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
|
/* mpfr_sin -- sine of a floating-point number
Copyright 2001, 2002, 2003, 2004, 2005 Free Software Foundation, Inc.
This file is part of the MPFR Library.
The 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 2.1 of the License, or (at your
option) any later version.
The 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include <stdio.h>
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* determine the sign of sin(x) using argument reduction.
Assumes x is not an exact multiple of Pi (this excludes x=0). */
static int
mpfr_sin_sign (mpfr_srcptr x)
{
mpfr_t c, k;
mp_exp_t K;
int sign;
mp_prec_t m;
mpfr_srcptr y;
MPFR_ZIV_DECL (loop);
K = MPFR_GET_EXP(x);
if (K < 0) /* Trivial case if ABS(x) < 1 */
return MPFR_SIGN (x);
m = K + BITS_PER_MP_LIMB;
mpfr_init2 (c, m);
mpfr_init2 (k, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
/* first determine round(x/Pi): does not have to be exact since
the result is an integer */
mpfr_const_pi (c, GMP_RNDN); /* err <= 1/2*ulp(c) = 2^(1-m) */
/* we need that k is not-to-badly rounded to an integer,
i.e. ulp(k) <= 1, so m >= EXP(k). */
mpfr_div (k, x, c, GMP_RNDN);
mpfr_round (k, k);
sign = 1;
if (!MPFR_IS_ZERO (k)) /* subtract k*approx(Pi) */
{
/* determine parity of k for sign */
if (MPFR_GET_EXP (k) <= 0 || (mpfr_uexp_t) MPFR_GET_EXP (k) <= m)
{
mp_size_t j = BITS_PER_MP_LIMB * MPFR_LIMB_SIZE(k)
- MPFR_GET_EXP(k);
mp_size_t l = j / BITS_PER_MP_LIMB;
/* parity bit is j-th bit starting from least significant bits */
if ((MPFR_MANT(k)[l] >> (j % BITS_PER_MP_LIMB)) & 1)
sign = -1; /* k is odd */
}
K = MPFR_GET_EXP (k); /* k is an integer, thus K >= 1, k < 2^K */
mpfr_mul (k, k, c, GMP_RNDN); /* err <= oldk*err(c) + 1/2*ulp(k)
<= 2^(K+2-m) */
mpfr_sub (k, x, k, GMP_RNDN);
/* assuming |k| <= Pi, err <= 2^(1-m)+2^(K+2-m) < 2^(K+3-m) */
MPFR_ASSERTN (MPFR_GET_EXP (k) <= 2);
y = k;
}
else
{
K = 1;
y = x;
}
/* sign of sign(y) is uncertain if |y| <= err < 2^(K+3-m),
thus EXP(y) < K+4-m */
if (MPFR_LIKELY (!MPFR_IS_ZERO (y)
&& MPFR_GET_EXP (y) >= K + 4 - (mp_exp_t) m))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (c, m);
mpfr_set_prec (k, m);
}
if (MPFR_IS_NEG (y))
sign = -sign;
mpfr_clear (k);
mpfr_clear (c);
return sign;
}
int
mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
int precy, m, inexact, sign;
mpfr_t c;
mp_exp_t e;
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
("y[%#R]=%R inexact=%d", y, y, inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
/* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -2*MPFR_GET_EXP (x)+2,0,rnd_mode, );
/* Compute initial precision */
precy = MPFR_PREC (y);
m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
e = MPFR_GET_EXP (x);
m += (e < 0) ? -2*e : e;
sign = mpfr_sin_sign (x);
mpfr_init2 (c, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_cos (c, x, GMP_RNDZ); /* can't be exact */
mpfr_nexttoinf (c); /* now c = cos(x) rounded away */
mpfr_mul (c, c, c, GMP_RNDU); /* away */
mpfr_ui_sub (c, 1, c, GMP_RNDZ);
mpfr_sqrt (c, c, GMP_RNDZ);
if (MPFR_IS_NEG_SIGN(sign))
MPFR_CHANGE_SIGN(c);
/* Warning c may be 0 ! */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
{
/* Huge cancellation: increase prec a lot! */
m = MAX (m, MPFR_PREC (x));
m = 2*m;
}
else
{
/* the absolute error on c is at most 2^(3-m-EXP(c)) */
e = 2 * MPFR_GET_EXP (c) + m - 3;
if (mpfr_can_round (c, e, GMP_RNDZ, rnd_mode, precy))
break;
/* check for huge cancellation (Near 0) */
if (e < (mp_exp_t) MPFR_PREC (y))
m += MPFR_PREC (y) - e;
/* Check if near 1 */
if (MPFR_GET_EXP (c) == 1)
m += m;
}
/* Else generic increase */
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (c, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, c, rnd_mode);
/* sin(x) is exact only for x = 0, which was treated apart above;
nevertheless, we can have inexact = 0 here if the approximation c
is exactly representable with PREC(y) bits. Since c is an approximation
towards zero, in that case the inexact flag should have the opposite sign
as y. */
if (MPFR_UNLIKELY (inexact == 0))
inexact = -MPFR_INT_SIGN (y);
mpfr_clear (c);
return inexact; /* inexact */
}
|
