contrib/mpfr/acos.c

   1 /* mpfr_acos -- arc-cosinus 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, and was contributed by Mathieu Dutour.
   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 #include "mpfr-impl.h"
  24
  25 int
  26 mpfr_acos (mpfr_ptr acos, mpfr_srcptr x, mp_rnd_t rnd_mode)
  27 {
  28   mpfr_t xp, arcc, tmp;
  29   mp_exp_t supplement;
  30   mp_prec_t prec;
  31   int sign, compared, inexact;
  32   MPFR_SAVE_EXPO_DECL (expo);
  33   MPFR_ZIV_DECL (loop);
  34
  35   MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
  36                  ("acos[%#R]=%R inexact=%d", acos, acos, inexact));
  37
  38   /* Singular cases */
  39   if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
  40     {
  41       if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
  42         {
  43           MPFR_SET_NAN (acos);
  44           MPFR_RET_NAN;
  45         }
  46       else /* necessarily x=0 */
  47         {
  48           MPFR_ASSERTD(MPFR_IS_ZERO(x));
  49           /* acos(0)=Pi/2 */
  50           inexact = mpfr_const_pi (acos, rnd_mode);
  51           mpfr_div_2ui (acos, acos, 1, rnd_mode); /* exact */
  52           MPFR_RET (inexact);
  53         }
  54     }
  55
  56   /* Set x_p=|x| */
  57   sign = MPFR_SIGN (x);
  58   mpfr_init2 (xp, MPFR_PREC (x));
  59   mpfr_abs (xp, x, GMP_RNDN); /* Exact */
  60
  61   compared = mpfr_cmp_ui (xp, 1);
  62
  63   if (MPFR_UNLIKELY (compared >= 0))
  64     {
  65       mpfr_clear (xp);
  66       if (compared > 0) /* acos(x) = NaN for x > 1 */
  67         {
  68           MPFR_SET_NAN(acos);
  69           MPFR_RET_NAN;
  70         }
  71       else
  72         {
  73           if (MPFR_IS_POS_SIGN (sign)) /* acos(+1) = 0 */
  74             return mpfr_set_ui (acos, 0, rnd_mode);
  75           else /* acos(-1) = Pi */
  76             return mpfr_const_pi (acos, rnd_mode);
  77         }
  78     }
  79
  80   MPFR_SAVE_EXPO_MARK (expo);
  81
  82   /* Compute the supplement */
  83   mpfr_ui_sub (xp, 1, xp, GMP_RNDD);
  84   if (MPFR_IS_POS_SIGN (sign))
  85     supplement = 2 - 2 * MPFR_GET_EXP (xp);
  86   else
  87     supplement = 2 - MPFR_GET_EXP (xp);
  88   mpfr_clear (xp);
  89
  90   prec = MPFR_PREC (acos) + 10 + supplement;
  91
  92   /* VL: The following change concerning prec comes from r3145
  93      "Optimize mpfr_acos by choosing a better initial precision."
  94      but it doesn't seem to be correct and leads to problems (assertion
  95      failure or very important inefficiency) with tiny arguments.
  96      Therefore, I've disabled it. */
  97   /* If x ~ 2^-N, acos(x) ~ PI/2 - x - x^3/6
  98      If Prec < 2*N, we can't round since x^3/6 won't be counted. */
  99 #if 0
 100   if (MPFR_PREC (acos) >= MPFR_PREC (x) && MPFR_GET_EXP (x) < 0)
 101     {
 102       mpfr_uexp_t pmin = (mpfr_uexp_t) (-2 * MPFR_GET_EXP (x)) + 5;
 103       MPFR_ASSERTN (pmin <= MPFR_PREC_MAX);
 104       if (prec < pmin)
 105         prec = pmin;
 106     }
 107 #endif
 108
 109   mpfr_init2 (tmp, prec);
 110   mpfr_init2 (arcc, prec);
 111
 112   MPFR_ZIV_INIT (loop, prec);
 113   for (;;)
 114     {
 115       /* acos(x) = Pi/2 - asin(x) = Pi/2 - atan(x/sqrt(1-x^2)) */
 116       mpfr_sqr (tmp, x, GMP_RNDN);
 117       mpfr_ui_sub (tmp, 1, tmp, GMP_RNDN);
 118       mpfr_sqrt (tmp, tmp, GMP_RNDN);
 119       mpfr_div (tmp, x, tmp, GMP_RNDN);
 120       mpfr_atan (arcc, tmp, GMP_RNDN);
 121       mpfr_const_pi (tmp, GMP_RNDN);
 122       mpfr_div_2ui (tmp, tmp, 1, GMP_RNDN);
 123       mpfr_sub (arcc, tmp, arcc, GMP_RNDN);
 124
 125       if (MPFR_LIKELY (MPFR_CAN_ROUND (arcc, prec-supplement,
 126                                        MPFR_PREC (acos), rnd_mode)))
 127         break;
 128       MPFR_ZIV_NEXT (loop, prec);
 129       mpfr_set_prec (tmp, prec);
 130       mpfr_set_prec (arcc, prec);
 131     }
 132   MPFR_ZIV_FREE (loop);
 133
 134   inexact = mpfr_set (acos, arcc, rnd_mode);
 135   mpfr_clear (tmp);
 136   mpfr_clear (arcc);
 137
 138   MPFR_SAVE_EXPO_FREE (expo);
 139   return mpfr_check_range (acos, inexact, rnd_mode);
 140 }