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
|
/* mpfr_mpn_exp -- auxiliary function for mpfr_get_str and mpfr_set_str
Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
Contributed by the AriC and Caramel projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR 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 3 of the License, or (at your
option) any later version.
The GNU MPFR 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 MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* this function computes an approximation of b^e in {a, n}, with exponent
stored in exp_r. The computed value is rounded toward zero (truncated).
It returns an integer f such that the final error is bounded by 2^f ulps,
that is:
a*2^exp_r <= b^e <= 2^exp_r (a + 2^f),
where a represents {a, n}, i.e. the integer
a[0] + a[1]*B + ... + a[n-1]*B^(n-1) where B=2^GMP_NUMB_BITS
Return -1 is the result is exact.
Return -2 if an overflow occurred in the computation of exp_r.
*/
long
mpfr_mpn_exp (mp_limb_t *a, mpfr_exp_t *exp_r, int b, mpfr_exp_t e, size_t n)
{
mp_limb_t *c, B;
mpfr_exp_t f, h;
int i;
unsigned long t; /* number of bits in e */
unsigned long bits;
size_t n1;
unsigned int error; /* (number - 1) of loop a^2b inexact */
/* error == t means no error */
int err_s_a2 = 0;
int err_s_ab = 0; /* number of error when shift A^2, AB */
MPFR_TMP_DECL(marker);
MPFR_ASSERTN(e > 0);
MPFR_ASSERTN((2 <= b) && (b <= 62));
MPFR_TMP_MARK(marker);
/* initialization of a, b, f, h */
/* normalize the base */
B = (mp_limb_t) b;
count_leading_zeros (h, B);
bits = GMP_NUMB_BITS - h;
B = B << h;
h = - h;
/* allocate space for A and set it to B */
c = MPFR_TMP_LIMBS_ALLOC (2 * n);
a [n - 1] = B;
MPN_ZERO (a, n - 1);
/* initial exponent for A: invariant is A = {a, n} * 2^f */
f = h - (n - 1) * GMP_NUMB_BITS;
/* determine number of bits in e */
count_leading_zeros (t, (mp_limb_t) e);
t = GMP_NUMB_BITS - t; /* number of bits of exponent e */
error = t; /* error <= GMP_NUMB_BITS */
MPN_ZERO (c, 2 * n);
for (i = t - 2; i >= 0; i--)
{
/* determine precision needed */
bits = n * GMP_NUMB_BITS - mpn_scan1 (a, 0);
n1 = (n * GMP_NUMB_BITS - bits) / GMP_NUMB_BITS;
/* square of A : {c+2n1, 2(n-n1)} = {a+n1, n-n1}^2 */
mpn_sqr_n (c + 2 * n1, a + n1, n - n1);
/* set {c+n, 2n1-n} to 0 : {c, n} = {a, n}^2*K^n */
/* check overflow on f */
if (MPFR_UNLIKELY(f < MPFR_EXP_MIN/2 || f > MPFR_EXP_MAX/2))
{
overflow:
MPFR_TMP_FREE(marker);
return -2;
}
/* FIXME: Could f = 2*f + n * GMP_NUMB_BITS be used? */
f = 2*f;
MPFR_SADD_OVERFLOW (f, f, n * GMP_NUMB_BITS,
mpfr_exp_t, mpfr_uexp_t,
MPFR_EXP_MIN, MPFR_EXP_MAX,
goto overflow, goto overflow);
if ((c[2*n - 1] & MPFR_LIMB_HIGHBIT) == 0)
{
/* shift A by one bit to the left */
mpn_lshift (a, c + n, n, 1);
a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
f --;
if (error != t)
err_s_a2 ++;
}
else
MPN_COPY (a, c + n, n);
if ((error == t) && (2 * n1 <= n) &&
(mpn_scan1 (c + 2 * n1, 0) < (n - 2 * n1) * GMP_NUMB_BITS))
error = i;
if (e & ((mpfr_exp_t) 1 << i))
{
/* multiply A by B */
c[2 * n - 1] = mpn_mul_1 (c + n - 1, a, n, B);
f += h + GMP_NUMB_BITS;
if ((c[2 * n - 1] & MPFR_LIMB_HIGHBIT) == 0)
{ /* shift A by one bit to the left */
mpn_lshift (a, c + n, n, 1);
a[0] |= mpn_lshift (c + n - 1, c + n - 1, 1, 1);
f --;
}
else
{
MPN_COPY (a, c + n, n);
if (error != t)
err_s_ab ++;
}
if ((error == t) && (c[n - 1] != 0))
error = i;
}
}
MPFR_TMP_FREE(marker);
*exp_r = f;
if (error == t)
return -1; /* result is exact */
else /* error <= t-2 <= GMP_NUMB_BITS-2
err_s_ab, err_s_a2 <= t-1 */
{
/* if there are p loops after the first inexact result, with
j shifts in a^2 and l shifts in a*b, then the final error is
at most 2^(p+ceil((j+1)/2)+l+1)*ulp(res).
This is bounded by 2^(5/2*t-1/2) where t is the number of bits of e.
*/
error = error + err_s_ab + err_s_a2 / 2 + 3; /* <= 5t/2-1/2 */
#if 0
if ((error - 1) >= ((n * GMP_NUMB_BITS - 1) / 2))
error = n * GMP_NUMB_BITS; /* result is completely wrong:
this is very unlikely since error is
at most 5/2*log_2(e), and
n * GMP_NUMB_BITS is at least
3*log_2(e) */
#endif
return error;
}
}
|
