1 /* mpfr_sin -- sine of a floating-point number
3 Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
4 Contributed by the Arenaire and Cacao projects, INRIA.
6 This file is part of the GNU MPFR Library.
8 The GNU MPFR Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 2.1 of the License, or (at your
11 option) any later version.
13 The GNU MPFR Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MPFR Library; see the file COPYING.LIB. If not, write to
20 the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
21 MA 02110-1301, USA. */
23 #define MPFR_NEED_LONGLONG_H
24 #include "mpfr-impl.h"
27 mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
33 int inexact, sign, reduce;
35 MPFR_SAVE_EXPO_DECL (expo);
37 MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
38 ("y[%#R]=%R inexact=%d", y, y, inexact));
40 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
42 if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
50 MPFR_ASSERTD (MPFR_IS_ZERO (x));
52 MPFR_SET_SAME_SIGN (y, x);
57 /* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
58 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -2 * MPFR_GET_EXP (x), 2, 0,
61 MPFR_SAVE_EXPO_MARK (expo);
63 /* Compute initial precision */
64 precy = MPFR_PREC (y);
65 m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
66 expx = MPFR_GET_EXP (x);
71 MPFR_ZIV_INIT (loop, m);
74 /* first perform argument reduction modulo 2*Pi (if needed),
75 also helps to determine the sign of sin(x) */
76 if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine
77 the sign of sin(x). For 2 <= |x| < Pi, we could avoid
81 /* As expx + m - 1 will silently be converted into mpfr_prec_t
82 in the mpfr_set_prec call, the assert below may be useful to
83 avoid undefined behavior. */
84 MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
85 mpfr_set_prec (c, expx + m - 1);
86 mpfr_set_prec (xr, m);
87 mpfr_const_pi (c, GMP_RNDN);
88 mpfr_mul_2ui (c, c, 1, GMP_RNDN);
89 mpfr_remainder (xr, x, c, GMP_RNDN);
90 /* The analysis is similar to that of cos.c:
91 |xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign
92 of sin(x) if xr is at distance at least 2^(2-m) of both
94 mpfr_div_2ui (c, c, 1, GMP_RNDN);
95 /* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m),
96 it suffices to check that c - |xr| >= 2^(2-m). */
97 if (MPFR_SIGN (xr) > 0)
98 mpfr_sub (c, c, xr, GMP_RNDZ);
100 mpfr_add (c, c, xr, GMP_RNDZ);
101 if (MPFR_IS_ZERO(xr) || MPFR_EXP(xr) < (mp_exp_t) 3 - (mp_exp_t) m
102 || MPFR_EXP(c) < (mp_exp_t) 3 - (mp_exp_t) m)
105 /* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */
108 else /* the input argument is already reduced */
114 sign = MPFR_SIGN(xx);
115 /* now that the argument is reduced, precision m is enough */
116 mpfr_set_prec (c, m);
117 mpfr_cos (c, xx, GMP_RNDZ); /* can't be exact */
118 mpfr_nexttoinf (c); /* now c = cos(x) rounded away */
119 mpfr_mul (c, c, c, GMP_RNDU); /* away */
120 mpfr_ui_sub (c, 1, c, GMP_RNDZ);
121 mpfr_sqrt (c, c, GMP_RNDZ);
122 if (MPFR_IS_NEG_SIGN(sign))
125 /* Warning: c may be 0! */
126 if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
128 /* Huge cancellation: increase prec a lot! */
129 m = MAX (m, MPFR_PREC (x));
134 /* the absolute error on c is at most 2^(3-m-EXP(c)),
135 plus 2^(2-m) if there was an argument reduction.
136 Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error
137 is at most 2^(3-m-EXP(c)) in case of argument reduction. */
138 err = 2 * MPFR_GET_EXP (c) + (mp_exp_t) m - 3 - (reduce != 0);
139 if (MPFR_CAN_ROUND (c, err, precy, rnd_mode))
142 /* check for huge cancellation (Near 0) */
143 if (err < (mp_exp_t) MPFR_PREC (y))
144 m += MPFR_PREC (y) - err;
145 /* Check if near 1 */
146 if (MPFR_GET_EXP (c) == 1)
151 /* Else generic increase */
152 MPFR_ZIV_NEXT (loop, m);
154 MPFR_ZIV_FREE (loop);
156 inexact = mpfr_set (y, c, rnd_mode);
157 /* inexact cannot be 0, since this would mean that c was representable
158 within the target precision, but in that case mpfr_can_round will fail */
163 MPFR_SAVE_EXPO_FREE (expo);
164 return mpfr_check_range (y, inexact, rnd_mode);