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

   1 /* mpfr_acosh -- inverse hyperbolic cosine
   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 /* The computation of acosh is done by   *
  27  *  acosh= ln(x + sqrt(x^2-1))           */
  28
  29 int
  30 mpfr_acosh (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode)
  31 {
  32   MPFR_SAVE_EXPO_DECL (expo);
  33   int inexact;
  34   int comp;
  35
  36   MPFR_LOG_FUNC (
  37     ("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
  38     ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
  39      inexact));
  40
  41   /* Deal with special cases */
  42   if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
  43     {
  44       /* Nan, or zero or -Inf */
  45       if (MPFR_IS_INF (x) && MPFR_IS_POS (x))
  46         {
  47           MPFR_SET_INF (y);
  48           MPFR_SET_POS (y);
  49           MPFR_RET (0);
  50         }
  51       else /* Nan, or zero or -Inf */
  52         {
  53           MPFR_SET_NAN (y);
  54           MPFR_RET_NAN;
  55         }
  56     }
  57   comp = mpfr_cmp_ui (x, 1);
  58   if (MPFR_UNLIKELY (comp < 0))
  59     {
  60       MPFR_SET_NAN (y);
  61       MPFR_RET_NAN;
  62     }
  63   else if (MPFR_UNLIKELY (comp == 0))
  64     {
  65       MPFR_SET_ZERO (y); /* acosh(1) = 0 */
  66       MPFR_SET_POS (y);
  67       MPFR_RET (0);
  68     }
  69   MPFR_SAVE_EXPO_MARK (expo);
  70
  71   /* General case */
  72   {
  73     /* Declaration of the intermediary variables */
  74     mpfr_t t;
  75     /* Declaration of the size variables */
  76     mpfr_prec_t Ny = MPFR_PREC(y);   /* Precision of output variable */
  77     mpfr_prec_t Nt;                  /* Precision of the intermediary variable */
  78     mpfr_exp_t  err, exp_te, d;      /* Precision of error */
  79     MPFR_ZIV_DECL (loop);
  80
  81     /* compute the precision of intermediary variable */
  82     /* the optimal number of bits : see algorithms.tex */
  83     Nt = Ny + 4 + MPFR_INT_CEIL_LOG2 (Ny);
  84
  85     /* initialization of intermediary variables */
  86     mpfr_init2 (t, Nt);
  87
  88     /* First computation of acosh */
  89     MPFR_ZIV_INIT (loop, Nt);
  90     for (;;)
  91       {
  92         MPFR_BLOCK_DECL (flags);
  93
  94         /* compute acosh */
  95         MPFR_BLOCK (flags, mpfr_mul (t, x, x, MPFR_RNDD));  /* x^2 */
  96         if (MPFR_OVERFLOW (flags))
  97           {
  98             mpfr_t ln2;
  99             mpfr_prec_t pln2;
 100
 101             /* As x is very large and the precision is not too large, we
 102                assume that we obtain the same result by evaluating ln(2x).
 103                We need to compute ln(x) + ln(2) as 2x can overflow. TODO:
 104                write a proof and add an MPFR_ASSERTN. */
 105             mpfr_log (t, x, MPFR_RNDN);  /* err(log) < 1/2 ulp(t) */
 106             pln2 = Nt - MPFR_PREC_MIN < MPFR_GET_EXP (t) ?
 107               MPFR_PREC_MIN : Nt - MPFR_GET_EXP (t);
 108             mpfr_init2 (ln2, pln2);
 109             mpfr_const_log2 (ln2, MPFR_RNDN);  /* err(ln2) < 1/2 ulp(t) */
 110             mpfr_add (t, t, ln2, MPFR_RNDN);  /* err <= 3/2 ulp(t) */
 111             mpfr_clear (ln2);
 112             err = 1;
 113           }
 114         else
 115           {
 116             exp_te = MPFR_GET_EXP (t);
 117             mpfr_sub_ui (t, t, 1, MPFR_RNDD);   /* x^2-1 */
 118             if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
 119               {
 120                 /* This means that x is very close to 1: x = 1 + t with
 121                    t < 2^(-Nt). We have: acosh(x) = sqrt(2t) (1 - eps(t))
 122                    with 0 < eps(t) < t / 12. */
 123                 mpfr_sub_ui (t, x, 1, MPFR_RNDD);   /* t = x - 1 */
 124                 mpfr_mul_2ui (t, t, 1, MPFR_RNDN);  /* 2t */
 125                 mpfr_sqrt (t, t, MPFR_RNDN);        /* sqrt(2t) */
 126                 err = 1;
 127               }
 128             else
 129               {
 130                 d = exp_te - MPFR_GET_EXP (t);
 131                 mpfr_sqrt (t, t, MPFR_RNDN);        /* sqrt(x^2-1) */
 132                 mpfr_add (t, t, x, MPFR_RNDN);      /* sqrt(x^2-1)+x */
 133                 mpfr_log (t, t, MPFR_RNDN);         /* ln(sqrt(x^2-1)+x) */
 134
 135                 /* error estimate -- see algorithms.tex */
 136                 err = 3 + MAX (1, d) - MPFR_GET_EXP (t);
 137                 /* error is bounded by 1/2 + 2^err <= 2^(max(0,1+err)) */
 138                 err = MAX (0, 1 + err);
 139               }
 140           }
 141
 142         if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Ny, rnd_mode)))
 143           break;
 144
 145         /* reactualisation of the precision */
 146         MPFR_ZIV_NEXT (loop, Nt);
 147         mpfr_set_prec (t, Nt);
 148       }
 149     MPFR_ZIV_FREE (loop);
 150
 151     inexact = mpfr_set (y, t, rnd_mode);
 152
 153     mpfr_clear (t);
 154   }
 155
 156   MPFR_SAVE_EXPO_FREE (expo);
 157   return mpfr_check_range (y, inexact, rnd_mode);
 158 }