contrib/mpfr/sin.c

   1 /* mpfr_sin -- sine of a floating-point number
   2
   3 Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
   4 Contributed by the Arenaire and Cacao 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 2.1 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.LIB.  If not, write to
  20 the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
  21 MA 02110-1301, USA. */
  22
  23 #define MPFR_NEED_LONGLONG_H
  24 #include "mpfr-impl.h"
  25
  26 int
  27 mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
  28 {
  29   mpfr_t c, xr;
  30   mpfr_srcptr xx;
  31   mp_exp_t expx, err;
  32   mp_prec_t precy, m;
  33   int inexact, sign, reduce;
  34   MPFR_ZIV_DECL (loop);
  35   MPFR_SAVE_EXPO_DECL (expo);
  36
  37   MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
  38                   ("y[%#R]=%R inexact=%d", y, y, inexact));
  39
  40   if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
  41     {
  42       if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
  43         {
  44           MPFR_SET_NAN (y);
  45           MPFR_RET_NAN;
  46
  47         }
  48       else /* x is zero */
  49         {
  50           MPFR_ASSERTD (MPFR_IS_ZERO (x));
  51           MPFR_SET_ZERO (y);
  52           MPFR_SET_SAME_SIGN (y, x);
  53           MPFR_RET (0);
  54         }
  55     }
  56
  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,
  59                                     rnd_mode, {});
  60
  61   MPFR_SAVE_EXPO_MARK (expo);
  62
  63   /* Compute initial precision */
  64   precy = MPFR_PREC (y);
  65   m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
  66   expx = MPFR_GET_EXP (x);
  67
  68   mpfr_init (c);
  69   mpfr_init (xr);
  70
  71   MPFR_ZIV_INIT (loop, m);
  72   for (;;)
  73     {
  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
  78                         the reduction. */
  79         {
  80           reduce = 1;
  81           mpfr_set_prec (c, expx + m - 1);
  82           mpfr_set_prec (xr, m);
  83           mpfr_const_pi (c, GMP_RNDN);
  84           mpfr_mul_2ui (c, c, 1, GMP_RNDN);
  85           mpfr_remainder (xr, x, c, GMP_RNDN);
  86           /* The analysis is similar to that of cos.c:
  87              |xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign
  88              of sin(x) if xr is at distance at least 2^(2-m) of both
  89              0 and +/-Pi. */
  90           mpfr_div_2ui (c, c, 1, GMP_RNDN);
  91           /* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m),
  92              it suffices to check that c - |xr| >= 2^(2-m). */
  93           if (MPFR_SIGN (xr) > 0)
  94             mpfr_sub (c, c, xr, GMP_RNDZ);
  95           else
  96             mpfr_add (c, c, xr, GMP_RNDZ);
  97           if (MPFR_IS_ZERO(xr) || MPFR_EXP(xr) < (mp_exp_t) 3 - (mp_exp_t) m
  98               || MPFR_EXP(c) < (mp_exp_t) 3 - (mp_exp_t) m)
  99             goto ziv_next;
 100
 101           /* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */
 102           xx = xr;
 103         }
 104       else /* the input argument is already reduced */
 105         {
 106           reduce = 0;
 107           xx = x;
 108         }
 109
 110       sign = MPFR_SIGN(xx);
 111       /* now that the argument is reduced, precision m is enough */
 112       mpfr_set_prec (c, m);
 113       mpfr_cos (c, xx, GMP_RNDZ);    /* can't be exact */
 114       mpfr_nexttoinf (c);           /* now c = cos(x) rounded away */
 115       mpfr_mul (c, c, c, GMP_RNDU); /* away */
 116       mpfr_ui_sub (c, 1, c, GMP_RNDZ);
 117       mpfr_sqrt (c, c, GMP_RNDZ);
 118       if (MPFR_IS_NEG_SIGN(sign))
 119         MPFR_CHANGE_SIGN(c);
 120
 121       /* Warning: c may be 0! */
 122       if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
 123         {
 124           /* Huge cancellation: increase prec a lot! */
 125           m = MAX (m, MPFR_PREC (x));
 126           m = 2 * m;
 127         }
 128       else
 129         {
 130           /* the absolute error on c is at most 2^(3-m-EXP(c)),
 131              plus 2^(2-m) if there was an argument reduction.
 132              Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error
 133              is at most 2^(3-m-EXP(c)) in case of argument reduction. */
 134           err = 2 * MPFR_GET_EXP (c) + (mp_exp_t) m - 3 - (reduce != 0);
 135           if (MPFR_CAN_ROUND (c, err, precy, rnd_mode))
 136             break;
 137
 138           /* check for huge cancellation (Near 0) */
 139           if (err < (mp_exp_t) MPFR_PREC (y))
 140             m += MPFR_PREC (y) - err;
 141           /* Check if near 1 */
 142           if (MPFR_GET_EXP (c) == 1)
 143             m += m;
 144         }
 145
 146     ziv_next:
 147       /* Else generic increase */
 148       MPFR_ZIV_NEXT (loop, m);
 149     }
 150   MPFR_ZIV_FREE (loop);
 151
 152   inexact = mpfr_set (y, c, rnd_mode);
 153   /* inexact cannot be 0, since this would mean that c was representable
 154      within the target precision, but in that case mpfr_can_round will fail */
 155
 156   mpfr_clear (c);
 157   mpfr_clear (xr);
 158
 159   MPFR_SAVE_EXPO_FREE (expo);
 160   return mpfr_check_range (y, inexact, rnd_mode);
 161 }