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
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
|
/* Copyright (C) 1997-2014 Free Software Foundation, Inc.
This file is part of the GNU C Library.
Contributed by Geoffrey Keating <Geoff.Keating@anu.edu.au>, 1997.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<http://www.gnu.org/licenses/>. */
#include <stdio.h>
#include <math.h>
#include <gmp.h>
#include <string.h>
#include <limits.h>
#include <assert.h>
#include <stdlib.h>
#define PRINT_ERRORS 0
#define TOL 80
#define N2 18
#define FRAC (32*4)
#define mpbpl (CHAR_BIT * sizeof (mp_limb_t))
#define SZ (FRAC / mpbpl + 1)
typedef mp_limb_t mp1[SZ], mp2[SZ * 2];
#if BITS_PER_MP_LIMB == 64
# define LIMB64(L, H) 0x ## H ## L
#elif BITS_PER_MP_LIMB == 32
# define LIMB64(L, H) 0x ## L, 0x ## H
#else
# error
#endif
/* Once upon a time these constants were generated to 400 bits.
We only need FRAC bits (128) at present, but we retain 384 bits
in the text Just In Case. */
#define CONSTSZ(INT, F1, F2, F3, F4, F5, F6, F7, F8, F9, Fa, Fb, Fc) \
LIMB64(F4, F3), LIMB64(F2, F1), INT
static const mp1 mp_exp1 = {
CONSTSZ (2, b7e15162, 8aed2a6a, bf715880, 9cf4f3c7, 62e7160f, 38b4da56,
a784d904, 5190cfef, 324e7738, 926cfbe5, f4bf8d8d, 8c31d763)
};
static const mp1 mp_exp_m1 = {
CONSTSZ (0, 5e2d58d8, b3bcdf1a, badec782, 9054f90d, da9805aa, b56c7733,
3024b9d0, a507daed, b16400bf, 472b4215, b8245b66, 9d90d27a)
};
static const mp1 mp_log2 = {
CONSTSZ (0, b17217f7, d1cf79ab, c9e3b398, 03f2f6af, 40f34326, 7298b62d,
8a0d175b, 8baafa2b, e7b87620, 6debac98, 559552fb, 4afa1b10)
};
static void
print_mpn_fp (const mp_limb_t *x, unsigned int dp, unsigned int base)
{
static const char hexdig[16] = "0123456789abcdef";
unsigned int i;
mp1 tx;
memcpy (tx, x, sizeof (mp1));
if (base == 16)
fputs ("0x", stdout);
assert (x[SZ-1] < base);
fputc (hexdig[x[SZ - 1]], stdout);
fputc ('.', stdout);
for (i = 0; i < dp; i++)
{
tx[SZ - 1] = 0;
mpn_mul_1 (tx, tx, SZ, base);
assert (tx[SZ - 1] < base);
fputc (hexdig[tx[SZ - 1]], stdout);
}
}
/* Compute e^x. */
static void
exp_mpn (mp1 ex, mp1 x)
{
unsigned int n;
mp1 xp;
mp2 tmp;
mp_limb_t chk;
mp1 tol;
memset (xp, 0, sizeof (mp1));
memset (ex, 0, sizeof (mp1));
xp[FRAC / mpbpl] = (mp_limb_t)1 << FRAC % mpbpl;
memset (tol, 0, sizeof (mp1));
tol[(FRAC - TOL) / mpbpl] = (mp_limb_t)1 << (FRAC - TOL) % mpbpl;
n = 0;
do
{
/* Calculate sum(x^n/n!) until the next term is sufficiently small. */
mpn_mul_n (tmp, xp, x, SZ);
assert(tmp[SZ * 2 - 1] == 0);
if (n > 0)
mpn_divmod_1 (xp, tmp + FRAC / mpbpl, SZ, n);
chk = mpn_add_n (ex, ex, xp, SZ);
assert (chk == 0);
++n;
assert (n < 80); /* Catch too-high TOL. */
}
while (n < 10 || mpn_cmp (xp, tol, SZ) >= 0);
}
/* Calculate 2^x. */
static void
exp2_mpn (mp1 ex, mp1 x)
{
mp2 tmp;
mpn_mul_n (tmp, x, mp_log2, SZ);
assert(tmp[SZ * 2 - 1] == 0);
exp_mpn (ex, tmp + FRAC / mpbpl);
}
static int
mpn_bitsize(const mp_limb_t *SRC_PTR, mp_size_t SIZE)
{
int i, j;
for (i = SIZE - 1; i > 0; --i)
if (SRC_PTR[i] != 0)
break;
for (j = mpbpl - 1; j >= 0; --j)
if ((SRC_PTR[i] & (mp_limb_t)1 << j) != 0)
break;
return i * mpbpl + j;
}
static int
do_test (void)
{
mp1 ex, x, xt, e2, e3;
int i;
int errors = 0;
int failures = 0;
mp1 maxerror;
int maxerror_s = 0;
const double sf = pow (2, mpbpl);
/* assert(mpbpl == mp_bits_per_limb); */
assert(FRAC / mpbpl * mpbpl == FRAC);
memset (maxerror, 0, sizeof (mp1));
memset (xt, 0, sizeof (mp1));
xt[(FRAC - N2) / mpbpl] = (mp_limb_t)1 << (FRAC - N2) % mpbpl;
for (i = 0; i < (1 << N2); ++i)
{
int e2s, e3s, j;
double de2;
mpn_mul_1 (x, xt, SZ, i);
exp2_mpn (ex, x);
de2 = exp2 (i / (double) (1 << N2));
for (j = SZ - 1; j >= 0; --j)
{
e2[j] = (mp_limb_t) de2;
de2 = (de2 - e2[j]) * sf;
}
if (mpn_cmp (ex, e2, SZ) >= 0)
mpn_sub_n (e3, ex, e2, SZ);
else
mpn_sub_n (e3, e2, ex, SZ);
e2s = mpn_bitsize (e2, SZ);
e3s = mpn_bitsize (e3, SZ);
if (e3s >= 0 && e2s - e3s < 54)
{
#if PRINT_ERRORS
printf ("%06x ", i * (0x100000 / (1 << N2)));
print_mpn_fp (ex, (FRAC / 4) + 1, 16);
putchar ('\n');
fputs (" ",stdout);
print_mpn_fp (e2, (FRAC / 4) + 1, 16);
putchar ('\n');
printf (" %c ",
e2s - e3s < 54 ? e2s - e3s == 53 ? 'e' : 'F' : 'P');
print_mpn_fp (e3, (FRAC / 4) + 1, 16);
putchar ('\n');
#endif
errors += (e2s - e3s == 53);
failures += (e2s - e3s < 53);
}
if (e3s >= maxerror_s
&& mpn_cmp (e3, maxerror, SZ) > 0)
{
memcpy (maxerror, e3, sizeof (mp1));
maxerror_s = e3s;
}
}
/* Check exp_mpn against precomputed value of exp(1). */
memset (x, 0, sizeof (mp1));
x[FRAC / mpbpl] = (mp_limb_t)1 << FRAC % mpbpl;
exp_mpn (ex, x);
if (mpn_cmp (ex, mp_exp1, SZ) >= 0)
mpn_sub_n (e3, ex, mp_exp1, SZ);
else
mpn_sub_n (e3, mp_exp1, ex, SZ);
printf ("%d failures; %d errors; error rate %0.2f%%\n", failures, errors,
errors * 100.0 / (double) (1 << N2));
fputs ("maximum error: ", stdout);
print_mpn_fp (maxerror, (FRAC / 4) + 1, 16);
putchar ('\n');
fputs ("error in exp(1): ", stdout);
print_mpn_fp (e3, (FRAC / 4) + 1, 16);
putchar ('\n');
return failures == 0 ? 0 : 1;
}
#define TIMEOUT 10
#define TEST_FUNCTION do_test ()
#include "../test-skeleton.c"
|
