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1/*
2 * atanhf.s
3 *
4 * by Stephen Canon
5 *
6 * Copyright (c) 2007, Apple Inc. All rights reserved
7 *
8 * Single precision implementation of C99 math library function atanhf for the OS X i386 and x86_64 abis.
9 *
10 */
11
12#include <machine/asm.h>
13#include "abi.h"
14
15.const
16.align 4
17// Polynomial coefficients used for the 7/8 < |x| < 1 case. These are addressed by offset from big_table,
18// so take care if you move things around.
19
20 .quad 0x3fc00400e807c4f5, 0xbf82304e7c64d56d // c4
21 .quad 0xbff5533310026652, 0x4002a85c54ffbc1c // c3/c4
22 .quad 0x3fcfffffaa90af21, 0xbfb0005bce0fa090 // c2
23 .quad 0xc0000000244fd260, 0x400fff47b17904a4 // c1/c2
24
25/*********************************************************
26void gen_big_table() {
27 union { double d; uint64_t x; } constant;
28
29 int i,p,q;
30 for (i=0; i<256; ++i) {
31 double a = (double)i / 256.0;
32 constant.d = -0.5*log1p(a); printf("\t\t\t\t.quad 0x%016llx,", constant.x);
33 constant.d = 1.0 / (1.0 + a); printf(" 0x%016llx", constant.x);
34 if (!i) {
35 printf("\t// a = 0\n");
36 } else {
37 p = i; q = 256;
38 while (!(p % 2)) {
39 p = p/2; q = q/2;
40 }
41 printf("\t// a = %d/%d\n", p, q);
42 }
43 }
44}
45*********************************************************/
46
47 // -0.5*log1p(a) 1/(1+a)
48big_table: .quad 0x8000000000000000, 0x3ff0000000000000 // a = 0
49 .quad 0xbf5ff00aa2b10bc0, 0x3fefe01fe01fe020 // a = 1/256
50 .quad 0xbf6fe02a6b106789, 0x3fefc07f01fc07f0 // a = 1/128
51 .quad 0xbf77dc475f810a77, 0x3fefa11caa01fa12 // a = 3/256
52 .quad 0xbf7fc0a8b0fc03e4, 0x3fef81f81f81f820 // a = 1/64
53 .quad 0xbf83cea44346a575, 0x3fef6310aca0dbb5 // a = 5/256
54 .quad 0xbf87b91b07d5b11b, 0x3fef44659e4a4271 // a = 3/128
55 .quad 0xbf8b9fc027af9198, 0x3fef25f644230ab5 // a = 7/256
56 .quad 0xbf8f829b0e783300, 0x3fef07c1f07c1f08 // a = 1/32
57 .quad 0xbf91b0d98923d980, 0x3feee9c7f8458e02 // a = 9/256
58 .quad 0xbf939e87b9febd60, 0x3feecc07b301ecc0 // a = 5/128
59 .quad 0xbf958a5bafc8e4d5, 0x3feeae807aba01eb // a = 11/256
60 .quad 0xbf977458f632dcfc, 0x3fee9131abf0b767 // a = 3/64
61 .quad 0xbf995c830ec8e3eb, 0x3fee741aa59750e4 // a = 13/256
62 .quad 0xbf9b42dd711971bf, 0x3fee573ac901e574 // a = 7/128
63 .quad 0xbf9d276b8adb0b52, 0x3fee3a9179dc1a73 // a = 15/256
64 .quad 0xbf9f0a30c01162a6, 0x3fee1e1e1e1e1e1e // a = 1/16
65 .quad 0xbfa075983598e471, 0x3fee01e01e01e01e // a = 17/256
66 .quad 0xbfa16536eea37ae1, 0x3fede5d6e3f8868a // a = 9/128
67 .quad 0xbfa253f62f0a1417, 0x3fedca01dca01dca // a = 19/256
68 .quad 0xbfa341d7961bd1d1, 0x3fedae6076b981db // a = 5/64
69 .quad 0xbfa42edcbea646f0, 0x3fed92f2231e7f8a // a = 21/256
70 .quad 0xbfa51b073f06183f, 0x3fed77b654b82c34 // a = 11/128
71 .quad 0xbfa60658a93750c4, 0x3fed5cac807572b2 // a = 23/256
72 .quad 0xbfa6f0d28ae56b4c, 0x3fed41d41d41d41d // a = 3/32
73 .quad 0xbfa7da766d7b12cc, 0x3fed272ca3fc5b1a // a = 25/256
74 .quad 0xbfa8c345d6319b21, 0x3fed0cb58f6ec074 // a = 13/128
75 .quad 0xbfa9ab42462033ad, 0x3fecf26e5c44bfc6 // a = 27/256
76 .quad 0xbfaa926d3a4ad563, 0x3fecd85689039b0b // a = 7/64
77 .quad 0xbfab78c82bb0eda1, 0x3fecbe6d9601cbe7 // a = 29/256
78 .quad 0xbfac5e548f5bc743, 0x3feca4b3055ee191 // a = 15/128
79 .quad 0xbfad4313d66cb35d, 0x3fec8b265afb8a42 // a = 31/256
80 .quad 0xbfae27076e2af2e6, 0x3fec71c71c71c71c // a = 1/8
81 .quad 0xbfaf0a30c01162a6, 0x3fec5894d10d4986 // a = 33/256
82 .quad 0xbfafec9131dbeabb, 0x3fec3f8f01c3f8f0 // a = 17/128
83 .quad 0xbfb0671512ca596e, 0x3fec26b5392ea01c // a = 35/256
84 .quad 0xbfb0d77e7cd08e59, 0x3fec0e070381c0e0 // a = 9/64
85 .quad 0xbfb14785846742ac, 0x3febf583ee868d8b // a = 37/256
86 .quad 0xbfb1b72ad52f67a0, 0x3febdd2b899406f7 // a = 19/128
87 .quad 0xbfb2266f190a5acc, 0x3febc4fd65883e7b // a = 39/256
88 .quad 0xbfb29552f81ff523, 0x3febacf914c1bad0 // a = 5/32
89 .quad 0xbfb303d718e47fd3, 0x3feb951e2b18ff23 // a = 41/256
90 .quad 0xbfb371fc201e8f74, 0x3feb7d6c3dda338b // a = 21/128
91 .quad 0xbfb3dfc2b0ecc62a, 0x3feb65e2e3beee05 // a = 43/256
92 .quad 0xbfb44d2b6ccb7d1e, 0x3feb4e81b4e81b4f // a = 11/64
93 .quad 0xbfb4ba36f39a55e5, 0x3feb37484ad806ce // a = 45/256
94 .quad 0xbfb526e5e3a1b438, 0x3feb2036406c80d9 // a = 23/128
95 .quad 0xbfb59338d9982086, 0x3feb094b31d922a4 // a = 47/256
96 .quad 0xbfb5ff3070a793d4, 0x3feaf286bca1af28 // a = 3/16
97 .quad 0xbfb66acd4272ad51, 0x3feadbe87f94905e // a = 49/256
98 .quad 0xbfb6d60fe719d21d, 0x3feac5701ac5701b // a = 25/128
99 .quad 0xbfb740f8f54037a5, 0x3feaaf1d2f87ebfd // a = 51/256
100 .quad 0xbfb7ab890210d909, 0x3fea98ef606a63be // a = 13/64
101 .quad 0xbfb815c0a14357eb, 0x3fea82e65130e159 // a = 53/256
102 .quad 0xbfb87fa06520c911, 0x3fea6d01a6d01a6d // a = 27/128
103 .quad 0xbfb8e928de886d41, 0x3fea574107688a4a // a = 55/256
104 .quad 0xbfb9525a9cf456b4, 0x3fea41a41a41a41a // a = 7/32
105 .quad 0xbfb9bb362e7dfb83, 0x3fea2c2a87c51ca0 // a = 57/256
106 .quad 0xbfba23bc1fe2b563, 0x3fea16d3f97a4b02 // a = 29/128
107 .quad 0xbfba8becfc882f19, 0x3fea01a01a01a01a // a = 59/256
108 .quad 0xbfbaf3c94e80bff3, 0x3fe9ec8e951033d9 // a = 15/64
109 .quad 0xbfbb5b519e8fb5a4, 0x3fe9d79f176b682d // a = 61/256
110 .quad 0xbfbbc286742d8cd6, 0x3fe9c2d14ee4a102 // a = 31/128
111 .quad 0xbfbc2968558c18c1, 0x3fe9ae24ea5510da // a = 63/256
112 .quad 0xbfbc8ff7c79a9a22, 0x3fe999999999999a // a = 1/4
113 .quad 0xbfbcf6354e09c5dc, 0x3fe9852f0d8ec0ff // a = 65/256
114 .quad 0xbfbd5c216b4fbb91, 0x3fe970e4f80cb872 // a = 33/128
115 .quad 0xbfbdc1bca0abec7d, 0x3fe95cbb0be377ae // a = 67/256
116 .quad 0xbfbe27076e2af2e6, 0x3fe948b0fcd6e9e0 // a = 17/64
117 .quad 0xbfbe8c0252aa5a60, 0x3fe934c67f9b2ce6 // a = 69/256
118 .quad 0xbfbef0adcbdc5936, 0x3fe920fb49d0e229 // a = 35/128
119 .quad 0xbfbf550a564b7b37, 0x3fe90d4f120190d5 // a = 71/256
120 .quad 0xbfbfb9186d5e3e2b, 0x3fe8f9c18f9c18fa // a = 9/32
121 .quad 0xbfc00e6c45ad501d, 0x3fe8e6527af1373f // a = 73/256
122 .quad 0xbfc0402594b4d041, 0x3fe8d3018d3018d3 // a = 37/128
123 .quad 0xbfc071b85fcd590d, 0x3fe8bfce8062ff3a // a = 75/256
124 .quad 0xbfc0a324e27390e3, 0x3fe8acb90f6bf3aa // a = 19/64
125 .quad 0xbfc0d46b579ab74b, 0x3fe899c0f601899c // a = 77/256
126 .quad 0xbfc1058bf9ae4ad5, 0x3fe886e5f0abb04a // a = 39/128
127 .quad 0xbfc136870293a8b0, 0x3fe87427bcc092b9 // a = 79/256
128 .quad 0xbfc1675cababa60e, 0x3fe8618618618618 // a = 5/16
129 .quad 0xbfc1980d2dd4236f, 0x3fe84f00c2780614 // a = 81/256
130 .quad 0xbfc1c898c16999fb, 0x3fe83c977ab2bedd // a = 41/128
131 .quad 0xbfc1f8ff9e48a2f3, 0x3fe82a4a0182a4a0 // a = 83/256
132 .quad 0xbfc22941fbcf7966, 0x3fe8181818181818 // a = 21/64
133 .quad 0xbfc2596010df763a, 0x3fe8060180601806 // a = 85/256
134 .quad 0xbfc2895a13de86a3, 0x3fe7f405fd017f40 // a = 43/128
135 .quad 0xbfc2b9303ab89d25, 0x3fe7e225515a4f1d // a = 87/256
136 .quad 0xbfc2e8e2bae11d31, 0x3fe7d05f417d05f4 // a = 11/32
137 .quad 0xbfc31871c9544185, 0x3fe7beb3922e017c // a = 89/256
138 .quad 0xbfc347dd9a987d55, 0x3fe7ad2208e0ecc3 // a = 45/128
139 .quad 0xbfc3772662bfd85b, 0x3fe79baa6bb6398b // a = 91/256
140 .quad 0xbfc3a64c556945ea, 0x3fe78a4c8178a4c8 // a = 23/64
141 .quad 0xbfc3d54fa5c1f710, 0x3fe77908119ac60d // a = 93/256
142 .quad 0xbfc404308686a7e4, 0x3fe767dce434a9b1 // a = 47/128
143 .quad 0xbfc432ef2a04e814, 0x3fe756cac201756d // a = 95/256
144 .quad 0xbfc4618bc21c5ec2, 0x3fe745d1745d1746 // a = 3/8
145 .quad 0xbfc49006804009d1, 0x3fe734f0c541fe8d // a = 97/256
146 .quad 0xbfc4be5f957778a1, 0x3fe724287f46debc // a = 49/128
147 .quad 0xbfc4ec973260026a, 0x3fe713786d9c7c09 // a = 99/256
148 .quad 0xbfc51aad872df82d, 0x3fe702e05c0b8170 // a = 25/64
149 .quad 0xbfc548a2c3add263, 0x3fe6f26016f26017 // a = 101/256
150 .quad 0xbfc5767717455a6c, 0x3fe6e1f76b4337c7 // a = 51/128
151 .quad 0xbfc5a42ab0f4cfe2, 0x3fe6d1a62681c861 // a = 103/256
152 .quad 0xbfc5d1bdbf5809ca, 0x3fe6c16c16c16c17 // a = 13/32
153 .quad 0xbfc5ff3070a793d4, 0x3fe6b1490aa31a3d // a = 105/256
154 .quad 0xbfc62c82f2b9c795, 0x3fe6a13cd1537290 // a = 53/128
155 .quad 0xbfc659b57303e1f3, 0x3fe691473a88d0c0 // a = 107/256
156 .quad 0xbfc686c81e9b14af, 0x3fe6816816816817 // a = 27/64
157 .quad 0xbfc6b3bb2235943e, 0x3fe6719f3601671a // a = 109/256
158 .quad 0xbfc6e08eaa2ba1e4, 0x3fe661ec6a5122f9 // a = 55/128
159 .quad 0xbfc70d42e2789236, 0x3fe6524f853b4aa3 // a = 111/256
160 .quad 0xbfc739d7f6bbd007, 0x3fe642c8590b2164 // a = 7/16
161 .quad 0xbfc7664e1239dbcf, 0x3fe63356b88ac0de // a = 113/256
162 .quad 0xbfc792a55fdd47a2, 0x3fe623fa77016240 // a = 57/128
163 .quad 0xbfc7bede0a37afc0, 0x3fe614b36831ae94 // a = 115/256
164 .quad 0xbfc7eaf83b82afc3, 0x3fe6058160581606 // a = 29/64
165 .quad 0xbfc816f41da0d496, 0x3fe5f66434292dfc // a = 117/256
166 .quad 0xbfc842d1da1e8b17, 0x3fe5e75bb8d015e7 // a = 59/128
167 .quad 0xbfc86e919a330ba0, 0x3fe5d867c3ece2a5 // a = 119/256
168 .quad 0xbfc89a3386c1425b, 0x3fe5c9882b931057 // a = 15/32
169 .quad 0xbfc8c5b7c858b48b, 0x3fe5babcc647fa91 // a = 121/256
170 .quad 0xbfc8f11e873662c7, 0x3fe5ac056b015ac0 // a = 61/128
171 .quad 0xbfc91c67eb45a83e, 0x3fe59d61f123ccaa // a = 123/256
172 .quad 0xbfc947941c2116fb, 0x3fe58ed2308158ed // a = 31/64
173 .quad 0xbfc972a341135158, 0x3fe5805601580560 // a = 125/256
174 .quad 0xbfc99d958117e08b, 0x3fe571ed3c506b3a // a = 63/128
175 .quad 0xbfc9c86b02dc0863, 0x3fe56397ba7c52e2 // a = 127/256
176 .quad 0xbfc9f323ecbf984c, 0x3fe5555555555555 // a = 1/2
177 .quad 0xbfca1dc064d5b995, 0x3fe54725e6bb82fe // a = 129/256
178 .quad 0xbfca484090e5bb0a, 0x3fe5390948f40feb // a = 65/128
179 .quad 0xbfca72a4966bd9ea, 0x3fe52aff56a8054b // a = 131/256
180 .quad 0xbfca9cec9a9a084a, 0x3fe51d07eae2f815 // a = 33/64
181 .quad 0xbfcac718c258b0e4, 0x3fe50f22e111c4c5 // a = 133/256
182 .quad 0xbfcaf1293247786b, 0x3fe5015015015015 // a = 67/128
183 .quad 0xbfcb1b1e0ebdfc5b, 0x3fe4f38f62dd4c9b // a = 135/256
184 .quad 0xbfcb44f77bcc8f63, 0x3fe4e5e0a72f0539 // a = 17/32
185 .quad 0xbfcb6eb59d3cf35e, 0x3fe4d843bedc2c4c // a = 137/256
186 .quad 0xbfcb9858969310fb, 0x3fe4cab88725af6e // a = 69/128
187 .quad 0xbfcbc1e08b0dad0a, 0x3fe4bd3edda68fe1 // a = 139/256
188 .quad 0xbfcbeb4d9da71b7c, 0x3fe4afd6a052bf5b // a = 35/64
189 .quad 0xbfcc149ff115f027, 0x3fe4a27fad76014a // a = 141/256
190 .quad 0xbfcc3dd7a7cdad4d, 0x3fe49539e3b2d067 // a = 71/128
191 .quad 0xbfcc66f4e3ff6ff8, 0x3fe4880522014880 // a = 143/256
192 .quad 0xbfcc8ff7c79a9a22, 0x3fe47ae147ae147b // a = 9/16
193 .quad 0xbfccb8e0744d7aca, 0x3fe46dce34596066 // a = 145/256
194 .quad 0xbfcce1af0b85f3eb, 0x3fe460cbc7f5cf9a // a = 73/128
195 .quad 0xbfcd0a63ae721e64, 0x3fe453d9e2c776ca // a = 147/256
196 .quad 0xbfcd32fe7e00ebd5, 0x3fe446f86562d9fb // a = 37/64
197 .quad 0xbfcd5b7f9ae2c684, 0x3fe43a2730abee4d // a = 149/256
198 .quad 0xbfcd83e7258a2f3e, 0x3fe42d6625d51f87 // a = 75/128
199 .quad 0xbfcdac353e2c5954, 0x3fe420b5265e5951 // a = 151/256
200 .quad 0xbfcdd46a04c1c4a1, 0x3fe4141414141414 // a = 19/32
201 .quad 0xbfcdfc859906d5b5, 0x3fe40782d10e6566 // a = 153/256
202 .quad 0xbfce24881a7c6c26, 0x3fe3fb013fb013fb // a = 77/128
203 .quad 0xbfce4c71a8687704, 0x3fe3ee8f42a5af07 // a = 155/256
204 .quad 0xbfce744261d68788, 0x3fe3e22cbce4a902 // a = 39/64
205 .quad 0xbfce9bfa659861f5, 0x3fe3d5d991aa75c6 // a = 157/256
206 .quad 0xbfcec399d2468cc0, 0x3fe3c995a47babe7 // a = 79/128
207 .quad 0xbfceeb20c640ddf4, 0x3fe3bd60d9232955 // a = 159/256
208 .quad 0xbfcf128f5faf06ed, 0x3fe3b13b13b13b14 // a = 5/8
209 .quad 0xbfcf39e5bc811e5c, 0x3fe3a524387ac822 // a = 161/256
210 .quad 0xbfcf6123fa7028ac, 0x3fe3991c2c187f63 // a = 81/128
211 .quad 0xbfcf884a36fe9ec2, 0x3fe38d22d366088e // a = 163/256
212 .quad 0xbfcfaf588f78f31f, 0x3fe3813813813814 // a = 41/64
213 .quad 0xbfcfd64f20f61572, 0x3fe3755bd1c945ee // a = 165/256
214 .quad 0xbfcffd2e0857f498, 0x3fe3698df3de0748 // a = 83/128
215 .quad 0xbfd011fab125ff8a, 0x3fe35dce5f9f2af8 // a = 167/256
216 .quad 0xbfd02552a5a5d0ff, 0x3fe3521cfb2b78c1 // a = 21/32
217 .quad 0xbfd0389eefce633b, 0x3fe34679ace01346 // a = 169/256
218 .quad 0xbfd04bdf9da926d2, 0x3fe33ae45b57bcb2 // a = 85/128
219 .quad 0xbfd05f14bd26459c, 0x3fe32f5ced6a1dfa // a = 171/256
220 .quad 0xbfd0723e5c1cdf40, 0x3fe323e34a2b10bf // a = 43/64
221 .quad 0xbfd0855c884b450e, 0x3fe3187758e9ebb6 // a = 173/256
222 .quad 0xbfd0986f4f573521, 0x3fe30d190130d190 // a = 87/128
223 .quad 0xbfd0ab76bece14d2, 0x3fe301c82ac40260 // a = 175/256
224 .quad 0xbfd0be72e4252a83, 0x3fe2f684bda12f68 // a = 11/16
225 .quad 0xbfd0d163ccb9d6b8, 0x3fe2eb4ea1fed14b // a = 177/256
226 .quad 0xbfd0e44985d1cc8c, 0x3fe2e025c04b8097 // a = 89/128
227 .quad 0xbfd0f7241c9b497d, 0x3fe2d50a012d50a0 // a = 179/256
228 .quad 0xbfd109f39e2d4c97, 0x3fe2c9fb4d812ca0 // a = 45/64
229 .quad 0xbfd11cb81787ccf8, 0x3fe2bef98e5a3711 // a = 181/256
230 .quad 0xbfd12f719593efbc, 0x3fe2b404ad012b40 // a = 91/128
231 .quad 0xbfd1422025243d45, 0x3fe2a91c92f3c105 // a = 183/256
232 .quad 0xbfd154c3d2f4d5ea, 0x3fe29e4129e4129e // a = 23/32
233 .quad 0xbfd1675cababa60e, 0x3fe293725bb804a5 // a = 185/256
234 .quad 0xbfd179eabbd899a1, 0x3fe288b01288b013 // a = 93/128
235 .quad 0xbfd18c6e0ff5cf06, 0x3fe27dfa38a1ce4d // a = 187/256
236 .quad 0xbfd19ee6b467c96f, 0x3fe27350b8812735 // a = 47/64
237 .quad 0xbfd1b154b57da29f, 0x3fe268b37cd60127 // a = 189/256
238 .quad 0xbfd1c3b81f713c25, 0x3fe25e22708092f1 // a = 95/128
239 .quad 0xbfd1d610fe677003, 0x3fe2539d7e9177b2 // a = 191/256
240 .quad 0xbfd1e85f5e7040d0, 0x3fe2492492492492 // a = 3/4
241 .quad 0xbfd1faa34b87094c, 0x3fe23eb79717605b // a = 193/256
242 .quad 0xbfd20cdcd192ab6e, 0x3fe23456789abcdf // a = 97/128
243 .quad 0xbfd21f0bfc65beec, 0x3fe22a0122a0122a // a = 195/256
244 .quad 0xbfd23130d7bebf43, 0x3fe21fb78121fb78 // a = 49/64
245 .quad 0xbfd2434b6f483934, 0x3fe21579804855e6 // a = 197/256
246 .quad 0xbfd2555bce98f7cb, 0x3fe20b470c67c0d9 // a = 99/128
247 .quad 0xbfd26762013430e0, 0x3fe2012012012012 // a = 199/256
248 .quad 0xbfd2795e1289b11b, 0x3fe1f7047dc11f70 // a = 25/32
249 .quad 0xbfd28b500df60783, 0x3fe1ecf43c7fb84c // a = 201/256
250 .quad 0xbfd29d37fec2b08b, 0x3fe1e2ef3b3fb874 // a = 101/128
251 .quad 0xbfd2af15f02640ad, 0x3fe1d8f5672e4abd // a = 203/256
252 .quad 0xbfd2c0e9ed448e8c, 0x3fe1cf06ada2811d // a = 51/64
253 .quad 0xbfd2d2b4012edc9e, 0x3fe1c522fc1ce059 // a = 205/256
254 .quad 0xbfd2e47436e40268, 0x3fe1bb4a4046ed29 // a = 103/128
255 .quad 0xbfd2f62a99509546, 0x3fe1b17c67f2bae3 // a = 207/256
256 .quad 0xbfd307d7334f10be, 0x3fe1a7b9611a7b96 // a = 13/16
257 .quad 0xbfd3197a0fa7fe6a, 0x3fe19e0119e0119e // a = 209/256
258 .quad 0xbfd32b1339121d71, 0x3fe19453808ca29c // a = 105/128
259 .quad 0xbfd33ca2ba328995, 0x3fe18ab083902bdb // a = 211/256
260 .quad 0xbfd34e289d9ce1d3, 0x3fe1811811811812 // a = 53/64
261 .quad 0xbfd35fa4edd36ea0, 0x3fe1778a191bd684 // a = 213/256
262 .quad 0xbfd37117b54747b6, 0x3fe16e0689427379 // a = 107/128
263 .quad 0xbfd38280fe58797f, 0x3fe1648d50fc3201 // a = 215/256
264 .quad 0xbfd393e0d3562a1a, 0x3fe15b1e5f75270d // a = 27/32
265 .quad 0xbfd3a5373e7ebdfa, 0x3fe151b9a3fdd5c9 // a = 217/256
266 .quad 0xbfd3b68449fffc23, 0x3fe1485f0e0acd3b // a = 109/128
267 .quad 0xbfd3c7c7fff73206, 0x3fe13f0e8d344724 // a = 219/256
268 .quad 0xbfd3d9026a7156fb, 0x3fe135c81135c811 // a = 55/64
269 .quad 0xbfd3ea33936b2f5c, 0x3fe12c8b89edc0ac // a = 221/256
270 .quad 0xbfd3fb5b84d16f42, 0x3fe12358e75d3033 // a = 111/128
271 .quad 0xbfd40c7a4880dce9, 0x3fe11a3019a74826 // a = 223/256
272 .quad 0xbfd41d8fe84672ae, 0x3fe1111111111111 // a = 7/8
273 .quad 0xbfd42e9c6ddf80bf, 0x3fe107fbbe011080 // a = 225/256
274 .quad 0xbfd43f9fe2f9ce67, 0x3fe0fef010fef011 // a = 113/128
275 .quad 0xbfd4509a5133bb0a, 0x3fe0f5edfab325a2 // a = 227/256
276 .quad 0xbfd4618bc21c5ec2, 0x3fe0ecf56be69c90 // a = 57/64
277 .quad 0xbfd472743f33aaad, 0x3fe0e40655826011 // a = 229/256
278 .quad 0xbfd48353d1ea88df, 0x3fe0db20a88f4696 // a = 115/128
279 .quad 0xbfd4942a83a2fc07, 0x3fe0d24456359e3a // a = 231/256
280 .quad 0xbfd4a4f85db03ebb, 0x3fe0c9714fbcda3b // a = 29/32
281 .quad 0xbfd4b5bd6956e274, 0x3fe0c0a7868b4171 // a = 233/256
282 .quad 0xbfd4c679afccee3a, 0x3fe0b7e6ec259dc8 // a = 117/128
283 .quad 0xbfd4d72d3a39fd00, 0x3fe0af2f722eecb5 // a = 235/256
284 .quad 0xbfd4e7d811b75bb1, 0x3fe0a6810a6810a7 // a = 59/64
285 .quad 0xbfd4f87a3f5026e9, 0x3fe09ddba6af8360 // a = 237/256
286 .quad 0xbfd50913cc01686b, 0x3fe0953f39010954 // a = 119/128
287 .quad 0xbfd519a4c0ba3446, 0x3fe08cabb37565e2 // a = 239/256
288 .quad 0xbfd52a2d265bc5ab, 0x3fe0842108421084 // a = 15/16
289 .quad 0xbfd53aad05b99b7d, 0x3fe07b9f29b8eae2 // a = 241/256
290 .quad 0xbfd54b2467999498, 0x3fe073260a47f7c6 // a = 121/128
291 .quad 0xbfd55b9354b40bcd, 0x3fe06ab59c7912fb // a = 243/256
292 .quad 0xbfd56bf9d5b3f399, 0x3fe0624dd2f1a9fc // a = 61/64
293 .quad 0xbfd57c57f336f191, 0x3fe059eea0727586 // a = 245/256
294 .quad 0xbfd58cadb5cd7989, 0x3fe05197f7d73404 // a = 123/128
295 .quad 0xbfd59cfb25fae87e, 0x3fe04949cc1664c5 // a = 247/256
296 .quad 0xbfd5ad404c359f2d, 0x3fe0410410410410 // a = 31/32
297 .quad 0xbfd5bd7d30e71c73, 0x3fe038c6b78247fc // a = 249/256
298 .quad 0xbfd5cdb1dc6c1765, 0x3fe03091b51f5e1a // a = 125/128
299 .quad 0xbfd5ddde57149923, 0x3fe02864fc7729e9 // a = 251/256
300 .quad 0xbfd5ee02a9241675, 0x3fe0204081020408 // a = 63/64
301 .quad 0xbfd5fe1edad18919, 0x3fe0182436517a37 // a = 253/256
302 .quad 0xbfd60e32f44788d9, 0x3fe0101010101010 // a = 127/128
303 .quad 0xbfd61e3efda46467, 0x3fe0080402010080 // a = 255/256
304
305
306
307/********************************************************
308void gen_small_table() {
309 union { double d; uint64_t x; } constant;
310
311 int i,p,q;
312 for (i=0; i<256; ++i) {
313 double a = (double)i / 256.0;
314 constant.d = atanh(a); printf("\t\t\t\t.quad 0x%016llx,", constant.x);
315 constant.d = 1.0 / (1.0 - a*a); printf(" 0x%016llx", constant.x);
316 if (!i) {
317 printf("\t// a = 0\n");
318 } else {
319 p = i; q = 256;
320 while (!(p % 2)) {
321 p = p/2; q = q/2;
322 }
323 printf("\t// a = %d/%d\n", p, q);
324 }
325 }
326}
327*********************************************************/
328
329 // atanh(hi) 1/(1-hi^2)
330small_table: .quad 0x0000000000000000, 0x3ff0000000000000 // a = 0
331 .quad 0x3f7000055558888b, 0x3ff0001000100010 // a = 1/256
332 .quad 0x3f8000155588891b, 0x3ff0004001000400 // a = 1/128
333 .quad 0x3f8800480184d690, 0x3ff0009005102d92 // a = 3/256
334 .quad 0x3f9000555888ad1c, 0x3ff0010010010010 // a = 1/64
335 .quad 0x3f9400a6b46f591b, 0x3ff001902713d0ef // a = 5/256
336 .quad 0x3f980120184f3ded, 0x3ff00240510b659a // a = 3/128
337 .quad 0x3f9c01c989e21e45, 0x3ff00310962cbe90 // a = 7/256
338 .quad 0x3fa001558891aee2, 0x3ff0040100401004 // a = 1/32
339 .quad 0x3fa201e65c5878df, 0x3ff005119a91e82a // a = 9/256
340 .quad 0x3fa4029b471650a5, 0x3ff0064271f48383 // a = 5/128
341 .quad 0x3fa60378514ed016, 0x3ff0079394c14f5f // a = 11/256
342 .quad 0x3fa8048185694820, 0x3ff0090512da9af7 // a = 3/64
343 .quad 0x3faa05baefe1fa74, 0x3ff00a96fdad7784 // a = 13/256
344 .quad 0x3fac07289f7b9245, 0x3ff00c496833c7a5 // a = 7/128
345 .quad 0x3fae08cea570e1e7, 0x3ff00e1c66f67ea5 // a = 15/256
346 .quad 0x3fb005588ad375ad, 0x3ff0101010101010 // a = 1/16
347 .quad 0x3fb1066a036f9cec, 0x3ff012247b2f1021 // a = 17/256
348 .quad 0x3fb2079dc9754942, 0x3ff01459c19905ab // a = 9/128
349 .quad 0x3fb308f5eb6e013e, 0x3ff016affe2d6e10 // a = 19/256
350 .quad 0x3fb40a74799e283e, 0x3ff019274d68f3fd // a = 5/64
351 .quad 0x3fb50c1b861eec5e, 0x3ff01bbfcd68d99f // a = 21/256
352 .quad 0x3fb60ded24f86c90, 0x3ff01e799dee9716 // a = 11/128
353 .quad 0x3fb70feb6c3c1a5a, 0x3ff02154e063adfb // a = 23/256
354 .quad 0x3fb81218741f5a6a, 0x3ff02451b7ddb2d2 // a = 3/32
355 .quad 0x3fb9147657166782, 0x3ff0277049228d5b // a = 25/256
356 .quad 0x3fba170731ef7b1e, 0x3ff02ab0baacf0ac // a = 13/128
357 .quad 0x3fbb19cd23ee3f68, 0x3ff02e1334b10c2c // a = 27/256
358 .quad 0x3fbc1cca4ee78e02, 0x3ff03197e121767b // a = 7/64
359 .quad 0x3fbd2000d75d7f60, 0x3ff0353eebb45366 // a = 29/256
360 .quad 0x3fbe2372e49bce55, 0x3ff0390881e8b620 // a = 15/128
361 .quad 0x3fbf2722a0d493b5, 0x3ff03cf4d30c4100 // a = 31/256
362 .quad 0x3fc015891c9eaef7, 0x3ff0410410410410 // a = 1/8
363 .quad 0x3fc097a1ef16543f, 0x3ff045366c839bda // a = 33/256
364 .quad 0x3fc119dce19bdbaf, 0x3ff0498c1cb191d7 // a = 17/128
365 .quad 0x3fc19c3b0fa86d54, 0x3ff04e0557900010 // a = 35/256
366 .quad 0x3fc21ebd96730f38, 0x3ff052a255d27987 // a = 9/64
367 .quad 0x3fc2a165950035bc, 0x3ff0576352223903 // a = 37/256
368 .quad 0x3fc324342c318e81, 0x3ff05c488925980e // a = 19/128
369 .quad 0x3fc3a72a7ed6082c, 0x3ff061523987cfea // a = 39/256
370 .quad 0x3fc42a49b1ba196a, 0x3ff06680a4010668 // a = 5/32
371 .quad 0x3fc4ad92ebb84987, 0x3ff06bd40b5ea891 // a = 41/256
372 .quad 0x3fc5310755c9fd18, 0x3ff0714cb48c1542 // a = 21/128
373 .quad 0x3fc5b4a81b18894f, 0x3ff076eae69b99da // a = 43/256
374 .quad 0x3fc63876690e9070, 0x3ff07caeeacfc334 // a = 11/64
375 .quad 0x3fc6bc736f69aa39, 0x3ff082990ca50557 // a = 45/256
376 .quad 0x3fc740a0604c5adb, 0x3ff088a999dbbc40 // a = 23/128
377 .quad 0x3fc7c4fe70505b75, 0x3ff08ee0e282885b // a = 47/256
378 .quad 0x3fc8498ed69936dc, 0x3ff0953f39010954 // a = 3/16
379 .quad 0x3fc8ce52cce73dc7, 0x3ff09bc4f222fa0b // a = 49/256
380 .quad 0x3fc9534b8faad565, 0x3ff0a2726523b088 // a = 25/128
381 .quad 0x3fc9d87a5e18238b, 0x3ff0a947ebba04fd // a = 51/256
382 .quad 0x3fca5de07a3b1bc2, 0x3ff0b045e224a2f9 // a = 13/64
383 .quad 0x3fcae37f290bf096, 0x3ff0b76ca736c81a // a = 53/256
384 .quad 0x3fcb6957b283ec92, 0x3ff0bebc9c657399 // a = 27/128
385 .quad 0x3fcbef6b61b2b693, 0x3ff0c63625d50a60 // a = 55/256
386 .quad 0x3fcc75bb84d40518, 0x3ff0cdd9aa677344 // a = 7/32
387 .quad 0x3fccfc496d65c453, 0x3ff0d5a793caaf5c // a = 57/256
388 .quad 0x3fcd8316703eb314, 0x3ff0dda04e87f26e // a = 29/128
389 .quad 0x3fce0a23e5a57a70, 0x3ff0e5c44a133fbe // a = 59/256
390 .quad 0x3fce917329684475, 0x3ff0ee13f8db8f93 // a = 15/64
391 .quad 0x3fcf19059af4d646, 0x3ff0f68fd05b8216 // a = 61/256
392 .quad 0x3fcfa0dc9d7131ff, 0x3ff0ff38492aa44b // a = 31/128
393 .quad 0x3fd0147ccbea629a, 0x3ff1080ddf0f4c2c // a = 63/256
394 .quad 0x3fd058aefa811451, 0x3ff1111111111111 // a = 1/4
395 .quad 0x3fd09d0591f0bb21, 0x3ff11a42618be5dd // a = 65/256
396 .quad 0x3fd0e1814bbd9d56, 0x3ff123a25643da93 // a = 33/128
397 .quad 0x3fd12622e38a03ab, 0x3ff12d3178798b4c // a = 67/256
398 .quad 0x3fd16aeb1724557c, 0x3ff136f054ff42a4 // a = 17/64
399 .quad 0x3fd1afdaa6958afb, 0x3ff140df7c4ed62d // a = 69/256
400 .quad 0x3fd1f4f2542ff85b, 0x3ff14aff82a0438d // a = 35/128
401 .quad 0x3fd23a32e49e74ec, 0x3ff1555100011555 // a = 71/256
402 .quad 0x3fd27f9d1ef3e177, 0x3ff15fd4906c96f1 // a = 9/32
403 .quad 0x3fd2c531ccbb110c, 0x3ff16a8ad3e4df4c // a = 73/256
404 .quad 0x3fd30af1ba0717b8, 0x3ff175746e8cba40 // a = 37/128
405 .quad 0x3fd350ddb58402ab, 0x3ff1809208c27917 // a = 75/256
406 .quad 0x3fd396f69087fd7c, 0x3ff18be44f3bb2f6 // a = 19/64
407 .quad 0x3fd3dd3d1f24e85f, 0x3ff1976bf321fe4a // a = 77/256
408 .quad 0x3fd423b2383a6343, 0x3ff1a329aa30acca // a = 39/128
409 .quad 0x3fd46a56b58851f7, 0x3ff1af1e2ed3940c // a = 79/256
410 .quad 0x3fd4b12b73c1dd95, 0x3ff1bb4a4046ed29 // a = 5/16
411 .quad 0x3fd4f83152a0f7b5, 0x3ff1c7aea2b8565d // a = 81/256
412 .quad 0x3fd53f6934fa63f8, 0x3ff1d44c1f69021b // a = 41/128
413 .quad 0x3fd586d400d24cbe, 0x3ff1e12384d11f8a // a = 83/256
414 .quad 0x3fd5ce729f71680a, 0x3ff1ee35a6c48900 // a = 21/64
415 .quad 0x3fd61645fd7ab1ba, 0x3ff1fb835e98c5a2 // a = 85/256
416 .quad 0x3fd65e4f0b01c08e, 0x3ff2090d8b4c6bdc // a = 43/128
417 .quad 0x3fd6a68ebba1bb84, 0x3ff216d511aff336 // a = 87/256
418 .quad 0x3fd6ef060694f581, 0x3ff224dadc900489 // a = 11/32
419 .quad 0x3fd737b5e6cd3547, 0x3ff2331fdce15884 // a = 89/256
420 .quad 0x3fd7809f5b0cb028, 0x3ff241a509ee3506 // a = 45/128
421 .quad 0x3fd7c9c365ffbdfc, 0x3ff2506b61859acc // a = 91/256
422 .quad 0x3fd813230e574d5a, 0x3ff25f73e82c35af // a = 23/64
423 .quad 0x3fd85cbf5ee41f2a, 0x3ff26ebfa94f2298 // a = 93/256
424 .quad 0x3fd8a69966b2d128, 0x3ff27e4fb7789f5c // a = 47/128
425 .quad 0x3fd8f0b23928bf15, 0x3ff28e252c86b994 // a = 95/256
426 .quad 0x3fd93b0aee21c2c8, 0x3ff29e4129e4129e // a = 3/8
427 .quad 0x3fd985a4a20edba2, 0x3ff2aea4d8c2d024 // a = 97/256
428 .quad 0x3fd9d0807615c643, 0x3ff2bf516a59d19c // a = 49/128
429 .quad 0x3fda1b9f90318dcb, 0x3ff2d04818244483 // a = 99/256
430 .quad 0x3fda67031b542059, 0x3ff2e18a2423b269 // a = 25/64
431 .quad 0x3fdab2ac4788f0dc, 0x3ff2f318d924a53c // a = 101/256
432 .quad 0x3fdafe9c4a18b0e3, 0x3ff304f58b05ffd0 // a = 51/128
433 .quad 0x3fdb4ad45dae2d59, 0x3ff3172197032a26 // a = 103/256
434 .quad 0x3fdb9755c27c59e0, 0x3ff3299e6401329a // a = 13/32
435 .quad 0x3fdbe421be6596cb, 0x3ff33c6d62df06fc // a = 105/256
436 .quad 0x3fdc31399d243e72, 0x3ff34f900ec8ea4b // a = 53/128
437 .quad 0x3fdc7e9eb074870b, 0x3ff36307ed8f4df6 // a = 107/256
438 .quad 0x3fdccc52503fc6fe, 0x3ff376d69001376d // a = 27/64
439 .quad 0x3fdd1a55dac92a26, 0x3ff38afd924a5d42 // a = 109/256
440 .quad 0x3fdd68aab4dbe74b, 0x3ff39f7e9c55292e // a = 55/128
441 .quad 0x3fddb75249fb05b3, 0x3ff3b45b6230cf21 // a = 111/256
442 .quad 0x3fde064e0c92c396, 0x3ff3c995a47babe7 // a = 7/16
443 .quad 0x3fde559f762baeee, 0x3ff3df2f30d22100 // a = 113/256
444 .quad 0x3fdea548079f8314, 0x3ff3f529e2422615 // a = 57/128
445 .quad 0x3fdef549494fde6b, 0x3ff40b87a1c3cbdf // a = 115/256
446 .quad 0x3fdf45a4cb5ee468, 0x3ff4224a66b6ef90 // a = 29/64
447 .quad 0x3fdf965c25e9e132, 0x3ff4397437666199 // a = 117/256
448 .quad 0x3fdfe770f9460541, 0x3ff451072990c667 // a = 59/128
449 .quad 0x3fe01c72771fa832, 0x3ff4690562f77bee // a = 119/256
450 .quad 0x3fe0455cdb2ce279, 0x3ff4817119f3d325 // a = 15/32
451 .quad 0x3fe06e78860a7e8c, 0x3ff49a4c9612f15e // a = 121/256
452 .quad 0x3fe097c659991ec1, 0x3ff4b39a30b8b264 // a = 61/128
453 .quad 0x3fe0c1473c7e911c, 0x3ff4cd5c55c9e98b // a = 123/256
454 .quad 0x3fe0eafc1a4b81ea, 0x3ff4e795845e65bf // a = 31/64
455 .quad 0x3fe114e5e3a29a89, 0x3ff502484f7b2291 // a = 125/256
456 .quad 0x3fe13f058e611d13, 0x3ff51d775ed516de // a = 63/128
457 .quad 0x3fe1695c15c90ea9, 0x3ff539256f9d18b1 // a = 127/256
458 .quad 0x3fe193ea7aad030b, 0x3ff5555555555555 // a = 1/2
459 .quad 0x3fe1beb1c39d9d1a, 0x3ff57209fab0e4c8 // a = 129/256
460 .quad 0x3fe1e9b2fd18d91c, 0x3ff58f46627e080b // a = 65/128
461 .quad 0x3fe214ef39bb369d, 0x3ff5ad0da89bab4f // a = 131/256
462 .quad 0x3fe240679272d92c, 0x3ff5cb6302face89 // a = 33/64
463 .quad 0x3fe26c1d26b4b850, 0x3ff5ea49c2ac81d8 // a = 133/256
464 .quad 0x3fe298111cb3f8ad, 0x3ff609c554fd2e40 // a = 67/128
465 .quad 0x3fe2c444a19b89a5, 0x3ff629d9449defb6 // a = 135/256
466 .quad 0x3fe2f0b8e9ca2471, 0x3ff64a893adcd25f // a = 17/32
467 .quad 0x3fe31d6f3110cb49, 0x3ff66bd900ecd324 // a = 137/256
468 .quad 0x3fe34a68baf3e921, 0x3ff68dcc813e92e9 // a = 69/128
469 .quad 0x3fe377a6d2ef3448, 0x3ff6b067c8eabc0a // a = 139/256
470 .quad 0x3fe3a52accbc786f, 0x3ff6d3af092f2b6d // a = 35/64
471 .quad 0x3fe3d2f6049d6eb3, 0x3ff6f7a69900016f // a = 141/256
472 .quad 0x3fe40109dfa8ccb5, 0x3ff71c52f6add38a // a = 71/128
473 .quad 0x3fe42f67cc1ab64c, 0x3ff741b8c9a24d98 // a = 143/256
474 .quad 0x3fe45e1141a8c010, 0x3ff767dce434a9b1 // a = 9/16
475 .quad 0x3fe48d07c1d9b3ea, 0x3ff78ec445977f4f // a = 145/256
476 .quad 0x3fe4bc4cd8614bf6, 0x3ff7b6741be18685 // a = 73/128
477 .quad 0x3fe4ebe21b801b3d, 0x3ff7def1c6330a52 // a = 147/256
478 .quad 0x3fe51bc92c67dfa3, 0x3ff80842d6f9e5e7 // a = 37/64
479 .quad 0x3fe54c03b7a47bf0, 0x3ff8326d16560c53 // a = 149/256
480 .quad 0x3fe57c937589dd3f, 0x3ff85d7684a0c0a3 // a = 75/128
481 .quad 0x3fe5ad7a2aa7137d, 0x3ff889655d18ce68 // a = 151/256
482 .quad 0x3fe5deb9a83ee95c, 0x3ff8b64018b64019 // a = 19/32
483 .quad 0x3fe61053ccc64d7b, 0x3ff8e40d7128425c // a = 153/256
484 .quad 0x3fe6424a8468e3d0, 0x3ff912d464001913 // a = 77/128
485 .quad 0x3fe6749fc9941cbb, 0x3ff9429c360c45bd // a = 155/256
486 .quad 0x3fe6a755a5893549, 0x3ff9736c76e73ebc // a = 39/64
487 .quad 0x3fe6da6e30f68b8e, 0x3ff9a54d04bd5ccc // a = 157/256
488 .quad 0x3fe70deb9498b947, 0x3ff9d846104df033 // a = 79/128
489 .quad 0x3fe741d009e3ef5a, 0x3ffa0c60212bc26a // a = 159/256
490 .quad 0x3fe7761ddbb61598, 0x3ffa41a41a41a41a // a = 5/8
491 .quad 0x3fe7aad767123bc7, 0x3ffa781b3ea00af6 // a = 161/256
492 .quad 0x3fe7dfff1be5f383, 0x3ffaafcf36993030 // a = 81/128
493 .quad 0x3fe815977dd935af, 0x3ffae8ca15319829 // a = 163/256
494 .quad 0x3fe84ba3252982b5, 0x3ffb23165deb6f69 // a = 41/64
495 .quad 0x3fe88224bf90fa20, 0x3ffb5ebf0af3b992 // a = 165/256
496 .quad 0x3fe8b91f113a34a3, 0x3ffb9bcf93b8ede9 // a = 83/128
497 .quad 0x3fe8f094f5c1bba9, 0x3ffbda53f3f34c2e // a = 167/256
498 .quad 0x3fe9288961460abe, 0x3ffc1a58b327f576 // a = 21/32
499 .quad 0x3fe960ff61871a12, 0x3ffc5beaecb0a99a // a = 169/256
500 .quad 0x3fe999fa1f16860c, 0x3ffc9f185852f521 // a = 85/128
501 .quad 0x3fe9d37cde997e64, 0x3ffce3ef53729f97 // a = 171/256
502 .quad 0x3fea0d8b021dc00c, 0x3ffd2a7eeaec4a48 // a = 43/64
503 .quad 0x3fea48280a82f83e, 0x3ffd72d6e5a66e97 // a = 173/256
504 .quad 0x3fea835798fa0cba, 0x3ffdbd07cfe84d5e // a = 87/128
505 .quad 0x3feabf1d709be5fb, 0x3ffe09230787ea78 // a = 175/256
506 .quad 0x3feafb7d78197bed, 0x3ffe573ac901e574 // a = 11/16
507 .quad 0x3feb387bbb870d4d, 0x3ffea7623d8fe82c // a = 177/256
508 .quad 0x3feb761c6e44955e, 0x3ffef9ad8a548440 // a = 89/128
509 .quad 0x3febb463ed05c38d, 0x3fff4e31e0b5b7e6 // a = 179/256
510 .quad 0x3febf356bffbedc0, 0x3fffa5059001fa50 // a = 45/64
511 .quad 0x3fec32f99d24b0c8, 0x3ffffe40187ea913 // a = 181/256
512 .quad 0x3fec73516ac03329, 0x40002cfd200102d0 // a = 91/128
513 .quad 0x3fecb46341f246c6, 0x40005c27141f5891 // a = 183/256
514 .quad 0x3fecf6347191f5b5, 0x40008cabb37565e2 // a = 23/32
515 .quad 0x3fed38ca812b5fa0, 0x4000be998ff8ce1a // a = 185/256
516 .quad 0x3fed7c2b3438300b, 0x4000f20010f20011 // a = 93/128
517 .quad 0x3fedc05c8d936455, 0x400126ef8270fab7 // a = 187/256
518 .quad 0x3fee0564d32d9391, 0x40015d79261f33f6 // a = 47/64
519 .quad 0x3fee4b4a92077457, 0x400195af45931bd8 // a = 189/256
520 .quad 0x3fee9214a278f736, 0x4001cfa5464e21e2 // a = 95/128
521 .quad 0x3feed9ca2ccbf9e9, 0x40020b6fbf932b06 // a = 191/256
522 .quad 0x3fef2272ae325a57, 0x4002492492492492 // a = 3/4
523 .quad 0x3fef6c15fe200bf7, 0x400288db0323f01c // a = 193/256
524 .quad 0x3fefb6bc5412c9c8, 0x4002caabd7556820 // a = 97/128
525 .quad 0x3ff0013726e90b80, 0x40030eb17410dc80 // a = 195/256
526 .quad 0x3ff0279a7b19be98, 0x4003550801355080 // a = 49/64
527 .quad 0x3ff04e8ce6382f3a, 0x40039dcd8f7e31ca // a = 197/256
528 .quad 0x3ff07613660e2537, 0x4003e92242a773b1 // a = 99/128
529 .quad 0x3ff09e333ae64b65, 0x4004372880014373 // a = 199/256
530 .quad 0x3ff0c6f1ec420786, 0x4004880522014880 // a = 25/32
531 .quad 0x3ff0f0554dfbf7da, 0x4004dbdfb17409a8 // a = 201/256
532 .quad 0x3ff11a6385e30cf6, 0x400532e2a5092677 // a = 101/128
533 .quad 0x3ff1452311dbc75f, 0x40058d3ba8114219 // a = 203/256
534 .quad 0x3ff1709ace96ee39, 0x4005eb1be9658b37 // a = 51/64
535 .quad 0x3ff19cd1feef2a07, 0x40064cb87397b01f // a = 205/256
536 .quad 0x3ff1c9d0540158c9, 0x4006b24a8fb6f220 // a = 103/128
537 .quad 0x3ff1f79df6163ed8, 0x40071c10342d5c96 // a = 207/256
538 .quad 0x3ff226438e777c99, 0x40078a4c8178a4c8 // a = 13/16
539 .quad 0x3ff255ca524d82c3, 0x4007fd484ecf128b // a = 209/256
540 .quad 0x3ff2863c0ea8b8cf, 0x40087552c91cb5b8 // a = 105/128
541 .quad 0x3ff2b7a335dd4c9f, 0x4008f2c227337192 // a = 211/256
542 .quad 0x3ff2ea0aee5f5810, 0x400975f4768d3a48 // a = 53/64
543 .quad 0x3ff31d7f23546375, 0x4009ff5084a080c5 // a = 213/256
544 .quad 0x3ff3520c9718089f, 0x400a8f46e989d487 // a = 107/128
545 .quad 0x3ff387c0f7fbe5f1, 0x400b265339bb9f3a // a = 215/256
546 .quad 0x3ff3beaaf7978c58, 0x400bc4fd65883e7b // a = 27/32
547 .quad 0x3ff3f6da650c1586, 0x400c6bdb4ec15ed3 // a = 217/256
548 .quad 0x3ff430604ab1313f, 0x400d1b929e6308d0 // a = 109/128
549 .quad 0x3ff46b4f0fb778a3, 0x400dd4dae66843b4 // a = 219/256
550 .quad 0x3ff4a7ba9e66a99b, 0x400e98801e98801f // a = 55/64
551 .quad 0x3ff4e5b88fbf4bf0, 0x400f67658e7f8c33 // a = 221/256
552 .quad 0x3ff525605d6fc1b9, 0x401021449d84e212 // a = 111/128
553 .quad 0x3ff566cb9b3ef248, 0x40109583f9d88406 // a = 223/256
554 // a = 7/8
555
556.literal8
557.align 3
558one_256th: .quad 0x3f70000000000000 // 1/256
559one: .quad 0x3ff0000000000000
560one_plus_eps: .quad 0x3ff0000000000001 // 1 + ulp
561one_third: .quad 0x3fd5555555555555
562one_fifth: .quad 0x3fc999999999999a
563minimax_c: .quad 0x3ff0a3d70a3d70a4 // = 1.04
564third_over_c: .quad 0x3fd4834834834834 // = 1/3.12
565a_mask: .quad 0x7ffff00000000000ULL // top 8 bits of mantissa
566exp_mask: .quad 0x7ff0000000000000ULL // exponent bits / +Inf
567neglog2_2: .quad 0xbfd62e42fefa39ef // -ln(2)/2
568
569.literal4
570.align 2
571f256: .long 0x43800000 // 256.0f
572
573.text
574#if defined( __x86_64__ )
575 #define RELATIVE_ADDR( _a) (_a)( %rip )
576 #define INDEX %r8
577#elif defined( __i386__ )
578 #define RELATIVE_ADDR( _a) (_a)-atanhf_body( CX_P )
579 #define INDEX %edi
580.align 4
581atanhf_pic:
582 movl (%esp), %ecx // copy address of local_addr to %ecx
583 ret
584#else
585 #error arch not supported
586#endif
587
588ENTRY( atanhf )
589#if defined( __i386__ )
590 movl FRAME_SIZE( STACKP ), %eax
591 movss FRAME_SIZE( STACKP ), %xmm0
592 calll atanhf_pic
593atanhf_body:
594#else
595 movd %xmm0, %eax
596#endif
597 andl $0x7fffffff, %eax
598 movd %eax, %xmm1 // xmm1 <-- |x|
599 subl $0x3d000000, %eax // |x| - 1/32 as integers
600 cmpl $0x02600000, %eax // if (|x| < 1/32 or |x| >= 7/8 or isnan(x))
601 jae 2f // goto 2
602
603 /*
604 (Set with tab = 4 spaces)
605
606 For 1/32 < x < 7/8:
607
608 Compute a = floor(x*256), set hi = a/256. a is an integer between 0 and 223. hi is an approximation to
609 x, with
610
611 (1) 0 < (x - a) < 1/256
612
613 We base our approximation on the identity
614
615 1 + x
616 (2) atanh(x) = 1/2 log ------- = 1/2 ( log(1+x) - log(1-x) )
617 1 - x
618
619 Expand 1�x as follows:
620
621 (1 + x) = (1 + hi)(1 - even + odd) (1 - x) = (1 - hi)(1 - even - odd)
622
623 and solve for b,c to get:
624
625 x - a
626 c = --------- b = -ac
627 1 - a^2
628
629 Substituting into (2), we get:
630
631 atanh(x) = 1/2 ( log(1 + hi) + log(1 + even + odd) - log(1 - hi) - log(1 + even - odd) )
632
633 (3) = 1/2 atanh(hi) + 1/2 ( log(1 + even + odd) - log(1 + even - odd) )
634
635 We can store atanh(hi), and also the 1/(1-hi^2) factor used to compute odd, in a lookup table indexed on a.
636 A fairly straighforward analysis using (1) and the fact that a < 7/8 shows that
637
638 |even|, |odd| < 1/60
639
640 So a 4th-order taylor series will approximate the second term of (3) to very nearly the desired accuracy.
641 When we compute the series, some really truly wonderful cancellation takes place, leaving us with a beautiful
642 approximation:
643
644 1/2 (log(1+even+odd) - log(1+even-odd)) = odd - odd*even + odd*even^2 + 1/3 odd^3 - odd*even^3 - odd^3*even
645 = odd((1 - even)(1 + even^2) + (1/3 - even)odd^2)
646
647 Unfortunately, this is not *quite* accurate enough - it results in errors of approximately .75 ulps. So we
648 tweak the approximation ever so slightly to achieve the necessary accuracy:
649
650 1/2 (log(1+even+odd) - log(1+even-odd)) = odd((1 - even)(1 + even^2) + 1.04(1/3.12 - even)odd^2)
651
652 Putting this all together, we get
653
654 (4) atanh(x) = atanh(a) + odd((1 - even)(1 + even^2) + (1/3 - 1.04 even)odd^2) + (< .51 ulp)
655
656 - Stephen Canon, July 2007
657
658 */
659
660 xorps %xmm1, %xmm0 // xmm0 <-- signbit(x)
661 movaps %xmm1, %xmm2
662 mulss RELATIVE_ADDR(f256), %xmm1 // xmm1 <-- x * 256.0
663 cvtss2sd %xmm2, %xmm2 // xmm2 <-- (double)x
664 cvttss2si %xmm1, AX_P // trunc(x * 256.0)
665 cvtsi2sd %eax, %xmm1 // xmm1 <-- (double)trunc(x * 256.0)
666
667 shl $4, AX_P // stride of table is 16 bytes
668 mulsd RELATIVE_ADDR(one_256th), %xmm1 // xmm1 <-- hi = trunc(x * 256.0) / 256.0
669 lea RELATIVE_ADDR(small_table),DX_P
670 subsd %xmm1, %xmm2 // xmm2 <-- x - hi
671 movsd (DX_P,AX_P,1), %xmm7 // xmm7 <-- atanh(a)
672 mulsd 8(DX_P,AX_P,1), %xmm2 // xmm2 <-- odd
673 movsd RELATIVE_ADDR(minimax_c), %xmm3 // xmm3 <-- 1.04
674 movsd RELATIVE_ADDR(third_over_c),%xmm5 // xmm5 <-- 1/3.12
675 movsd RELATIVE_ADDR(one), %xmm6 // xmm6 <-- 1.0
676 mulsd %xmm2, %xmm1 // xmm1 <-- -even
677 mulsd %xmm2, %xmm3 // xmm3 <-- 1.04*odd
678 movapd %xmm1, %xmm4 // xmm4 <-- -even
679 mulsd %xmm1, %xmm1 // xmm1 <-- even*even
680 addsd %xmm4, %xmm5 // xmm5 <-- 1/3.12 - even
681 mulsd %xmm2, %xmm3 // xmm3 <-- 1.04*odd*odd
682 addsd %xmm6, %xmm4 // xmm4 <-- 1.0 - even
683 addsd %xmm6, %xmm1 // xmm1 <-- 1 + even*even
684 mulsd %xmm3, %xmm5 // xmm5 <-- (1/3 - 1.04even)*odd^2
685 mulsd %xmm1, %xmm4 // xmm4 <-- (1.0 - even)(1.0 + even^2)
686 addsd %xmm5, %xmm4 // xmm4 <-- (1.0 - even)(1.0 + even^2) + (1/3 - 1.04even)*odd^2
687 mulsd %xmm2, %xmm4 // xmm4 <-- odd*(1.0 - even)(1.0 + even^2) + (1/3 - 1.04even)*odd^2
688 addsd %xmm7, %xmm4 // xmm4 <-- atanh(x)
689 cvtsd2ss %xmm4, %xmm1
690 orps %xmm1, %xmm0 // signbit
691#if defined( __i386__ )
692 movss %xmm0, FRAME_SIZE( STACKP )
693 flds FRAME_SIZE( STACKP )
694#endif
695 ret
696
6972:
698 cmpl $0x02800000, %eax // if (|x| < 1/32 or |x| >= 1 or isnan(x))
699 jae 3f // goto 3
700
701 /*
702 (Set with tab = 4 spaces)
703
704 For 7/8 <= |x| < 1:
705
706 On this range, we use an approximation of the form:
707
708 (1) atanh(x) = -1/2 log((1-x)/2) - minimax polynomial in (1-x)
709
710 The second term of (1) is just a straightforward 4-term minimax polynomial. To compute the first term, consider
711
712 (1-x) = mantissa * 2^exponent
713
714 with 1.0 <= mantissa < 2. Set a = floor(256(mantissa - 1)), and let hi = a/256 and lo = (mantissa-(1+hi))/(1+hi).
715 Then:
716
717 (2) -1/2 log((1-x)/2) = -ln2/2 * (exponent-1) - 1/2 log(1+hi) - 1/2 log(1+lo)
718
719 The first term can be computed directly; the second term is looked up in the table indexed on a, and the final
720 term is approximated via another minimax polynomial.
721
722 - Stephen Canon, July 2007
723
724 */
725
726 xorps %xmm1, %xmm0 // xmm0 <-- signbit(x)
727 cvtss2sd %xmm1, %xmm1 // xmm1 <-- (double)|x|
728 movsd RELATIVE_ADDR(one), %xmm2 // xmm2 <-- 1.0
729 movsd RELATIVE_ADDR(exp_mask), %xmm3 // xmm3 <-- 0x7ff0000000000000ULL
730 movapd %xmm2, %xmm4 // xmm4 <-- 1.0
731 subsd %xmm1, %xmm2 // xmm2 <-- 1.0 - x
732 movsd RELATIVE_ADDR(a_mask), %xmm5 // xmm5 <-- 0x7ffff00000000000ULL
733 andnpd %xmm2, %xmm3
734 movapd %xmm3, %xmm1
735 orpd %xmm4, %xmm3 // xmm3 <-- mantissa(1-x)
736 andpd %xmm3, %xmm5 // xmm5 <-- 1 + hi
737
738 movlhps %xmm2, %xmm3
739
740 subsd %xmm5, %xmm3 // xmm3 <-- mantissa(1-x) - (1+hi)
741 lea RELATIVE_ADDR(big_table), DX_P
742 psrlq $52-(8+4), %xmm1 // xmm1 <-- 8 bit integer a | 4 bit junk
743 movd %xmm1, %eax
744 and $0xff0, AX_P
745 mulsd 8(DX_P,AX_P,1), %xmm3 // xmm3 <-- u = [ lo , 1-x ]
746 movsd (DX_P,AX_P,1), %xmm1 // xmm1 <-- -1/2 log(1+hi)
747
748 movapd %xmm3, %xmm4 // xmm4 <-- u
749 mulpd %xmm3, %xmm3 // xmm3 <-- uu
750 movapd -48(DX_P), %xmm6 // xmm6 <-- c3/c4
751 movapd -32(DX_P), %xmm5 // xmm5 <-- c2
752
753 psrlq $52, %xmm2 // xmm2 <-- biased exponent of (1-x)
754 movd %xmm2, %eax
755 subl $0x400, %eax // unbiased exponent - 1
756 cvtsi2sd %eax, %xmm2 // xmm2 <-- unbiased exponent - 1
757 mulsd RELATIVE_ADDR(neglog2_2), %xmm2 // xmm2 <-- -ln2/2 * (exponent-1)
758 addsd %xmm2, %xmm1 // xmm1 <-- -ln2/2 * (exponent-1) - 1/2 log(1+hi)
759
760 mulpd %xmm4, %xmm6 // xmm6 <-- c3/c4 u
761 mulpd %xmm4, %xmm5 // xmm5 <-- c2 u
762 addpd -16(DX_P), %xmm4 // xmm4 <-- c1/c2 + u
763 addpd %xmm3, %xmm6 // xmm6 <-- c3/c4 u + uu
764 mulpd -64(DX_P), %xmm3 // xmm3 <-- c4uu
765 mulpd %xmm5, %xmm4 // xmm4 <-- c1 u + c2 u
766 mulpd %xmm3, %xmm6 // xmm3 <-- c3 uuu + c4 uuuu
767
768 // Now add up all the terms
769 addsd %xmm1, %xmm4
770 addpd %xmm6, %xmm4
771 movhlps %xmm4, %xmm1
772 addsd %xmm4, %xmm1
773
774 // Convert to single precision
775 cvtsd2ss %xmm1, %xmm1
776 orps %xmm1, %xmm0 // signbit
777#if defined( __i386__ )
778 movss %xmm0, FRAME_SIZE( STACKP )
779 flds FRAME_SIZE( STACKP )
780#endif
781 ret
782
7833: // if (|x| >= 1 or isnan(x))
784 jge 5f // goto 5
785
786 // Now we know |x| < 1/32
787 addl $0x3d000000, %eax // |x|
788 cmpl $0x38000000, %eax // if (|x| <= 2^-13)
789 jle 4f // goto 4
790
791 cvtss2sd %xmm0, %xmm1
792 movapd %xmm1, %xmm2
793 mulsd %xmm2, %xmm2 // xmm2 <-- x*x
794 movsd RELATIVE_ADDR(one_fifth), %xmm3
795 mulsd %xmm2, %xmm3 // xmm3 <-- .2xx
796 addsd RELATIVE_ADDR(one_third), %xmm3 // xmm3 <-- .333 + .2xx
797 mulsd %xmm3, %xmm2 // xmm2 <-- .333xx + .2xxxx
798 mulsd %xmm1, %xmm2 // xmm2 <-- .333xxx + .2xxxxx
799 addsd %xmm2, %xmm1 // xmm1 <-- x + .333xxx + .2xxxxx
800 cvtsd2ss %xmm1, %xmm0
801
802#if defined( __i386__ )
803 movss %xmm0, FRAME_SIZE( STACKP )
804 flds FRAME_SIZE( STACKP )
805#endif
806 ret
807
8084: // very small case, just multiply by (1+ulp) and return
809 cvtss2sd %xmm0, %xmm1
810 mulsd RELATIVE_ADDR(one_plus_eps),%xmm1 // xmm1 <-- x * (1 + ulp) for rounding and inexact.
811 cvtsd2ss %xmm1, %xmm0
812#if defined( __i386__ )
813 movss %xmm0, FRAME_SIZE( STACKP )
814 flds FRAME_SIZE( STACKP )
815#endif
816 ret
817
818
8195: // if x = �1
820 je 7f // goto 7
821 ucomiss %xmm0, %xmm0 // if isnan(x)
822 jp 6f // goto 6
823
824 pcmpeqb %xmm0, %xmm0
825 pslld $23, %xmm0
8266:
827 movaps %xmm0, %xmm1
828 subss %xmm1, %xmm0
829#if defined( __i386__ )
830 movss %xmm0, FRAME_SIZE( STACKP )
831 flds FRAME_SIZE( STACKP )
832#endif
833 ret
834
8357:
836 xorps %xmm1, %xmm1 // xmm1 <-- 0
837 divss %xmm1, %xmm0 // xmm0 <-- �1/0 = �inf, div by zero
838#if defined( __i386__ )
839 movss %xmm0, FRAME_SIZE( STACKP )
840 flds FRAME_SIZE( STACKP )
841#endif
842 ret
843