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      1 /* mpfr_acos -- arc-cosinus of a floating-point number
      2 
      3 Copyright 2001-2023 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 int
     27 mpfr_acos (mpfr_ptr acos, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
     28 {
     29   mpfr_t xp, arcc, tmp;
     30   mpfr_exp_t supplement;
     31   mpfr_prec_t prec;
     32   int sign, compared, inexact;
     33   MPFR_SAVE_EXPO_DECL (expo);
     34   MPFR_ZIV_DECL (loop);
     35 
     36   MPFR_LOG_FUNC
     37     (("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
     38      ("acos[%Pd]=%.*Rg inexact=%d",
     39       mpfr_get_prec(acos), mpfr_log_prec, acos, inexact));
     40 
     41   /* Singular cases */
     42   if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
     43     {
     44       if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
     45         {
     46           MPFR_SET_NAN (acos);
     47           MPFR_RET_NAN;
     48         }
     49       else /* necessarily x=0 */
     50         {
     51           MPFR_ASSERTD(MPFR_IS_ZERO(x));
     52           /* acos(0)=Pi/2 */
     53           MPFR_SAVE_EXPO_MARK (expo);
     54           inexact = mpfr_const_pi (acos, rnd_mode);
     55           mpfr_div_2ui (acos, acos, 1, rnd_mode); /* exact */
     56           MPFR_SAVE_EXPO_FREE (expo);
     57           return mpfr_check_range (acos, inexact, rnd_mode);
     58         }
     59     }
     60 
     61   /* Set x_p=|x| */
     62   sign = MPFR_SIGN (x);
     63   mpfr_init2 (xp, MPFR_PREC (x));
     64   mpfr_abs (xp, x, MPFR_RNDN); /* Exact */
     65 
     66   compared = mpfr_cmp_ui (xp, 1);
     67 
     68   if (MPFR_UNLIKELY (compared >= 0))
     69     {
     70       mpfr_clear (xp);
     71       if (compared > 0) /* acos(x) = NaN for x > 1 */
     72         {
     73           MPFR_SET_NAN(acos);
     74           MPFR_RET_NAN;
     75         }
     76       else
     77         {
     78           if (MPFR_IS_POS_SIGN (sign)) /* acos(+1) = +0 */
     79             return mpfr_set_ui (acos, 0, rnd_mode);
     80           else /* acos(-1) = Pi */
     81             return mpfr_const_pi (acos, rnd_mode);
     82         }
     83     }
     84 
     85   MPFR_SAVE_EXPO_MARK (expo);
     86 
     87   /* Compute the supplement */
     88   mpfr_ui_sub (xp, 1, xp, MPFR_RNDD);
     89   if (MPFR_IS_POS_SIGN (sign))
     90     supplement = 2 - 2 * MPFR_GET_EXP (xp);
     91   else
     92     supplement = 2 - MPFR_GET_EXP (xp);
     93   mpfr_clear (xp);
     94 
     95   prec = MPFR_PREC (acos);
     96   prec += MPFR_INT_CEIL_LOG2(prec) + 10 + supplement;
     97 
     98   /* VL: The following change concerning prec comes from r3145
     99      "Optimize mpfr_acos by choosing a better initial precision."
    100      but it doesn't seem to be correct and leads to problems (assertion
    101      failure or very important inefficiency) with tiny arguments.
    102      Therefore, I've disabled it. */
    103   /* If x ~ 2^-N, acos(x) ~ PI/2 - x - x^3/6
    104      If Prec < 2*N, we can't round since x^3/6 won't be counted. */
    105 #if 0
    106   if (MPFR_PREC (acos) >= MPFR_PREC (x) && MPFR_GET_EXP (x) < 0)
    107     {
    108       mpfr_uexp_t pmin = (mpfr_uexp_t) (-2 * MPFR_GET_EXP (x)) + 5;
    109       MPFR_ASSERTN (pmin <= MPFR_PREC_MAX);
    110       if (prec < pmin)
    111         prec = pmin;
    112     }
    113 #endif
    114 
    115   mpfr_init2 (tmp, prec);
    116   mpfr_init2 (arcc, prec);
    117 
    118   MPFR_ZIV_INIT (loop, prec);
    119   for (;;)
    120     {
    121       /* acos(x) = Pi/2 - asin(x) = Pi/2 - atan(x/sqrt(1-x^2)) */
    122       mpfr_sqr (tmp, x, MPFR_RNDN);
    123       mpfr_ui_sub (tmp, 1, tmp, MPFR_RNDN);
    124       mpfr_sqrt (tmp, tmp, MPFR_RNDN);
    125       mpfr_div (tmp, x, tmp, MPFR_RNDN);
    126       mpfr_atan (arcc, tmp, MPFR_RNDN);
    127       mpfr_const_pi (tmp, MPFR_RNDN);
    128       mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDN);
    129       mpfr_sub (arcc, tmp, arcc, MPFR_RNDN);
    130 
    131       if (MPFR_LIKELY (MPFR_CAN_ROUND (arcc, prec - supplement,
    132                                        MPFR_PREC (acos), rnd_mode)))
    133         break;
    134       MPFR_ZIV_NEXT (loop, prec);
    135       mpfr_set_prec (tmp, prec);
    136       mpfr_set_prec (arcc, prec);
    137     }
    138   MPFR_ZIV_FREE (loop);
    139 
    140   inexact = mpfr_set (acos, arcc, rnd_mode);
    141   mpfr_clear (tmp);
    142   mpfr_clear (arcc);
    143 
    144   MPFR_SAVE_EXPO_FREE (expo);
    145   return mpfr_check_range (acos, inexact, rnd_mode);
    146 }
    147