source/libs/mpfr/mpfr-3.1.3/src/sin.c

   1 /* mpfr_sin -- sine of a floating-point number
   2
   3 Copyright 2001-2015 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     {
  80       inexact = mpfr_sin_fast (y, x, rnd_mode);
  81       goto end;
  82     }
  83
  84   m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
  85   expx = MPFR_GET_EXP (x);
  86
  87   mpfr_init (c);
  88   mpfr_init (xr);
  89
  90   MPFR_ZIV_INIT (loop, m);
  91   for (;;)
  92     {
  93       /* first perform argument reduction modulo 2*Pi (if needed),
  94          also helps to determine the sign of sin(x) */
  95       if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine
  96                         the sign of sin(x). For 2 <= |x| < Pi, we could avoid
  97                         the reduction. */
  98         {
  99           reduce = 1;
 100           /* As expx + m - 1 will silently be converted into mpfr_prec_t
 101              in the mpfr_set_prec call, the assert below may be useful to
 102              avoid undefined behavior. */
 103           MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
 104           mpfr_set_prec (c, expx + m - 1);
 105           mpfr_set_prec (xr, m);
 106           mpfr_const_pi (c, MPFR_RNDN);
 107           mpfr_mul_2ui (c, c, 1, MPFR_RNDN);
 108           mpfr_remainder (xr, x, c, MPFR_RNDN);
 109           /* The analysis is similar to that of cos.c:
 110              |xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign
 111              of sin(x) if xr is at distance at least 2^(2-m) of both
 112              0 and +/-Pi. */
 113           mpfr_div_2ui (c, c, 1, MPFR_RNDN);
 114           /* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m),
 115              it suffices to check that c - |xr| >= 2^(2-m). */
 116           if (MPFR_SIGN (xr) > 0)
 117             mpfr_sub (c, c, xr, MPFR_RNDZ);
 118           else
 119             mpfr_add (c, c, xr, MPFR_RNDZ);
 120           if (MPFR_IS_ZERO(xr)
 121               || MPFR_GET_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m
 122               || MPFR_IS_ZERO(c)
 123               || MPFR_GET_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m)
 124             goto ziv_next;
 125
 126           /* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */
 127           xx = xr;
 128         }
 129       else /* the input argument is already reduced */
 130         {
 131           reduce = 0;
 132           xx = x;
 133         }
 134
 135       sign = MPFR_SIGN(xx);
 136       /* now that the argument is reduced, precision m is enough */
 137       mpfr_set_prec (c, m);
 138       mpfr_cos (c, xx, MPFR_RNDZ);    /* can't be exact */
 139       mpfr_nexttoinf (c);           /* now c = cos(x) rounded away */
 140       mpfr_mul (c, c, c, MPFR_RNDU); /* away */
 141       mpfr_ui_sub (c, 1, c, MPFR_RNDZ);
 142       mpfr_sqrt (c, c, MPFR_RNDZ);
 143       if (MPFR_IS_NEG_SIGN(sign))
 144         MPFR_CHANGE_SIGN(c);
 145
 146       /* Warning: c may be 0! */
 147       if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
 148         {
 149           /* Huge cancellation: increase prec a lot! */
 150           m = MAX (m, MPFR_PREC (x));
 151           m = 2 * m;
 152         }
 153       else
 154         {
 155           /* the absolute error on c is at most 2^(3-m-EXP(c)),
 156              plus 2^(2-m) if there was an argument reduction.
 157              Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error
 158              is at most 2^(3-m-EXP(c)) in case of argument reduction. */
 159           err = 2 * MPFR_GET_EXP (c) + (mpfr_exp_t) m - 3 - (reduce != 0);
 160           if (MPFR_CAN_ROUND (c, err, precy, rnd_mode))
 161             break;
 162
 163           /* check for huge cancellation (Near 0) */
 164           if (err < (mpfr_exp_t) MPFR_PREC (y))
 165             m += MPFR_PREC (y) - err;
 166           /* Check if near 1 */
 167           if (MPFR_GET_EXP (c) == 1)
 168             m += m;
 169         }
 170
 171     ziv_next:
 172       /* Else generic increase */
 173       MPFR_ZIV_NEXT (loop, m);
 174     }
 175   MPFR_ZIV_FREE (loop);
 176
 177   inexact = mpfr_set (y, c, rnd_mode);
 178   /* inexact cannot be 0, since this would mean that c was representable
 179      within the target precision, but in that case mpfr_can_round will fail */
 180
 181   mpfr_clear (c);
 182   mpfr_clear (xr);
 183
 184  end:
 185   MPFR_SAVE_EXPO_FREE (expo);
 186   return mpfr_check_range (y, inexact, rnd_mode);
 187 }