1 /* mpfr_sin -- sine of a floating-point number 2 3 Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. 4 Contributed by the AriC and Caramel projects, INRIA. 5 6 This file is part of the GNU MPFR Library. 7 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 3 of the License, or (at your 11 option) any later version. 12 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. 17 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.LESSER. If not, see 20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23 #define MPFR_NEED_LONGLONG_H 24 #include "mpfr-impl.h" 25 26 static int 27 mpfr_sin_fast (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 28 { 29 int inex; 30 31 inex = mpfr_sincos_fast (y, NULL, x, rnd_mode); 32 inex = inex & 3; /* 0: exact, 1: rounded up, 2: rounded down */ 33 return (inex == 2) ? -1 : inex; 34 } 35 36 int 37 mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) 38 { 39 mpfr_t c, xr; 40 mpfr_srcptr xx; 41 mpfr_exp_t expx, err; 42 mpfr_prec_t precy, m; 43 int inexact, sign, reduce; 44 MPFR_ZIV_DECL (loop); 45 MPFR_SAVE_EXPO_DECL (expo); 46 47 MPFR_LOG_FUNC 48 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), 49 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, 50 inexact)); 51 52 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) 53 { 54 if (MPFR_IS_NAN (x) || MPFR_IS_INF (x)) 55 { 56 MPFR_SET_NAN (y); 57 MPFR_RET_NAN; 58 59 } 60 else /* x is zero */ 61 { 62 MPFR_ASSERTD (MPFR_IS_ZERO (x)); 63 MPFR_SET_ZERO (y); 64 MPFR_SET_SAME_SIGN (y, x); 65 MPFR_RET (0); 66 } 67 } 68 69 /* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */ 70 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -2 * MPFR_GET_EXP (x), 2, 0, 71 rnd_mode, {}); 72 73 MPFR_SAVE_EXPO_MARK (expo); 74 75 /* Compute initial precision */ 76 precy = MPFR_PREC (y); 77 78 if (precy >= MPFR_SINCOS_THRESHOLD) 79 return mpfr_sin_fast (y, x, rnd_mode); 80 81 m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13; 82 expx = MPFR_GET_EXP (x); 83 84 mpfr_init (c); 85 mpfr_init (xr); 86 87 MPFR_ZIV_INIT (loop, m); 88 for (;;) 89 { 90 /* first perform argument reduction modulo 2*Pi (if needed), 91 also helps to determine the sign of sin(x) */ 92 if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine 93 the sign of sin(x). For 2 <= |x| < Pi, we could avoid 94 the reduction. */ 95 { 96 reduce = 1; 97 /* As expx + m - 1 will silently be converted into mpfr_prec_t 98 in the mpfr_set_prec call, the assert below may be useful to 99 avoid undefined behavior. */ 100 MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX); 101 mpfr_set_prec (c, expx + m - 1); 102 mpfr_set_prec (xr, m); 103 mpfr_const_pi (c, MPFR_RNDN); 104 mpfr_mul_2ui (c, c, 1, MPFR_RNDN); 105 mpfr_remainder (xr, x, c, MPFR_RNDN); 106 /* The analysis is similar to that of cos.c: 107 |xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign 108 of sin(x) if xr is at distance at least 2^(2-m) of both 109 0 and +/-Pi. */ 110 mpfr_div_2ui (c, c, 1, MPFR_RNDN); 111 /* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m), 112 it suffices to check that c - |xr| >= 2^(2-m). */ 113 if (MPFR_SIGN (xr) > 0) 114 mpfr_sub (c, c, xr, MPFR_RNDZ); 115 else 116 mpfr_add (c, c, xr, MPFR_RNDZ); 117 if (MPFR_IS_ZERO(xr) 118 || MPFR_GET_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m 119 || MPFR_IS_ZERO(c) 120 || MPFR_GET_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m) 121 goto ziv_next; 122 123 /* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */ 124 xx = xr; 125 } 126 else /* the input argument is already reduced */ 127 { 128 reduce = 0; 129 xx = x; 130 } 131 132 sign = MPFR_SIGN(xx); 133 /* now that the argument is reduced, precision m is enough */ 134 mpfr_set_prec (c, m); 135 mpfr_cos (c, xx, MPFR_RNDZ); /* can't be exact */ 136 mpfr_nexttoinf (c); /* now c = cos(x) rounded away */ 137 mpfr_mul (c, c, c, MPFR_RNDU); /* away */ 138 mpfr_ui_sub (c, 1, c, MPFR_RNDZ); 139 mpfr_sqrt (c, c, MPFR_RNDZ); 140 if (MPFR_IS_NEG_SIGN(sign)) 141 MPFR_CHANGE_SIGN(c); 142 143 /* Warning: c may be 0! */ 144 if (MPFR_UNLIKELY (MPFR_IS_ZERO (c))) 145 { 146 /* Huge cancellation: increase prec a lot! */ 147 m = MAX (m, MPFR_PREC (x)); 148 m = 2 * m; 149 } 150 else 151 { 152 /* the absolute error on c is at most 2^(3-m-EXP(c)), 153 plus 2^(2-m) if there was an argument reduction. 154 Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error 155 is at most 2^(3-m-EXP(c)) in case of argument reduction. */ 156 err = 2 * MPFR_GET_EXP (c) + (mpfr_exp_t) m - 3 - (reduce != 0); 157 if (MPFR_CAN_ROUND (c, err, precy, rnd_mode)) 158 break; 159 160 /* check for huge cancellation (Near 0) */ 161 if (err < (mpfr_exp_t) MPFR_PREC (y)) 162 m += MPFR_PREC (y) - err; 163 /* Check if near 1 */ 164 if (MPFR_GET_EXP (c) == 1) 165 m += m; 166 } 167 168 ziv_next: 169 /* Else generic increase */ 170 MPFR_ZIV_NEXT (loop, m); 171 } 172 MPFR_ZIV_FREE (loop); 173 174 inexact = mpfr_set (y, c, rnd_mode); 175 /* inexact cannot be 0, since this would mean that c was representable 176 within the target precision, but in that case mpfr_can_round will fail */ 177 178 mpfr_clear (c); 179 mpfr_clear (xr); 180 181 MPFR_SAVE_EXPO_FREE (expo); 182 return mpfr_check_range (y, inexact, rnd_mode); 183 } 184