1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
|
/*
* Shared data between exp, exp2 and pow.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include "exp_data.h"
#define N (1 << EXP_TABLE_BITS)
const struct exp_data __exp_data = {
// N/ln2
.invln2N = 0x1.71547652b82fep0 * N,
// -ln2/N
.negln2hiN = -0x1.62e42fefa0000p-8,
.negln2loN = -0x1.cf79abc9e3b3ap-47,
// Used for rounding when !TOINT_INTRINSICS
#if EXP_USE_TOINT_NARROW
.shift = 0x1800000000.8p0,
#else
.shift = 0x1.8p52,
#endif
// exp polynomial coefficients.
.poly = {
// abs error: 1.555*2^-66
// ulp error: 0.509 (0.511 without fma)
// if |x| < ln2/256+eps
// abs error if |x| < ln2/256+0x1p-15: 1.09*2^-65
// abs error if |x| < ln2/128: 1.7145*2^-56
0x1.ffffffffffdbdp-2,
0x1.555555555543cp-3,
0x1.55555cf172b91p-5,
0x1.1111167a4d017p-7,
},
.exp2_shift = 0x1.8p52 / N,
// exp2 polynomial coefficients.
.exp2_poly = {
// abs error: 1.2195*2^-65
// ulp error: 0.507 (0.511 without fma)
// if |x| < 1/256
// abs error if |x| < 1/128: 1.9941*2^-56
0x1.62e42fefa39efp-1,
0x1.ebfbdff82c424p-3,
0x1.c6b08d70cf4b5p-5,
0x1.3b2abd24650ccp-7,
0x1.5d7e09b4e3a84p-10,
},
// 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N)
// tab[2*k] = asuint64(T[k])
// tab[2*k+1] = asuint64(H[k]) - (k << 52)/N
.tab = {
0x0, 0x3ff0000000000000,
0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335,
0xbc7160139cd8dc5d, 0x3fefec9a3e778061,
0xbc905e7a108766d1, 0x3fefe315e86e7f85,
0x3c8cd2523567f613, 0x3fefd9b0d3158574,
0xbc8bce8023f98efa, 0x3fefd06b29ddf6de,
0x3c60f74e61e6c861, 0x3fefc74518759bc8,
0x3c90a3e45b33d399, 0x3fefbe3ecac6f383,
0x3c979aa65d837b6d, 0x3fefb5586cf9890f,
0x3c8eb51a92fdeffc, 0x3fefac922b7247f7,
0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2,
0xbc6a033489906e0b, 0x3fef9b66affed31b,
0xbc9556522a2fbd0e, 0x3fef9301d0125b51,
0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc,
0xbc91c923b9d5f416, 0x3fef829aaea92de0,
0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51,
0xbc801b15eaa59348, 0x3fef72b83c7d517b,
0xbc8f1ff055de323d, 0x3fef6af9388c8dea,
0x3c8b898c3f1353bf, 0x3fef635beb6fcb75,
0xbc96d99c7611eb26, 0x3fef5be084045cd4,
0x3c9aecf73e3a2f60, 0x3fef54873168b9aa,
0xbc8fe782cb86389d, 0x3fef4d5022fcd91d,
0x3c8a6f4144a6c38d, 0x3fef463b88628cd6,
0x3c807a05b0e4047d, 0x3fef3f49917ddc96,
0x3c968efde3a8a894, 0x3fef387a6e756238,
0x3c875e18f274487d, 0x3fef31ce4fb2a63f,
0x3c80472b981fe7f2, 0x3fef2b4565e27cdd,
0xbc96b87b3f71085e, 0x3fef24dfe1f56381,
0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1,
0xbc3d219b1a6fbffa, 0x3fef187fd0dad990,
0x3c8b3782720c0ab4, 0x3fef1285a6e4030b,
0x3c6e149289cecb8f, 0x3fef0cafa93e2f56,
0x3c834d754db0abb6, 0x3fef06fe0a31b715,
0x3c864201e2ac744c, 0x3fef0170fc4cd831,
0x3c8fdd395dd3f84a, 0x3feefc08b26416ff,
0xbc86a3803b8e5b04, 0x3feef6c55f929ff1,
0xbc924aedcc4b5068, 0x3feef1a7373aa9cb,
0xbc9907f81b512d8e, 0x3feeecae6d05d866,
0xbc71d1e83e9436d2, 0x3feee7db34e59ff7,
0xbc991919b3ce1b15, 0x3feee32dc313a8e5,
0x3c859f48a72a4c6d, 0x3feedea64c123422,
0xbc9312607a28698a, 0x3feeda4504ac801c,
0xbc58a78f4817895b, 0x3feed60a21f72e2a,
0xbc7c2c9b67499a1b, 0x3feed1f5d950a897,
0x3c4363ed60c2ac11, 0x3feece086061892d,
0x3c9666093b0664ef, 0x3feeca41ed1d0057,
0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0,
0x3c93ff8e3f0f1230, 0x3feec32af0d7d3de,
0x3c7690cebb7aafb0, 0x3feebfdad5362a27,
0x3c931dbdeb54e077, 0x3feebcb299fddd0d,
0xbc8f94340071a38e, 0x3feeb9b2769d2ca7,
0xbc87deccdc93a349, 0x3feeb6daa2cf6642,
0xbc78dec6bd0f385f, 0x3feeb42b569d4f82,
0xbc861246ec7b5cf6, 0x3feeb1a4ca5d920f,
0x3c93350518fdd78e, 0x3feeaf4736b527da,
0x3c7b98b72f8a9b05, 0x3feead12d497c7fd,
0x3c9063e1e21c5409, 0x3feeab07dd485429,
0x3c34c7855019c6ea, 0x3feea9268a5946b7,
0x3c9432e62b64c035, 0x3feea76f15ad2148,
0xbc8ce44a6199769f, 0x3feea5e1b976dc09,
0xbc8c33c53bef4da8, 0x3feea47eb03a5585,
0xbc845378892be9ae, 0x3feea34634ccc320,
0xbc93cedd78565858, 0x3feea23882552225,
0x3c5710aa807e1964, 0x3feea155d44ca973,
0xbc93b3efbf5e2228, 0x3feea09e667f3bcd,
0xbc6a12ad8734b982, 0x3feea012750bdabf,
0xbc6367efb86da9ee, 0x3fee9fb23c651a2f,
0xbc80dc3d54e08851, 0x3fee9f7df9519484,
0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74,
0xbc86ee4ac08b7db0, 0x3fee9f9a48a58174,
0xbc8619321e55e68a, 0x3fee9feb564267c9,
0x3c909ccb5e09d4d3, 0x3feea0694fde5d3f,
0xbc7b32dcb94da51d, 0x3feea11473eb0187,
0x3c94ecfd5467c06b, 0x3feea1ed0130c132,
0x3c65ebe1abd66c55, 0x3feea2f336cf4e62,
0xbc88a1c52fb3cf42, 0x3feea427543e1a12,
0xbc9369b6f13b3734, 0x3feea589994cce13,
0xbc805e843a19ff1e, 0x3feea71a4623c7ad,
0xbc94d450d872576e, 0x3feea8d99b4492ed,
0x3c90ad675b0e8a00, 0x3feeaac7d98a6699,
0x3c8db72fc1f0eab4, 0x3feeace5422aa0db,
0xbc65b6609cc5e7ff, 0x3feeaf3216b5448c,
0x3c7bf68359f35f44, 0x3feeb1ae99157736,
0xbc93091fa71e3d83, 0x3feeb45b0b91ffc6,
0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5,
0xbc6c23f97c90b959, 0x3feeba44cbc8520f,
0xbc92434322f4f9aa, 0x3feebd829fde4e50,
0xbc85ca6cd7668e4b, 0x3feec0f170ca07ba,
0x3c71affc2b91ce27, 0x3feec49182a3f090,
0x3c6dd235e10a73bb, 0x3feec86319e32323,
0xbc87c50422622263, 0x3feecc667b5de565,
0x3c8b1c86e3e231d5, 0x3feed09bec4a2d33,
0xbc91bbd1d3bcbb15, 0x3feed503b23e255d,
0x3c90cc319cee31d2, 0x3feed99e1330b358,
0x3c8469846e735ab3, 0x3feede6b5579fdbf,
0xbc82dfcd978e9db4, 0x3feee36bbfd3f37a,
0x3c8c1a7792cb3387, 0x3feee89f995ad3ad,
0xbc907b8f4ad1d9fa, 0x3feeee07298db666,
0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb,
0xbc90a40e3da6f640, 0x3feef9728de5593a,
0xbc68d6f438ad9334, 0x3feeff76f2fb5e47,
0xbc91eee26b588a35, 0x3fef05b030a1064a,
0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2,
0xbc91bdfbfa9298ac, 0x3fef12c25bd71e09,
0x3c736eae30af0cb3, 0x3fef199bdd85529c,
0x3c8ee3325c9ffd94, 0x3fef20ab5fffd07a,
0x3c84e08fd10959ac, 0x3fef27f12e57d14b,
0x3c63cdaf384e1a67, 0x3fef2f6d9406e7b5,
0x3c676b2c6c921968, 0x3fef3720dcef9069,
0xbc808a1883ccb5d2, 0x3fef3f0b555dc3fa,
0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c,
0xbc900dae3875a949, 0x3fef4f87080d89f2,
0x3c74a385a63d07a7, 0x3fef5818dcfba487,
0xbc82919e2040220f, 0x3fef60e316c98398,
0x3c8e5a50d5c192ac, 0x3fef69e603db3285,
0x3c843a59ac016b4b, 0x3fef7321f301b460,
0xbc82d52107b43e1f, 0x3fef7c97337b9b5f,
0xbc892ab93b470dc9, 0x3fef864614f5a129,
0x3c74b604603a88d3, 0x3fef902ee78b3ff6,
0x3c83c5ec519d7271, 0x3fef9a51fbc74c83,
0xbc8ff7128fd391f0, 0x3fefa4afa2a490da,
0xbc8dae98e223747d, 0x3fefaf482d8e67f1,
0x3c8ec3bc41aa2008, 0x3fefba1bee615a27,
0x3c842b94c3a9eb32, 0x3fefc52b376bba97,
0x3c8a64a931d185ee, 0x3fefd0765b6e4540,
0xbc8e37bae43be3ed, 0x3fefdbfdad9cbe14,
0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8,
0x3c5305c14160cc89, 0x3feff3c22b8f71f1,
},
};
|
