1 /* mpfr_sin -- sine of a floating-point number 2 3 Copyright 2001-2020 Free Software Foundation, Inc. 4 Contributed by the AriC and Caramba 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 https://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, err1, 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 else /* x is zero */ 60 { 61 MPFR_ASSERTD (MPFR_IS_ZERO (x)); 62 MPFR_SET_ZERO (y); 63 MPFR_SET_SAME_SIGN (y, x); 64 MPFR_RET (0); 65 } 66 } 67 68 expx = MPFR_GET_EXP (x); 69 err1 = -2 * expx; 70 71 /* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */ 72 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, err1, 2, 0, rnd_mode, {}); 73 74 MPFR_SAVE_EXPO_MARK (expo); 75 76 /* Compute initial precision */ 77 precy = MPFR_PREC (y); 78 79 if (precy >= MPFR_SINCOS_THRESHOLD) 80 { 81 inexact = mpfr_sin_fast (y, x, rnd_mode); 82 goto end; 83 } 84 85 /* for x large, since argument reduction is expensive, we want to avoid 86 any failure in Ziv's strategy, thus we take into account expx too */ 87 m = precy + MPFR_INT_CEIL_LOG2 (MAX(precy,expx)) + 8; 88 89 /* since we compute sin(x) as sqrt(1-cos(x)^2), and for x small we have 90 cos(x)^2 ~ 1 - x^2, when subtracting cos(x)^2 from 1 we will lose 91 about -2*expx bits if expx < 0 */ 92 if (expx < 0) 93 { 94 /* The following assertion includes a check for integer overflow. 95 At this point, precy < MPFR_SINCOS_THRESHOLD, so that both m and 96 err1 should be small enough. But the assertion makes the code 97 safer (a smart compiler might be able to remove it). */ 98 MPFR_ASSERTN (err1 <= MPFR_PREC_MAX - m); 99 m += err1; 100 } 101 102 mpfr_init (c); 103 mpfr_init (xr); 104 105 MPFR_ZIV_INIT (loop, m); 106 for (;;) 107 { 108 /* first perform argument reduction modulo 2*Pi (if needed), 109 also helps to determine the sign of sin(x) */ 110 if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine 111 the sign of sin(x). For 2 <= |x| < Pi, we could avoid 112 the reduction. */ 113 { 114 reduce = 1; 115 /* As expx + m - 1 will silently be converted into mpfr_prec_t 116 in the mpfr_set_prec call, the assert below may be useful to 117 avoid undefined behavior. */ 118 MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX); 119 mpfr_set_prec (c, expx + m - 1); 120 mpfr_set_prec (xr, m); 121 mpfr_const_pi (c, MPFR_RNDN); 122 mpfr_mul_2ui (c, c, 1, MPFR_RNDN); 123 mpfr_remainder (xr, x, c, MPFR_RNDN); 124 /* The analysis is similar to that of cos.c: 125 |xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign 126 of sin(x) if xr is at distance at least 2^(2-m) of both 127 0 and +/-Pi. */ 128 mpfr_div_2ui (c, c, 1, MPFR_RNDN); 129 /* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m), 130 it suffices to check that c - |xr| >= 2^(2-m). */ 131 if (MPFR_IS_POS (xr)) 132 mpfr_sub (c, c, xr, MPFR_RNDZ); 133 else 134 mpfr_add (c, c, xr, MPFR_RNDZ); 135 if (MPFR_IS_ZERO(xr) 136 || MPFR_GET_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m 137 || MPFR_IS_ZERO(c) 138 || MPFR_GET_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m) 139 goto ziv_next; 140 141 /* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */ 142 xx = xr; 143 } 144 else /* the input argument is already reduced */ 145 { 146 reduce = 0; 147 xx = x; 148 } 149 150 sign = MPFR_SIGN(xx); 151 /* now that the argument is reduced, precision m is enough */ 152 mpfr_set_prec (c, m); 153 mpfr_cos (c, xx, MPFR_RNDA); /* c = cos(x) rounded away */ 154 mpfr_sqr (c, c, MPFR_RNDU); /* away */ 155 mpfr_ui_sub (c, 1, c, MPFR_RNDZ); 156 mpfr_sqrt (c, c, MPFR_RNDZ); 157 if (MPFR_IS_NEG_SIGN(sign)) 158 MPFR_CHANGE_SIGN(c); 159 160 /* Warning: c may be 0! */ 161 if (MPFR_UNLIKELY (MPFR_IS_ZERO (c))) 162 { 163 /* Huge cancellation: increase prec a lot! */ 164 m = MAX (m, MPFR_PREC (x)); 165 m = 2 * m; 166 } 167 else 168 { 169 /* the absolute error on c is at most 2^(3-m-EXP(c)), 170 plus 2^(2-m) if there was an argument reduction. 171 Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error 172 is at most 2^(3-m-EXP(c)) in case of argument reduction. */ 173 err = 2 * MPFR_GET_EXP (c) + (mpfr_exp_t) m - 3 - (reduce != 0); 174 if (MPFR_CAN_ROUND (c, err, precy, rnd_mode)) 175 break; 176 177 /* check for huge cancellation (Near 0) */ 178 if (err < (mpfr_exp_t) MPFR_PREC (y)) 179 m += MPFR_PREC (y) - err; 180 /* Check if near 1 */ 181 if (MPFR_GET_EXP (c) == 1) 182 m += m; 183 } 184 185 ziv_next: 186 /* Else generic increase */ 187 MPFR_ZIV_NEXT (loop, m); 188 } 189 MPFR_ZIV_FREE (loop); 190 191 inexact = mpfr_set (y, c, rnd_mode); 192 /* inexact cannot be 0, since this would mean that c was representable 193 within the target precision, but in that case mpfr_can_round will fail */ 194 195 mpfr_clear (c); 196 mpfr_clear (xr); 197 198 end: 199 MPFR_SAVE_EXPO_FREE (expo); 200 return mpfr_check_range (y, inexact, rnd_mode); 201 } 202