c73, 0x3ff33ae45b57bcb2 // log1p(-0.16797), 1/0.83203
138 .quad 0xbfc6f0128b756abc, 0x3ff323e34a2b10bf // log1p(-0.16406), 1/0.83594
139 .quad 0xbfc6574ebe8c133a, 0x3ff30d190130d190 // log1p(-0.16016), 1/0.83984
140 .quad 0xbfc5bf406b543db2, 0x3ff2f684bda12f68 // log1p(-0.15625), 1/0.84375
141 .quad 0xbfc527e5e4a1b58d, 0x3ff2e025c04b8097 // log1p(-0.15234), 1/0.84766
142 .quad 0xbfc4913d8333b561, 0x3ff2c9fb4d812ca0 // log1p(-0.14844), 1/0.85156
143 .quad 0xbfc3fb45a59928cc, 0x3ff2b404ad012b40 // log1p(-0.14453), 1/0.85547
144 .quad 0xbfc365fcb0159016, 0x3ff29e4129e4129e // log1p(-0.14062), 1/0.85938
145 .quad 0xbfc2d1610c86813a, 0x3ff288b01288b013 // log1p(-0.13672), 1/0.86328
146 .quad 0xbfc23d712a49c202, 0x3ff27350b8812735 // log1p(-0.13281), 1/0.86719
147 .quad 0xbfc1aa2b7e23f72a, 0x3ff25e22708092f1 // log1p(-0.12891), 1/0.87109
148 .quad 0xbfc1178e8227e47c, 0x3ff2492492492492 // log1p(-0.12500), 1/0.87500
149 .quad 0xbfc08598b59e3a07, 0x3ff23456789abcdf // log1p(-0.12109), 1/0.87891
150 .quad 0xbfbfe89139dbd566, 0x3ff21fb78121fb78 // log1p(-0.11719), 1/0.88281
151 .quad 0xbfbec739830a1120, 0x3ff20b470c67c0d9 // log1p(-0.11328), 1/0.88672
152 .quad 0xbfbda727638446a2, 0x3ff1f7047dc11f70 // log1p(-0.10938), 1/0.89062
153 .quad 0xbfbc885801bc4b23, 0x3ff1e2ef3b3fb874 // log1p(-0.10547), 1/0.89453
154 .quad 0xbfbb6ac88dad5b1c, 0x3ff1cf06ada2811d // log1p(-0.10156), 1/0.89844
155 .quad 0xbfba4e7640b1bc38, 0x3ff1bb4a4046ed29 // log1p(-0.09766), 1/0.90234
156 .quad 0xbfb9335e5d594989, 0x3ff1a7b9611a7b96 // log1p(-0.09375), 1/0.90625
157 .quad 0xbfb8197e2f40e3f0, 0x3ff19453808ca29c // log1p(-0.08984), 1/0.91016
158 .quad 0xbfb700d30aeac0e1, 0x3ff1811811811812 // log1p(-0.08594), 1/0.91406
159 .quad 0xbfb5e95a4d9791cb, 0x3ff16e0689427379 // log1p(-0.08203), 1/0.91797
160 .quad 0xbfb4d3115d207eac, 0x3ff15b1e5f75270d // log1p(-0.07812), 1/0.92188
161 .quad 0xbfb3bdf5a7d1ee64, 0x3ff1485f0e0acd3b // log1p(-0.07422), 1/0.92578
162 .quad 0xbfb2aa04a44717a5, 0x3ff135c81135c811 // log1p(-0.07031), 1/0.92969
163 .quad 0xbfb1973bd1465567, 0x3ff12358e75d3033 // log1p(-0.06641), 1/0.93359
164 .quad 0xbfb08598b59e3a07, 0x3ff1111111111111 // log1p(-0.06250), 1/0.93750
165 .quad 0xbfaeea31c006b87c, 0x3ff0fef010fef011 // log1p(-0.05859), 1/0.94141
166 .quad 0xbfaccb73cdddb2cc, 0x3ff0ecf56be69c90 // log1p(-0.05469), 1/0.94531
167 .quad 0xbfaaaef2d0fb10fc, 0x3ff0db20a88f4696 // log1p(-0.05078), 1/0.94922
168 .quad 0xbfa894aa149fb343, 0x3ff0c9714fbcda3b // log1p(-0.04688), 1/0.95312
169 .quad 0xbfa67c94f2d4bb58, 0x3ff0b7e6ec259dc8 // log1p(-0.04297), 1/0.95703
170 .quad 0xbfa466aed42de3ea, 0x3ff0a6810a6810a7 // log1p(-0.03906), 1/0.96094
171 .quad 0xbfa252f32f8d183f, 0x3ff0953f39010954 // log1p(-0.03516), 1/0.96484
172 .quad 0xbfa0415d89e74444, 0x3ff0842108421084 // log1p(-0.03125), 1/0.96875
173 .quad 0xbf9c63d2ec14aaf2, 0x3ff073260a47f7c6 // log1p(-0.02734), 1/0.97266
174 .quad 0xbf98492528c8cabf, 0x3ff0624dd2f1a9fc // log1p(-0.02344), 1/0.97656
175 .quad 0xbf9432a925980cc1, 0x3ff05197f7d73404 // log1p(-0.01953), 1/0.98047
176 .quad 0xbf90205658935847, 0x3ff0410410410410 // log1p(-0.01562), 1/0.98438
177 .quad 0xbf882448a388a2aa, 0x3ff03091b51f5e1a // log1p(-0.01172), 1/0.98828
178 .quad 0xbf8010157588de71, 0x3ff0204081020408 // log1p(-0.00781), 1/0.99219
179 .quad 0xbf70080559588b35, 0x3ff0101010101010 // log1p(-0.00391), 1/0.99609
180log_m_table: .quad 0x0000000000000000, 0x3ff0000000000000 // log1p(0.00000), 1/1.00000
181 .quad 0x3f6ff00aa2b10bc0, 0x3fefe01fe01fe020 // log1p(0.00391), 1/1.00391
182 .quad 0x3f7fe02a6b106789, 0x3fefc07f01fc07f0 // log1p(0.00781), 1/1.00781
183 .quad 0x3f87dc475f810a77, 0x3fefa11caa01fa12 // log1p(0.01172), 1/1.01172
184 .quad 0x3f8fc0a8b0fc03e4, 0x3fef81f81f81f820 // log1p(0.01562), 1/1.01562
185 .quad 0x3f93cea44346a575, 0x3fef6310aca0dbb5 // log1p(0.01953), 1/1.01953
186 .quad 0x3f97b91b07d5b11b, 0x3fef44659e4a4271 // log1p(0.02344), 1/1.02344
187 .quad 0x3f9b9fc027af9198, 0x3fef25f644230ab5 // log1p(0.02734), 1/1.02734
188 .quad 0x3f9f829b0e783300, 0x3fef07c1f07c1f08 // log1p(0.03125), 1/1.03125
189 .quad 0x3fa1b0d98923d980, 0x3feee9c7f8458e02 // log1p(0.03516), 1/1.03516
190 .quad 0x3fa39e87b9febd60, 0x3feecc07b301ecc0 // log1p(0.03906), 1/1.03906
191 .quad 0x3fa58a5bafc8e4d5, 0x3feeae807aba01eb // log1p(0.04297), 1/1.04297
192 .quad 0x3fa77458f632dcfc, 0x3fee9131abf0b767 // log1p(0.04688), 1/1.04688
193 .quad 0x3fa95c830ec8e3eb, 0x3fee741aa59750e4 // log1p(0.05078), 1/1.05078
194 .quad 0x3fab42dd711971bf, 0x3fee573ac901e574 // log1p(0.05469), 1/1.05469
195 .quad 0x3fad276b8adb0b52, 0x3fee3a9179dc1a73 // log1p(0.05859), 1/1.05859
196 .quad 0x3faf0a30c01162a6, 0x3fee1e1e1e1e1e1e // log1p(0.06250), 1/1.06250
197 .quad 0x3fb075983598e471, 0x3fee01e01e01e01e // log1p(0.06641), 1/1.06641
198 .quad 0x3fb16536eea37ae1, 0x3fede5d6e3f8868a // log1p(0.07031), 1/1.07031
199 .quad 0x3fb253f62f0a1417, 0x3fedca01dca01dca // log1p(0.07422), 1/1.07422
200 .quad 0x3fb341d7961bd1d1, 0x3fedae6076b981db // log1p(0.07812), 1/1.07812
201 .quad 0x3fb42edcbea646f0, 0x3fed92f2231e7f8a // log1p(0.08203), 1/1.08203
202 .quad 0x3fb51b073f06183f, 0x3fed77b654b82c34 // log1p(0.08594), 1/1.08594
203 .quad 0x3fb60658a93750c4, 0x3fed5cac807572b2 // log1p(0.08984), 1/1.08984
204 .quad 0x3fb6f0d28ae56b4c, 0x3fed41d41d41d41d // log1p(0.09375), 1/1.09375
205 .quad 0x3fb7da766d7b12cd, 0x3fed272ca3fc5b1a // log1p(0.09766), 1/1.09766
206 .quad 0x3fb8c345d6319b21, 0x3fed0cb58f6ec074 // log1p(0.10156), 1/1.10156
207 .quad 0x3fb9ab42462033ad, 0x3fecf26e5c44bfc6 // log1p(0.10547), 1/1.10547
208 .quad 0x3fba926d3a4ad563, 0x3fecd85689039b0b // log1p(0.10938), 1/1.10938
209 .quad 0x3fbb78c82bb0eda1, 0x3fecbe6d9601cbe7 // log1p(0.11328), 1/1.11328
210 .quad 0x3fbc5e548f5bc743, 0x3feca4b3055ee191 // log1p(0.11719), 1/1.11719
211 .quad 0x3fbd4313d66cb35d, 0x3fec8b265afb8a42 // log1p(0.12109), 1/1.12109
212 .quad 0x3fbe27076e2af2e6, 0x3fec71c71c71c71c // log1p(0.12500), 1/1.12500
213 .quad 0x3fbf0a30c01162a6, 0x3fec5894d10d4986 // log1p(0.12891), 1/1.12891
214 .quad 0x3fbfec9131dbeabb, 0x3fec3f8f01c3f8f0 // log1p(0.13281), 1/1.13281
215 .quad 0x3fc0671512ca596e, 0x3fec26b5392ea01c // log1p(0.13672), 1/1.13672
216 .quad 0x3fc0d77e7cd08e59, 0x3fec0e070381c0e0 // log1p(0.14062), 1/1.14062
217 .quad 0x3fc14785846742ac, 0x3febf583ee868d8b // log1p(0.14453), 1/1.14453
218 .quad 0x3fc1b72ad52f67a0, 0x3febdd2b899406f7 // log1p(0.14844), 1/1.14844
219 .quad 0x3fc2266f190a5acb, 0x3febc4fd65883e7b // log1p(0.15234), 1/1.15234
220 .quad 0x3fc29552f81ff523, 0x3febacf914c1bad0 // log1p(0.15625), 1/1.15625
221 .quad 0x3fc303d718e47fd3, 0x3feb951e2b18ff23 // log1p(0.16016), 1/1.16016
222 .quad 0x3fc371fc201e8f74, 0x3feb7d6c3dda338b // log1p(0.16406), 1/1.16406
223 .quad 0x3fc3dfc2b0ecc62a, 0x3feb65e2e3beee05 // log1p(0.16797), 1/1.16797
224 .quad 0x3fc44d2b6ccb7d1e, 0x3feb4e81b4e81b4f // log1p(0.17188), 1/1.17188
225 .quad 0x3fc4ba36f39a55e5, 0x3feb37484ad806ce // log1p(0.17578), 1/1.17578
226 .quad 0x3fc526e5e3a1b438, 0x3feb2036406c80d9 // log1p(0.17969), 1/1.17969
227 .quad 0x3fc59338d9982086, 0x3feb094b31d922a4 // log1p(0.18359), 1/1.18359
228 .quad 0x3fc5ff3070a793d4, 0x3feaf286bca1af28 // log1p(0.18750), 1/1.18750
229 .quad 0x3fc66acd4272ad51, 0x3feadbe87f94905e // log1p(0.19141), 1/1.19141
230 .quad 0x3fc6d60fe719d21d, 0x3feac5701ac5701b // log1p(0.19531), 1/1.19531
231 .quad 0x3fc740f8f54037a5, 0x3feaaf1d2f87ebfd // log1p(0.19922), 1/1.19922
232 .quad 0x3fc7ab890210d909, 0x3fea98ef606a63be // log1p(0.20312), 1/1.20312
233 .quad 0x3fc815c0a14357eb, 0x3fea82e65130e159 // log1p(0.20703), 1/1.20703
234 .quad 0x3fc87fa06520c911, 0x3fea6d01a6d01a6d // log1p(0.21094), 1/1.21094
235 .quad 0x3fc8e928de886d41, 0x3fea574107688a4a // log1p(0.21484), 1/1.21484
236 .quad 0x3fc9525a9cf456b4, 0x3fea41a41a41a41a // log1p(0.21875), 1/1.21875
237 .quad 0x3fc9bb362e7dfb83, 0x3fea2c2a87c51ca0 // log1p(0.22266), 1/1.22266
238 .quad 0x3fca23bc1fe2b563, 0x3fea16d3f97a4b02 // log1p(0.22656), 1/1.22656
239 .quad 0x3fca8becfc882f19, 0x3fea01a01a01a01a // log1p(0.23047), 1/1.23047
240 .quad 0x3fcaf3c94e80bff3, 0x3fe9ec8e951033d9 // log1p(0.23438), 1/1.23438
241 .quad 0x3fcb5b519e8fb5a4, 0x3fe9d79f176b682d // log1p(0.23828), 1/1.23828
242 .quad 0x3fcbc286742d8cd6, 0x3fe9c2d14ee4a102 // log1p(0.24219), 1/1.24219
243 .quad 0x3fcc2968558c18c1, 0x3fe9ae24ea5510da // log1p(0.24609), 1/1.24609
244 .quad 0x3fcc8ff7c79a9a22, 0x3fe999999999999a // log1p(0.25000), 1/1.25000
245 .quad 0x3fccf6354e09c5dc, 0x3fe9852f0d8ec0ff // log1p(0.25391), 1/1.25391
246 .quad 0x3fcd5c216b4fbb91, 0x3fe970e4f80cb872 // log1p(0.25781), 1/1.25781
247 .quad 0x3fcdc1bca0abec7d, 0x3fe95cbb0be377ae // log1p(0.26172), 1/1.26172
248 .quad 0x3fce27076e2af2e6, 0x3fe948b0fcd6e9e0 // log1p(0.26562), 1/1.26562
249 .quad 0x3fce8c0252aa5a60, 0x3fe934c67f9b2ce6 // log1p(0.26953), 1/1.26953
250 .quad 0x3fcef0adcbdc5936, 0x3fe920fb49d0e229 // log1p(0.27344), 1/1.27344
251 .quad 0x3fcf550a564b7b37, 0x3fe90d4f120190d5 // log1p(0.27734), 1/1.27734
252 .quad 0x3fcfb9186d5e3e2b, 0x3fe8f9c18f9c18fa // log1p(0.28125), 1/1.28125
253 .quad 0x3fd00e6c45ad501d, 0x3fe8e6527af1373f // log1p(0.28516), 1/1.28516
254 .quad 0x3fd0402594b4d041, 0x3fe8d3018d3018d3 // log1p(0.28906), 1/1.28906
255 .quad 0x3fd071b85fcd590d, 0x3fe8bfce8062ff3a // log1p(0.29297), 1/1.29297
256 .quad 0x3fd0a324e27390e3, 0x3fe8acb90f6bf3aa // log1p(0.29688), 1/1.29688
257 .quad 0x3fd0d46b579ab74b, 0x3fe899c0f601899c // log1p(0.30078), 1/1.30078
258 .quad 0x3fd1058bf9ae4ad5, 0x3fe886e5f0abb04a // log1p(0.30469), 1/1.30469
259 .quad 0x3fd136870293a8b0, 0x3fe87427bcc092b9 // log1p(0.30859), 1/1.30859
260 .quad 0x3fd1675cababa60e, 0x3fe8618618618618 // log1p(0.31250), 1/1.31250
261 .quad 0x3fd1980d2dd4236f, 0x3fe84f00c2780614 // log1p(0.31641), 1/1.31641
262 .quad 0x3fd1c898c16999fb, 0x3fe83c977ab2bedd // log1p(0.32031), 1/1.32031
263 .quad 0x3fd1f8ff9e48a2f3, 0x3fe82a4a0182a4a0 // log1p(0.32422), 1/1.32422
264 .quad 0x3fd22941fbcf7966, 0x3fe8181818181818 // log1p(0.32812), 1/1.32812
265 .quad 0x3fd2596010df763a, 0x3fe8060180601806 // log1p(0.33203), 1/1.33203
266 .quad 0x3fd2895a13de86a3, 0x3fe7f405fd017f40 // log1p(0.33594), 1/1.33594
267 .quad 0x3fd2b9303ab89d25, 0x3fe7e225515a4f1d // log1p(0.33984), 1/1.33984
268 .quad 0x3fd2e8e2bae11d31, 0x3fe7d05f417d05f4 // log1p(0.34375), 1/1.34375
269 .quad 0x3fd31871c9544185, 0x3fe7beb3922e017c // log1p(0.34766), 1/1.34766
270 .quad 0x3fd347dd9a987d55, 0x3fe7ad2208e0ecc3 // log1p(0.35156), 1/1.35156
271 .quad 0x3fd3772662bfd85b, 0x3fe79baa6bb6398b // log1p(0.35547), 1/1.35547
272 .quad 0x3fd3a64c556945ea, 0x3fe78a4c8178a4c8 // log1p(0.35938), 1/1.35938
273 .quad 0x3fd3d54fa5c1f710, 0x3fe77908119ac60d // log1p(0.36328), 1/1.36328
274 .quad 0x3fd404308686a7e4, 0x3fe767dce434a9b1 // log1p(0.36719), 1/1.36719
275 .quad 0x3fd432ef2a04e814, 0x3fe756cac201756d // log1p(0.37109), 1/1.37109
276 .quad 0x3fd4618bc21c5ec2, 0x3fe745d1745d1746 // log1p(0.37500), 1/1.37500
277 .quad 0x3fd49006804009d1, 0x3fe734f0c541fe8d // log1p(0.37891), 1/1.37891
278 .quad 0x3fd4be5f957778a1, 0x3fe724287f46debc // log1p(0.38281), 1/1.38281
279 .quad 0x3fd4ec973260026a, 0x3fe713786d9c7c09 // log1p(0.38672), 1/1.38672
280 .quad 0x3fd51aad872df82d, 0x3fe702e05c0b8170 // log1p(0.39062), 1/1.39062
281 .quad 0x3fd548a2c3add263, 0x3fe6f26016f26017 // log1p(0.39453), 1/1.39453
282 .quad 0x3fd5767717455a6c, 0x3fe6e1f76b4337c7 // log1p(0.39844), 1/1.39844
283 .quad 0x3fd5a42ab0f4cfe2, 0x3fe6d1a62681c861 // log1p(0.40234), 1/1.40234
284 .quad 0x3fd5d1bdbf5809ca, 0x3fe6c16c16c16c17 // log1p(0.40625), 1/1.40625
285 .quad 0x3fd5ff3070a793d4, 0x3fe6b1490aa31a3d // log1p(0.41016), 1/1.41016
286 .quad 0x3fd62c82f2b9c795, 0x3fe6a13cd1537290 // log1p(0.41406), 1/1.41406
287 .quad 0x3fd659b57303e1f3, 0x3fe691473a88d0c0 // log1p(0.41797), 1/1.41797
288 .quad 0x3fd686c81e9b14af, 0x3fe6816816816817 // log1p(0.42188), 1/1.42188
289 .quad 0x3fd6b3bb2235943e, 0x3fe6719f3601671a // log1p(0.42578), 1/1.42578
290 .quad 0x3fd6e08eaa2ba1e4, 0x3fe661ec6a5122f9 // log1p(0.42969), 1/1.42969
291 .quad 0x3fd70d42e2789236, 0x3fe6524f853b4aa3 // log1p(0.43359), 1/1.43359
292 .quad 0x3fd739d7f6bbd007, 0x3fe642c8590b2164 // log1p(0.43750), 1/1.43750
293 .quad 0x3fd7664e1239dbcf, 0x3fe63356b88ac0de // log1p(0.44141), 1/1.44141
294 .quad 0x3fd792a55fdd47a2, 0x3fe623fa77016240 // log1p(0.44531), 1/1.44531
295 .quad 0x3fd7bede0a37afc0, 0x3fe614b36831ae94 // log1p(0.44922), 1/1.44922
296 .quad 0x3fd7eaf83b82afc3, 0x3fe6058160581606 // log1p(0.45312), 1/1.45312
297 .quad 0x3fd816f41da0d496, 0x3fe5f66434292dfc // log1p(0.45703), 1/1.45703
298 .quad 0x3fd842d1da1e8b17, 0x3fe5e75bb8d015e7 // log1p(0.46094), 1/1.46094
299 .quad 0x3fd86e919a330ba0, 0x3fe5d867c3ece2a5 // log1p(0.46484), 1/1.46484
300 .quad 0x3fd89a3386c1425b, 0x3fe5c9882b931057 // log1p(0.46875), 1/1.46875
301 .quad 0x3fd8c5b7c858b48b, 0x3fe5babcc647fa91 // log1p(0.47266), 1/1.47266
302 .quad 0x3fd8f11e873662c7, 0x3fe5ac056b015ac0 // log1p(0.47656), 1/1.47656
303 .quad 0x3fd91c67eb45a83e, 0x3fe59d61f123ccaa // log1p(0.48047), 1/1.48047
304 .quad 0x3fd947941c2116fb, 0x3fe58ed2308158ed // log1p(0.48438), 1/1.48438
305 .quad 0x3fd972a341135158, 0x3fe5805601580560 // log1p(0.48828), 1/1.48828
306 .quad 0x3fd99d958117e08b, 0x3fe571ed3c506b3a // log1p(0.49219), 1/1.49219
307 .quad 0x3fd9c86b02dc0863, 0x3fe56397ba7c52e2 // log1p(0.49609), 1/1.49609
308 .quad 0x3fd9f323ecbf984c, 0x3fe5555555555555 // log1p(0.50000), 1/1.50000
309 .quad 0x3fda1dc064d5b995, 0x3fe54725e6bb82fe // log1p(0.50391), 1/1.50391
310 .quad 0x3fda484090e5bb0a, 0x3fe5390948f40feb // log1p(0.50781), 1/1.50781
311 .quad 0x3fda72a4966bd9ea, 0x3fe52aff56a8054b // log1p(0.51172), 1/1.51172
312 .quad 0x3fda9cec9a9a084a, 0x3fe51d07eae2f815 // log1p(0.51562), 1/1.51562
313 .quad 0x3fdac718c258b0e4, 0x3fe50f22e111c4c5 // log1p(0.51953), 1/1.51953
314 .quad 0x3fdaf1293247786b, 0x3fe5015015015015 // log1p(0.52344), 1/1.52344
315 .quad 0x3fdb1b1e0ebdfc5b, 0x3fe4f38f62dd4c9b // log1p(0.52734), 1/1.52734
316 .quad 0x3fdb44f77bcc8f63, 0x3fe4e5e0a72f0539 // log1p(0.53125), 1/1.53125
317 .quad 0x3fdb6eb59d3cf35e, 0x3fe4d843bedc2c4c // log1p(0.53516), 1/1.53516
318 .quad 0x3fdb9858969310fb, 0x3fe4cab88725af6e // log1p(0.53906), 1/1.53906
319 .quad 0x3fdbc1e08b0dad0a, 0x3fe4bd3edda68fe1 // log1p(0.54297), 1/1.54297
320 .quad 0x3fdbeb4d9da71b7c, 0x3fe4afd6a052bf5b // log1p(0.54688), 1/1.54688
321 .quad 0x3fdc149ff115f027, 0x3fe4a27fad76014a // log1p(0.55078), 1/1.55078
322 .quad 0x3fdc3dd7a7cdad4d, 0x3fe49539e3b2d067 // log1p(0.55469), 1/1.55469
323 .quad 0x3fdc66f4e3ff6ff8, 0x3fe4880522014880 // log1p(0.55859), 1/1.55859
324 .quad 0x3fdc8ff7c79a9a22, 0x3fe47ae147ae147b // log1p(0.56250), 1/1.56250
325 .quad 0x3fdcb8e0744d7aca, 0x3fe46dce34596066 // log1p(0.56641), 1/1.56641
326 .quad 0x3fdce1af0b85f3eb, 0x3fe460cbc7f5cf9a // log1p(0.57031), 1/1.57031
327 .quad 0x3fdd0a63ae721e64, 0x3fe453d9e2c776ca // log1p(0.57422), 1/1.57422
328 .quad 0x3fdd32fe7e00ebd5, 0x3fe446f86562d9fb // log1p(0.57812), 1/1.57812
329 .quad 0x3fdd5b7f9ae2c684, 0x3fe43a2730abee4d // log1p(0.58203), 1/1.58203
330 .quad 0x3fdd83e7258a2f3e, 0x3fe42d6625d51f87 // log1p(0.58594), 1/1.58594
331 .quad 0x3fddac353e2c5954, 0x3fe420b5265e5951 // log1p(0.58984), 1/1.58984
332 .quad 0x3fddd46a04c1c4a1, 0x3fe4141414141414 // log1p(0.59375), 1/1.59375
333 .quad 0x3fddfc859906d5b5, 0x3fe40782d10e6566 // log1p(0.59766), 1/1.59766
334 .quad 0x3fde24881a7c6c26, 0x3fe3fb013fb013fb // log1p(0.60156), 1/1.60156
335 .quad 0x3fde4c71a8687704, 0x3fe3ee8f42a5af07 // log1p(0.60547), 1/1.60547
336 .quad 0x3fde744261d68788, 0x3fe3e22cbce4a902 // log1p(0.60938), 1/1.60938
337 .quad 0x3fde9bfa659861f5, 0x3fe3d5d991aa75c6 // log1p(0.61328), 1/1.61328
338 .quad 0x3fdec399d2468cc0, 0x3fe3c995a47babe7 // log1p(0.61719), 1/1.61719
339 .quad 0x3fdeeb20c640ddf4, 0x3fe3bd60d9232955 // log1p(0.62109), 1/1.62109
340 .quad 0x3fdf128f5faf06ed, 0x3fe3b13b13b13b14 // log1p(0.62500), 1/1.62500
341 .quad 0x3fdf39e5bc811e5c, 0x3fe3a524387ac822 // log1p(0.62891), 1/1.62891
342 .quad 0x3fdf6123fa7028ac, 0x3fe3991c2c187f63 // log1p(0.63281), 1/1.63281
343 .quad 0x3fdf884a36fe9ec2, 0x3fe38d22d366088e // log1p(0.63672), 1/1.63672
344 .quad 0x3fdfaf588f78f31f, 0x3fe3813813813814 // log1p(0.64062), 1/1.64062
345 .quad 0x3fdfd64f20f61572, 0x3fe3755bd1c945ee // log1p(0.64453), 1/1.64453
346 .quad 0x3fdffd2e0857f498, 0x3fe3698df3de0748 // log1p(0.64844), 1/1.64844
347 .quad 0x3fe011fab125ff8a, 0x3fe35dce5f9f2af8 // log1p(0.65234), 1/1.65234
348 .quad 0x3fe02552a5a5d0ff, 0x3fe3521cfb2b78c1 // log1p(0.65625), 1/1.65625
349 .quad 0x3fe0389eefce633b, 0x3fe34679ace01346 // log1p(0.66016), 1/1.66016
350 .quad 0x3fe04bdf9da926d2, 0x3fe33ae45b57bcb2 // log1p(0.66406), 1/1.66406
351 .quad 0x3fe05f14bd26459c, 0x3fe32f5ced6a1dfa // log1p(0.66797), 1/1.66797
352 .quad 0x3fe0723e5c1cdf40, 0x3fe323e34a2b10bf // log1p(0.67188), 1/1.67188
353 .quad 0x3fe0855c884b450e, 0x3fe3187758e9ebb6 // log1p(0.67578), 1/1.67578
354 .quad 0x3fe0986f4f573521, 0x3fe30d190130d190 // log1p(0.67969), 1/1.67969
355 .quad 0x3fe0ab76bece14d2, 0x3fe301c82ac40260 // log1p(0.68359), 1/1.68359
356 .quad 0x3fe0be72e4252a83, 0x3fe2f684bda12f68 // log1p(0.68750), 1/1.68750
357 .quad 0x3fe0d163ccb9d6b8, 0x3fe2eb4ea1fed14b // log1p(0.69141), 1/1.69141
358 .quad 0x3fe0e44985d1cc8c, 0x3fe2e025c04b8097 // log1p(0.69531), 1/1.69531
359 .quad 0x3fe0f7241c9b497d, 0x3fe2d50a012d50a0 // log1p(0.69922), 1/1.69922
360 .quad 0x3fe109f39e2d4c97, 0x3fe2c9fb4d812ca0 // log1p(0.70312), 1/1.70312
361 .quad 0x3fe11cb81787ccf8, 0x3fe2bef98e5a3711 // log1p(0.70703), 1/1.70703
362 .quad 0x3fe12f719593efbc, 0x3fe2b404ad012b40 // log1p(0.71094), 1/1.71094
363 .quad 0x3fe1422025243d45, 0x3fe2a91c92f3c105 // log1p(0.71484), 1/1.71484
364 .quad 0x3fe154c3d2f4d5ea, 0x3fe29e4129e4129e // log1p(0.71875), 1/1.71875
365 .quad 0x3fe1675cababa60e, 0x3fe293725bb804a5 // log1p(0.72266), 1/1.72266
366 .quad 0x3fe179eabbd899a1, 0x3fe288b01288b013 // log1p(0.72656), 1/1.72656
367 .quad 0x3fe18c6e0ff5cf06, 0x3fe27dfa38a1ce4d // log1p(0.73047), 1/1.73047
368 .quad 0x3fe19ee6b467c96f, 0x3fe27350b8812735 // log1p(0.73438), 1/1.73438
369 .quad 0x3fe1b154b57da29f, 0x3fe268b37cd60127 // log1p(0.73828), 1/1.73828
370 .quad 0x3fe1c3b81f713c25, 0x3fe25e22708092f1 // log1p(0.74219), 1/1.74219
371 .quad 0x3fe1d610fe677003, 0x3fe2539d7e9177b2 // log1p(0.74609), 1/1.74609
372 .quad 0x3fe1e85f5e7040d0, 0x3fe2492492492492 // log1p(0.75000), 1/1.75000
373 .quad 0x3fe1faa34b87094c, 0x3fe23eb79717605b // log1p(0.75391), 1/1.75391
374 .quad 0x3fe20cdcd192ab6e, 0x3fe23456789abcdf // log1p(0.75781), 1/1.75781
375 .quad 0x3fe21f0bfc65beec, 0x3fe22a0122a0122a // log1p(0.76172), 1/1.76172
376 .quad 0x3fe23130d7bebf43, 0x3fe21fb78121fb78 // log1p(0.76562), 1/1.76562
377 .quad 0x3fe2434b6f483934, 0x3fe21579804855e6 // log1p(0.76953), 1/1.76953
378 .quad 0x3fe2555bce98f7cb, 0x3fe20b470c67c0d9 // log1p(0.77344), 1/1.77344
379 .quad 0x3fe26762013430e0, 0x3fe2012012012012 // log1p(0.77734), 1/1.77734
380 .quad 0x3fe2795e1289b11b, 0x3fe1f7047dc11f70 // log1p(0.78125), 1/1.78125
381 .quad 0x3fe28b500df60783, 0x3fe1ecf43c7fb84c // log1p(0.78516), 1/1.78516
382 .quad 0x3fe29d37fec2b08b, 0x3fe1e2ef3b3fb874 // log1p(0.78906), 1/1.78906
383 .quad 0x3fe2af15f02640ad, 0x3fe1d8f5672e4abd // log1p(0.79297), 1/1.79297
384 .quad 0x3fe2c0e9ed448e8c, 0x3fe1cf06ada2811d // log1p(0.79688), 1/1.79688
385 .quad 0x3fe2d2b4012edc9e, 0x3fe1c522fc1ce059 // log1p(0.80078), 1/1.80078
386 .quad 0x3fe2e47436e40268, 0x3fe1bb4a4046ed29 // log1p(0.80469), 1/1.80469
387 .quad 0x3fe2f62a99509546, 0x3fe1b17c67f2bae3 // log1p(0.80859), 1/1.80859
388 .quad 0x3fe307d7334f10be, 0x3fe1a7b9611a7b96 // log1p(0.81250), 1/1.81250
389 .quad 0x3fe3197a0fa7fe6a, 0x3fe19e0119e0119e // log1p(0.81641), 1/1.81641
390 .quad 0x3fe32b1339121d71, 0x3fe19453808ca29c // log1p(0.82031), 1/1.82031
391 .quad 0x3fe33ca2ba328995, 0x3fe18ab083902bdb // log1p(0.82422), 1/1.82422
392 .quad 0x3fe34e289d9ce1d3, 0x3fe1811811811812 // log1p(0.82812), 1/1.82812
393 .quad 0x3fe35fa4edd36ea0, 0x3fe1778a191bd684 // log1p(0.83203), 1/1.83203
394 .quad 0x3fe37117b54747b6, 0x3fe16e0689427379 // log1p(0.83594), 1/1.83594
395 .quad 0x3fe38280fe58797f, 0x3fe1648d50fc3201 // log1p(0.83984), 1/1.83984
396 .quad 0x3fe393e0d3562a1a, 0x3fe15b1e5f75270d // log1p(0.84375), 1/1.84375
397 .quad 0x3fe3a5373e7ebdfa, 0x3fe151b9a3fdd5c9 // log1p(0.84766), 1/1.84766
398 .quad 0x3fe3b68449fffc23, 0x3fe1485f0e0acd3b // log1p(0.85156), 1/1.85156
399 .quad 0x3fe3c7c7fff73206, 0x3fe13f0e8d344724 // log1p(0.85547), 1/1.85547
400 .quad 0x3fe3d9026a7156fb, 0x3fe135c81135c811 // log1p(0.85938), 1/1.85938
401 .quad 0x3fe3ea33936b2f5c, 0x3fe12c8b89edc0ac // log1p(0.86328), 1/1.86328
402 .quad 0x3fe3fb5b84d16f42, 0x3fe12358e75d3033 // log1p(0.86719), 1/1.86719
403 .quad 0x3fe40c7a4880dce9, 0x3fe11a3019a74826 // log1p(0.87109), 1/1.87109
404 .quad 0x3fe41d8fe84672ae, 0x3fe1111111111111 // log1p(0.87500), 1/1.87500
405 .quad 0x3fe42e9c6ddf80bf, 0x3fe107fbbe011080 // log1p(0.87891), 1/1.87891
406 .quad 0x3fe43f9fe2f9ce67, 0x3fe0fef010fef011 // log1p(0.88281), 1/1.88281
407 .quad 0x3fe4509a5133bb0a, 0x3fe0f5edfab325a2 // log1p(0.88672), 1/1.88672
408 .quad 0x3fe4618bc21c5ec2, 0x3fe0ecf56be69c90 // log1p(0.89062), 1/1.89062
409 .quad 0x3fe472743f33aaad, 0x3fe0e40655826011 // log1p(0.89453), 1/1.89453
410 .quad 0x3fe48353d1ea88df, 0x3fe0db20a88f4696 // log1p(0.89844), 1/1.89844
411 .quad 0x3fe4942a83a2fc07, 0x3fe0d24456359e3a // log1p(0.90234), 1/1.90234
412 .quad 0x3fe4a4f85db03ebb, 0x3fe0c9714fbcda3b // log1p(0.90625), 1/1.90625
413 .quad 0x3fe4b5bd6956e274, 0x3fe0c0a7868b4171 // log1p(0.91016), 1/1.91016
414 .quad 0x3fe4c679afccee3a, 0x3fe0b7e6ec259dc8 // log1p(0.91406), 1/1.91406
415 .quad 0x3fe4d72d3a39fd00, 0x3fe0af2f722eecb5 // log1p(0.91797), 1/1.91797
416 .quad 0x3fe4e7d811b75bb1, 0x3fe0a6810a6810a7 // log1p(0.92188), 1/1.92188
417 .quad 0x3fe4f87a3f5026e9, 0x3fe09ddba6af8360 // log1p(0.92578), 1/1.92578
418 .quad 0x3fe50913cc01686b, 0x3fe0953f39010954 // log1p(0.92969), 1/1.92969
419 .quad 0x3fe519a4c0ba3446, 0x3fe08cabb37565e2 // log1p(0.93359), 1/1.93359
420 .quad 0x3fe52a2d265bc5ab, 0x3fe0842108421084 // log1p(0.93750), 1/1.93750
421 .quad 0x3fe53aad05b99b7d, 0x3fe07b9f29b8eae2 // log1p(0.94141), 1/1.94141
422 .quad 0x3fe54b2467999498, 0x3fe073260a47f7c6 // log1p(0.94531), 1/1.94531
423 .quad 0x3fe55b9354b40bcd, 0x3fe06ab59c7912fb // log1p(0.94922), 1/1.94922
424 .quad 0x3fe56bf9d5b3f399, 0x3fe0624dd2f1a9fc // log1p(0.95312), 1/1.95312
425 .quad 0x3fe57c57f336f191, 0x3fe059eea0727586 // log1p(0.95703), 1/1.95703
426 .quad 0x3fe58cadb5cd7989, 0x3fe05197f7d73404 // log1p(0.96094), 1/1.96094
427 .quad 0x3fe59cfb25fae87e, 0x3fe04949cc1664c5 // log1p(0.96484), 1/1.96484
428 .quad 0x3fe5ad404c359f2d, 0x3fe0410410410410 // log1p(0.96875), 1/1.96875
429 .quad 0x3fe5bd7d30e71c73, 0x3fe038c6b78247fc // log1p(0.97266), 1/1.97266
430 .quad 0x3fe5cdb1dc6c1765, 0x3fe03091b51f5e1a // log1p(0.97656), 1/1.97656
431 .quad 0x3fe5ddde57149923, 0x3fe02864fc7729e9 // log1p(0.98047), 1/1.98047
432 .quad 0x3fe5ee02a9241675, 0x3fe0204081020408 // log1p(0.98438), 1/1.98438
433 .quad 0x3fe5fe1edad18919, 0x3fe0182436517a37 // log1p(0.98828), 1/1.98828
434 .quad 0x3fe60e32f44788d9, 0x3fe0101010101010 // log1p(0.99219), 1/1.99219
435 .quad 0x3fe61e3efda46467, 0x3fe0080402010080 // log1p(0.99609), 1/1.99609
436
437
438.literal8
439.align 3
440one: .double 1.0
441onehalf: .double 0.5
442onethird: .quad 0x3fd5555555555555 // 1/3
443onequarter: .double 0.25
444onefifth: .double 0.2
445log1p_amask: .quad 0x7ffff00000000000ULL // top 8 bits of mantissa
446exp_maskd: .quad 0x7ff0000000000000ULL // exponent bits / +Inf
447almost1: .quad 0x3fefffffffffffffULL // 1.0 - DBL_EPSILON/2
448log2: .quad 0x3fe62e42fefa39efULL // ln(2)
449
450.literal4
451.align 2
452f256: .long 0x43800000 //256.0f
453r256: .long 0x3b800000 //1.0f/256.0f
454
455
456.text
457
458#if defined( __x86_64__ )
459 #define RELATIVE_ADDR( _a) (_a)( %rip )
460 #define RELATIVE_ADDR_B( _a) (_a)( %rip )
461 #define RELATIVE_ADDR2( _a, _i, _step) ( %r8, _i, _step )
462#elif defined( __i386__ )
463 #define RELATIVE_ADDR( _a) (_a)-rel_addr( CX_P )
464 #define RELATIVE_ADDR_B( _a) (_a)-rel_addr_b( CX_P )
465 #define RELATIVE_ADDR2( _a, _i, _step) (_a)-rel_addr( CX_P, _i, _step )
466
467//a short routine to get the local address
468.align 4
469logf_pic: movl (%esp), %ecx //copy address of local_addr to %ecx
470 ret
471#else
472 #error arch not supported
473#endif
474
475
476//
477// logf -- overall approach
478//
479// We break up logf(x) as follows:
480//
481// x = 2**i * m 1.0 <= m < 2.0
482// log(x) = log(2**i) + log(m)
483//
484// log(2**i) is simply read from log_e_table.
485// To obtain log(m), we further break down m as :
486//
487// m = (1+a/256.0)(1+r) a = high 8 explicit bits of mantissa(m)
488// log(m) = log(1+a/256.0) + log2(1+r)
489//
490// We use the high 8 bits of the mantissa to look up log(1+a/256.0) in log_m_table above
491// We calculate 1+r as:
492//
493// 1+r = m * (1 /(1+a/256.0))
494//
495// We can lookup (from the same table) the value of 1/(1+a/256.0) based on a too.
496//
497// So the whole calculation is:
498//
499// logf(x) = log(2**i) + log(1+a/256.0) + log(1+r) = log_e_table[i+149] + log_m_table[a][0] + log( m * log_m_table[a][1] )
500//
501// The third term is calculated using the Taylor series:
502//
503// log(x+1) = x - x**2/2 + x**3/3 - x**4/4 + x**5/5
504//
505// The edge case code is done in integer to avoid setting flags, and because it is faster that way.
506// In the particular case of 1.0:
507// i = 0 -> log_e_table[0+149] = 0.0
508// a = 0 -> log_m_table[a][0] = 0.0
509// a = 0 -> log_m_table[a][1] = 1.0
510// 1+r = 1.0 * 1.0
511// r = 0
512// log(1+r) = 0 - 0/2 + 0/3 - 0/4 + 0/5
513//
514// so we get logf(x) = 0.0, with no inexact flag set. Other values should set inexact when we round to single precision,
515// either because the mantissa is not 1.0 or log_e_table[i+149] is non-zero.
516//
517
518
519ENTRY( logf )
520// Load input value
521#if defined( __i386__ )
522 movl FRAME_SIZE( STACKP ), %eax
523 movss FRAME_SIZE( STACKP ), %xmm0
524#else
525 movd %xmm0, %eax
526#endif
527
528 movl %eax, %ecx //set aside x
529 addl $0x00800000, %eax // push Infs, NaNs negative
530 xorps %xmm1, %xmm1 // 0.0f
531 cmpl $0x00800000, %eax // if( x <= 0 or Inf or NaN )
532 jle 2f
533
534 mov $(-127), DX_P // negative single precision bias
535 cmpl $0x00800000, %ecx // if( isnormal( x ))
536 jae 1f // skip denormal handling
537
538 //denormal whatnot
539 SUBP $126, DX_P // accumulate in -126 as pseudo exponent of denormal
540 movl $0x3f800000, %eax // 1.0
541 orl %eax, %ecx // multiply low bits by 2**126 by oring in 1.0
542 movd %eax, %xmm1 // 1.0
543 movd %ecx, %xmm0 // 1.0 | denorm bits
544 subss %xmm1, %xmm0 // (1.0 | denorm bits) - 1.0
545 movd %xmm0, %ecx
546
547 //find log_e_table[i] and load into %xmm0
5481: shrl $23, %ecx
549 addl %ecx, %edx
550
551#if defined( __i386__ )
552 calll logf_pic // set %ecx to point to local_addr
553rel_addr:
554#endif
555 cvtsi2sd %edx, %xmm7
556 mulsd RELATIVE_ADDR(log2), %xmm7 // log_e_table[i]
557
558 //find a, and load values from log_m_table
559 movd %xmm0, %eax // x
560 movl %eax, %edx // x
561 andl $0x00007fff, %eax // low bits of x
562 andl $0x007f8000, %edx // a << 15
563 orl $0x3f800000, %eax // 1.0 | low bits of x
564 shrl $11, %edx // a << 4
565 movd %eax, %xmm1 // 1.0 | low bits of x
566 cvtss2sd %xmm1, %xmm1 // 1.0 | low bits of x
567 subsd RELATIVE_ADDR( one ), %xmm1 // (1.0 | low bits of x) - 1.0
568 lea RELATIVE_ADDR( log_m_table), AX_P
569 mulsd 8( AX_P, DX_P, 1 ), %xmm1 // r = ((1.0 | low bits of x) - 1.0) / (1+a/256)
570 movapd %xmm1, %xmm2 // r
571 mulsd %xmm2, %xmm2 // rr
572 movsd ( AX_P, DX_P, 1 ), %xmm6 // log(1+a/256.0)
573
574 //Taylor series approximation: log(1+r) = r - rr/2 + rrr/3 - rrrr/4 + rrrrr/5
575 movsd RELATIVE_ADDR( onefifth ), %xmm3 // 0.2
576 movsd RELATIVE_ADDR( onequarter ), %xmm4 // 0.25
577 mulsd %xmm2, %xmm3 // 0.2rr
578 mulsd %xmm2, %xmm4 // 0.25rr
579 addsd RELATIVE_ADDR( onethird), %xmm3 // (1./3.) + 0.2rr
580 addsd RELATIVE_ADDR( onehalf), %xmm4 // 0.5 + 0.25rr
581 mulsd %xmm2, %xmm3 // (1./3.)rr + 0.2rrrr
582 mulsd %xmm2, %xmm4 // 0.5rr + 0.25rrrr
583 mulsd %xmm1, %xmm3 // (1./3.)rrr + 0.2rrrrr
584 subsd %xmm4, %xmm1 // r - 0.5rr - 0.25rrrr
585 addsd %xmm3, %xmm1 // r - 0.5rr + (1./3.)rrr - 0.25rrrr + 0.2rrrrr
586
587 //add up major terms
588 addsd %xmm6, %xmm1 // log(1+a/256.0) + (r - 0.5rr + (1./3.)rrr - 0.25rrrr + 0.2rrrrr)
589 addsd %xmm7, %xmm1 // log_e_table[i] + log(1+a/256.0) + (r - 0.5rr + (1./3.)rrr - 0.25rrrr + 0.2rrrrr)
590
591 //round to float
592 xorps %xmm0, %xmm0
593 cvtsd2ss %xmm1, %xmm0 // result
594
595#if defined( __i386__ )
596 movss %xmm0, FRAME_SIZE(STACKP)
597 flds FRAME_SIZE(STACKP)
598#endif
599 ret
600
6012: //special case code for negative, NaN or 0
602#if defined( __i386__ )
603 movss FRAME_SIZE( STACKP ), %xmm0
604#endif
605 ucomiss %xmm1, %xmm0 //test for x == 0
606 jb 4f //NaN or negative
607
608 //We imagine that 0.0, Inf are the most common cases, so those falls through
609 ja 3f //Infinity, just return inf
610
611 //We need to return -Inf for zero and set div/0 flag
612 pcmpeqb %xmm0, %xmm0 // -1U
613 cvtdq2ps %xmm0, %xmm0 // -1.0f
614 divss %xmm1, %xmm0 // -1.0f / 0 = -Inf + div/0 flag
615#if defined( __i386__ )
616 movss %xmm0, FRAME_SIZE(STACKP)
617#endif
6183:
619#if defined( __i386__ )
620 flds FRAME_SIZE(STACKP)
621#endif
622 ret
623
6244: jp 5f // handle NaN elsewhere
625
626 //negative number, return NaN and set invalid
627 pcmpeqb %xmm0, %xmm0 // -1U
628 pslld $23, %xmm0 // 0xff800000, -inf
629 mulss %xmm1, %xmm0 // 0 * -inf = NaN, set invalid
630#if defined( __i386__ )
631 movss %xmm0, FRAME_SIZE(STACKP)
632 flds FRAME_SIZE(STACKP)
633#endif
634 ret
635
636 //Its a NaN
6375:
638#if defined( __i386__ )
639 flds FRAME_SIZE(STACKP ) //load the NaN
640 fadd %st(0), %st(0) //quiet it
641#else
642 addss %xmm0, %xmm0
643#endif
644 ret
645
646
647
648
649// ------------------------------------- log1pf --------------------------------------
650
651#if defined( __i386__ )
652 #define INDEX %edi
653#else
654 #define INDEX %r8
655#endif
656
657ENTRY( log1pf )
658// Load input value
659#if defined( __i386__ )
660 movl FRAME_SIZE( STACKP ), %eax
661 movss FRAME_SIZE( STACKP ), %xmm0
662#else
663 movd %xmm0, %eax
664#endif
665
666 //move special cases ( x <= -1.0, +-Inf, NaN) to special case code
667 cmpl $0xbf800000, %eax // if( x <= -1.0f || x is a negative NaN )
668 jae 2f // goto 2
669
670 //Deal with +Inf, +NaN
671 movl %eax, %edx
672 andl $0x7f800000, %eax
673 cmpl $0x7f800000, %eax // if( x == +inf || x is positive NaN )
674 je 4f // goto 3
675
676#if defined( __i386__ )
677 pushl %edi
678 calll logf_pic // set %ecx to point to local_addr
679rel_addr_b:
680#endif
681
682 //Normal number in range
683 //Check for |x| < 0.5 cutoff between algorithms determined empirically
684 cmpl $0x3f000000, %eax // if( |x| < 1.0f )
685 jb 1f // goto 1
686
687 // x >= 1.0f. We just add 1.0 (in double precision) and do things much like log above
688 cvtss2sd %xmm0, %xmm1 // (double) x
689#if defined( __x86_64__ )
690 xorq %rax, %rax // zero top 32-bits of register a (and bottom 32-bits too)
691#endif
692 movsd RELATIVE_ADDR_B( one ), %xmm3 // 1.0
693 addsd %xmm3, %xmm1 // x + 1.0 //Note: because double precision, cant overflow here for round to +Inf
694 movsd RELATIVE_ADDR_B( exp_maskd), %xmm2 // 0x7ff0000000000000ULL
695 andnpd %xmm1, %xmm2 // (x + 1.0) mantissa, exponent all zero Note: sign bit always 0 here
696 psrlq $(52-8-4), %xmm1 // (x + 1.0) >> (52-8-4) 52 bit significand - 8 bits for a - 4 bits for size of log_m_table entries
697 movsd RELATIVE_ADDR_B(onethird), %xmm6 // 1/3
698 orpd %xmm3, %xmm2 // mantissa 1.0 <= m < 2.0
699 movd %xmm1, %eax // (x + 1.0) >> (52-8)
700 movsd RELATIVE_ADDR_B(log1p_amask ), %xmm1 // 0x7ffff00000000000ULL
701 MOVP AX_P, INDEX // set aside a bits
702 lea RELATIVE_ADDR_B( log_m_table ), DX_P // get pointer to log_m_table
703 andpd %xmm2, %xmm1 // 1 + a/256.0
704 shrl $(8+4), %eax // biased exponent
705 subsd %xmm1, %xmm2
706 and $0xff0, INDEX // a << 4 index into log_m_table clears high 32 bits too for x86_64
707 mulsd 8(DX_P, INDEX, 1), %xmm2 // r = (reduced mantissa) / (1+a/256.0)
708 movsd (DX_P, INDEX, 1), %xmm7 // log1p(a/256.0)
709 movapd %xmm2, %xmm1 // r
710 mulsd %xmm1, %xmm6 // r/3
711 mulsd %xmm2, %xmm2 // rr
712 subl $(1023), %eax // unbiased exponent, always positive so 32-bit subtract ok. Top 32-bits zeroed above. 149 added because table starts at 2**-149.
713 subsd RELATIVE_ADDR_B(onehalf), %xmm6 // -0.5 + r/3
714 mulsd %xmm2, %xmm6 // -0.5rr + rrr/3
715 cvtsi2sd %eax, %xmm5 // exponent
716 addsd %xmm1, %xmm6 // r + -0.5rr + rrr/3
717 mulsd RELATIVE_ADDR_B( log2 ), %xmm5 // ln(2) * exponent
718 addsd %xmm7, %xmm6 // r + -0.5rr + rrr/3 + log1p(a/256.0)
719 addsd %xmm5, %xmm6 // (r + -0.5rr + rrr/3 + log1p(a/256.0)) + log( 2**exponent )
720 cvtsd2ss %xmm6, %xmm0 // round to single precision
721
722#if defined( __i386__ )
723 popl %edi
724 movss %xmm0, FRAME_SIZE( STACKP )
725 flds FRAME_SIZE( STACKP )
726#endif
727 ret
728
729 // |x| < 0.5
730 //
7311: // check for |x| < 0x1.0p-24
732 cmpl $0x33800000, %eax // if( exponent is < -24 )
733 jb 6f // goto 6f
734
735 // Here we approximate as follows
736 //
737 // log(1+x) = log( (1+a) * (1+r ) )
738 // r = (1+x) / (1+a) - 1 = (x-a)/(1+a)
739 //
740 // This works well, as long as x > -0.5. For values below -0.5, the 1/(1+a) term blows up
741 // and the reduction fails to produce a r sufficiently near zero such that it will quickly
742 // converge with only a few terms.
743 //
744
745 // find the closest a/256.0f which is not greater in magnitide than f.
746 // and reduce f as r = (f - a/256.0f) / (1+a/256.0f)
747 movaps %xmm0, %xmm1 // f
748 mulss RELATIVE_ADDR_B(f256), %xmm0 // f * 256.0f
749 cvttss2si %xmm0, AX_P // (int)(f * 256.0f )
750 cvtsi2ss %eax, %xmm0 // trunc( f * 256.0f )
751
752#if defined( __x86_64__ )
753 cdqe // sign extend eax -> rax
754#endif
755
756 shl $4, AX_P // stride of log_m_table is 16 bytes
757 mulss RELATIVE_ADDR_B(r256), %xmm0 // fa = trunc( f * 256.0f ) / 256.0f
758 lea RELATIVE_ADDR_B( log_m_table), DX_P
759 subss %xmm0, %xmm1 // f - fa
760
761 // z = (1.0 + (-0.5 + (1./3. + -0.25*r)*r)*r)*r
762 movsd RELATIVE_ADDR_B( onequarter), %xmm4 // 0.25
763 cvtss2sd %xmm1, %xmm2 // (double) (f-fa)
764 movsd RELATIVE_ADDR_B( onethird ), %xmm3 // 1./3.
765 mulsd 8(DX_P, AX_P, 1), %xmm2 // r = log_m_table[a][1] * (f-fa)
766 movapd %xmm2, %xmm1 // r
767 mulsd %xmm2, %xmm2 // rr
768 mulsd %xmm2, %xmm4 // 0.25rr
769 mulsd %xmm2, %xmm3 // rr/3. (almost)
770 addsd RELATIVE_ADDR_B( onehalf ), %xmm4 // 0.5 + 0.25rr
771 addsd RELATIVE_ADDR_B( one ), %xmm3 // 1.0 + rr/3
772 mulsd %xmm2, %xmm4 // 0.5rr + 0.25rrrr
773 mulsd %xmm1, %xmm3 // r + rrr/3
774 subsd %xmm4, %xmm3 // r - 0.5rr + rrr/3 - 0.25rrrr
775
776 // Add in log1p( a / 256.0f)
777 addsd (DX_P, AX_P, 1), %xmm3 // log_m_table[a][0] + r - 0.5rr + rrr/3 - 0.25rrrr
778
779 // round to single precision -- this sets inexact if r happened to be 0, because log1p(a/256.0) is not exact in single precision
780 cvtsd2ss %xmm3, %xmm0
781
782#if defined( __i386__ )
783 popl %edi
784 movss %xmm0, FRAME_SIZE( STACKP )
785 flds FRAME_SIZE( STACKP )
786#endif
787 ret
788
789
7902: // x <= -1.0f || x is a negative NaN
791 je 3f // x == -1.0f
792 ucomiss %xmm0, %xmm0 // if( isnan(x) )
793 jp 5f // goto 4
794
795 // x < -1.0f, return NaN and set invalid
796 pcmpeqb %xmm0, %xmm0 // -1U
797 pslld $23, %xmm0 // 0xff800000, -inf
798 xorps %xmm1, %xmm1
799 mulss %xmm1, %xmm0 // set invalid, create NaN
800#if defined( __i386__ )
801 movss %xmm0, FRAME_SIZE(STACKP)
802 flds FRAME_SIZE( STACKP )
803#endif
804 ret
805
806
8073: // x == -1.0
808 //We need to return -Inf and set div/0 flag
809 xorps %xmm1, %xmm1 // 0.0f
810 divss %xmm1, %xmm0 // -1.0f / 0 = -Inf + div/0 flag
811#if defined( __i386__ )
812 movss %xmm0, FRAME_SIZE(STACKP)
813 flds FRAME_SIZE( STACKP )
814#endif
815 ret
816
8174: // x == +inf || x is positive NaN
818 ucomiss %xmm0, %xmm0 // if( isnan(x) )
819 jp 5f // goto 4
820
821 // x == +inf, return +inf
822#if defined( __i386__ )
823 movss %xmm0, FRAME_SIZE(STACKP)
824 flds FRAME_SIZE( STACKP )
825#endif
826 ret
827
828
8295: // x is NaN
830 addss %xmm0, %xmm0 //silence NaN
831#if defined( __i386__ )
832 movss %xmm0, FRAME_SIZE(STACKP)
833 flds FRAME_SIZE( STACKP )
834#endif
835 ret
836
8376: // |x| < 0x1.0p-24f
838#if defined( __i386__ )
839 popl %edi
840#endif
841 andl $0x7fffffff, %edx // |x|
842 jz 7f // if( |x| == 0.0f ) goto 7
843
844 // non-zero
845 cvtss2sd %xmm0, %xmm1
846 mulsd RELATIVE_ADDR_B( almost1), %xmm1 // multiply by 1ulp less than 1.0. Sets inexact.
847 cvtsd2ss %xmm1, %xmm0 // convert back to float, underflow as necessary, round appropriately
848#if defined( __i386__ )
849 movss %xmm0, FRAME_SIZE( STACKP )
850#endif
8517:
852#if defined( __i386__ )
853 flds FRAME_SIZE( STACKP ) // { x }
854#endif
855 ret