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
|
/* mpfr_sin -- sine of a floating-point number
Copyright 2001-2016 Free Software Foundation, Inc.
Contributed by the AriC and Caramba 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"
static int
mpfr_sin_fast (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
int inex;
inex = mpfr_sincos_fast (y, NULL, x, rnd_mode);
inex = inex & 3; /* 0: exact, 1: rounded up, 2: rounded down */
return (inex == 2) ? -1 : inex;
}
int
mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_t c, xr;
mpfr_srcptr xx;
mpfr_exp_t expx, err;
mpfr_prec_t precy, m;
int inexact, sign, reduce;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, 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, {});
MPFR_SAVE_EXPO_MARK (expo);
/* Compute initial precision */
precy = MPFR_PREC (y);
if (precy >= MPFR_SINCOS_THRESHOLD)
{
inexact = mpfr_sin_fast (y, x, rnd_mode);
goto end;
}
m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
expx = MPFR_GET_EXP (x);
mpfr_init (c);
mpfr_init (xr);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
/* first perform argument reduction modulo 2*Pi (if needed),
also helps to determine the sign of sin(x) */
if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine
the sign of sin(x). For 2 <= |x| < Pi, we could avoid
the reduction. */
{
reduce = 1;
/* As expx + m - 1 will silently be converted into mpfr_prec_t
in the mpfr_set_prec call, the assert below may be useful to
avoid undefined behavior. */
MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
mpfr_set_prec (c, expx + m - 1);
mpfr_set_prec (xr, m);
mpfr_const_pi (c, MPFR_RNDN);
mpfr_mul_2ui (c, c, 1, MPFR_RNDN);
mpfr_remainder (xr, x, c, MPFR_RNDN);
/* The analysis is similar to that of cos.c:
|xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign
of sin(x) if xr is at distance at least 2^(2-m) of both
0 and +/-Pi. */
mpfr_div_2ui (c, c, 1, MPFR_RNDN);
/* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m),
it suffices to check that c - |xr| >= 2^(2-m). */
if (MPFR_SIGN (xr) > 0)
mpfr_sub (c, c, xr, MPFR_RNDZ);
else
mpfr_add (c, c, xr, MPFR_RNDZ);
if (MPFR_IS_ZERO(xr)
|| MPFR_GET_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m
|| MPFR_IS_ZERO(c)
|| MPFR_GET_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m)
goto ziv_next;
/* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */
xx = xr;
}
else /* the input argument is already reduced */
{
reduce = 0;
xx = x;
}
sign = MPFR_SIGN(xx);
/* now that the argument is reduced, precision m is enough */
mpfr_set_prec (c, m);
mpfr_cos (c, xx, MPFR_RNDZ); /* can't be exact */
mpfr_nexttoinf (c); /* now c = cos(x) rounded away */
mpfr_mul (c, c, c, MPFR_RNDU); /* away */
mpfr_ui_sub (c, 1, c, MPFR_RNDZ);
mpfr_sqrt (c, c, MPFR_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)),
plus 2^(2-m) if there was an argument reduction.
Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error
is at most 2^(3-m-EXP(c)) in case of argument reduction. */
err = 2 * MPFR_GET_EXP (c) + (mpfr_exp_t) m - 3 - (reduce != 0);
if (MPFR_CAN_ROUND (c, err, precy, rnd_mode))
break;
/* check for huge cancellation (Near 0) */
if (err < (mpfr_exp_t) MPFR_PREC (y))
m += MPFR_PREC (y) - err;
/* Check if near 1 */
if (MPFR_GET_EXP (c) == 1)
m += m;
}
ziv_next:
/* Else generic increase */
MPFR_ZIV_NEXT (loop, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, c, rnd_mode);
/* inexact cannot be 0, since this would mean that c was representable
within the target precision, but in that case mpfr_can_round will fail */
mpfr_clear (c);
mpfr_clear (xr);
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
|
