/* mpfr_sin -- sine of a floating-point number Copyright 2001-2023 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 https://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, err1, err; mpfr_prec_t precy, m; int inexact, sign, reduce; MPFR_ZIV_DECL (loop); MPFR_SAVE_EXPO_DECL (expo); MPFR_LOG_FUNC (("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), ("y[%Pd]=%.*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); } } expx = MPFR_GET_EXP (x); err1 = -2 * expx; /* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */ MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, err1, 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; } /* for x large, since argument reduction is expensive, we want to avoid any failure in Ziv's strategy, thus we take into account expx too */ m = precy + MPFR_INT_CEIL_LOG2 (MAX(precy,expx)) + 8; /* since we compute sin(x) as sqrt(1-cos(x)^2), and for x small we have cos(x)^2 ~ 1 - x^2, when subtracting cos(x)^2 from 1 we will lose about -2*expx bits if expx < 0 */ if (expx < 0) { /* The following assertion includes a check for integer overflow. At this point, precy < MPFR_SINCOS_THRESHOLD, so that both m and err1 should be small enough. But the assertion makes the code safer (a smart compiler might be able to remove it). */ MPFR_ASSERTN (err1 <= MPFR_PREC_MAX - m); m += err1; } if (expx >= 2) { mpfr_init2 (c, expx + m - 1); mpfr_init2 (xr, m); } else mpfr_init2 (c, m); MPFR_ZIV_INIT (loop, m); for (;;) { /* first perform argument reduction modulo 2*Pi (if needed), also helps to determine the sign of sin(x) */ /* TODO: Perform range reduction in a way so that the sine can be computed directly from the cosine with sin(x)=cos(pi/2-x), without the need of sqrt(1 - x^2). */ 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_IS_POS (xr)) 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_RNDA); /* c = cos(x) rounded away */ mpfr_sqr (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); if (expx >= 2) mpfr_clear (xr); end: MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inexact, rnd_mode); }