| /* mpfr_sin -- sine of a floating-point number |
| |
| Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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" |
| |
| int |
| mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode) |
| { |
| mpfr_t c, xr; |
| mpfr_srcptr xx; |
| mp_exp_t expx, err; |
| mp_prec_t precy, m; |
| int inexact, sign, reduce; |
| MPFR_ZIV_DECL (loop); |
| MPFR_SAVE_EXPO_DECL (expo); |
| |
| 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, {}); |
| |
| MPFR_SAVE_EXPO_MARK (expo); |
| |
| /* Compute initial precision */ |
| precy = MPFR_PREC (y); |
| 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; |
| mpfr_set_prec (c, expx + m - 1); |
| mpfr_set_prec (xr, m); |
| mpfr_const_pi (c, GMP_RNDN); |
| mpfr_mul_2ui (c, c, 1, GMP_RNDN); |
| mpfr_remainder (xr, x, c, GMP_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, GMP_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, GMP_RNDZ); |
| else |
| mpfr_add (c, c, xr, GMP_RNDZ); |
| if (MPFR_IS_ZERO(xr) || MPFR_EXP(xr) < (mp_exp_t) 3 - (mp_exp_t) m |
| || MPFR_EXP(c) < (mp_exp_t) 3 - (mp_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, 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)), |
| 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) + (mp_exp_t) m - 3 - (reduce != 0); |
| if (MPFR_CAN_ROUND (c, err, precy, rnd_mode)) |
| break; |
| |
| /* check for huge cancellation (Near 0) */ |
| if (err < (mp_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); |
| |
| MPFR_SAVE_EXPO_FREE (expo); |
| return mpfr_check_range (y, inexact, rnd_mode); |
| } |