1/* mpfr_acosh -- inverse hyperbolic cosine
2
3Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4Contributed by the Arenaire and Cacao projects, INRIA.
5
6This file is part of the GNU MPFR Library.
7
8The GNU MPFR Library is free software; you can redistribute it and/or modify
9it under the terms of the GNU Lesser General Public License as published by
10the Free Software Foundation; either version 3 of the License, or (at your
11option) any later version.
12
13The GNU MPFR Library is distributed in the hope that it will be useful, but
14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
16License for more details.
17
18You should have received a copy of the GNU Lesser General Public License
19along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
2151 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
29int
30mpfr_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 (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
37                 ("y[%#R]=%R inexact=%d", y, y, inexact));
38
39  /* Deal with special cases */
40  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
41    {
42      /* Nan, or zero or -Inf */
43      if (MPFR_IS_INF (x) && MPFR_IS_POS (x))
44        {
45          MPFR_SET_INF (y);
46          MPFR_SET_POS (y);
47          MPFR_RET (0);
48        }
49      else /* Nan, or zero or -Inf */
50        {
51          MPFR_SET_NAN (y);
52          MPFR_RET_NAN;
53        }
54    }
55  comp = mpfr_cmp_ui (x, 1);
56  if (MPFR_UNLIKELY (comp < 0))
57    {
58      MPFR_SET_NAN (y);
59      MPFR_RET_NAN;
60    }
61  else if (MPFR_UNLIKELY (comp == 0))
62    {
63      MPFR_SET_ZERO (y); /* acosh(1) = 0 */
64      MPFR_SET_POS (y);
65      MPFR_RET (0);
66    }
67  MPFR_SAVE_EXPO_MARK (expo);
68
69  /* General case */
70  {
71    /* Declaration of the intermediary variables */
72    mpfr_t t;
73    /* Declaration of the size variables */
74    mpfr_prec_t Ny = MPFR_PREC(y);   /* Precision of output variable */
75    mpfr_prec_t Nt;                  /* Precision of the intermediary variable */
76    mpfr_exp_t  err, exp_te, d;      /* Precision of error */
77    MPFR_ZIV_DECL (loop);
78
79    /* compute the precision of intermediary variable */
80    /* the optimal number of bits : see algorithms.tex */
81    Nt = Ny + 4 + MPFR_INT_CEIL_LOG2 (Ny);
82
83    /* initialization of intermediary variables */
84    mpfr_init2 (t, Nt);
85
86    /* First computation of acosh */
87    MPFR_ZIV_INIT (loop, Nt);
88    for (;;)
89      {
90        MPFR_BLOCK_DECL (flags);
91
92        /* compute acosh */
93        MPFR_BLOCK (flags, mpfr_mul (t, x, x, MPFR_RNDD));  /* x^2 */
94        if (MPFR_OVERFLOW (flags))
95          {
96            mpfr_t ln2;
97            mpfr_prec_t pln2;
98
99            /* As x is very large and the precision is not too large, we
100               assume that we obtain the same result by evaluating ln(2x).
101               We need to compute ln(x) + ln(2) as 2x can overflow. TODO:
102               write a proof and add an MPFR_ASSERTN. */
103            mpfr_log (t, x, MPFR_RNDN);  /* err(log) < 1/2 ulp(t) */
104            pln2 = Nt - MPFR_PREC_MIN < MPFR_GET_EXP (t) ?
105              MPFR_PREC_MIN : Nt - MPFR_GET_EXP (t);
106            mpfr_init2 (ln2, pln2);
107            mpfr_const_log2 (ln2, MPFR_RNDN);  /* err(ln2) < 1/2 ulp(t) */
108            mpfr_add (t, t, ln2, MPFR_RNDN);  /* err <= 3/2 ulp(t) */
109            mpfr_clear (ln2);
110            err = 1;
111          }
112        else
113          {
114            exp_te = MPFR_GET_EXP (t);
115            mpfr_sub_ui (t, t, 1, MPFR_RNDD);   /* x^2-1 */
116            if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
117              {
118                /* This means that x is very close to 1: x = 1 + t with
119                   t < 2^(-Nt). We have: acosh(x) = sqrt(2t) (1 - eps(t))
120                   with 0 < eps(t) < t / 12. */
121                mpfr_sub_ui (t, x, 1, MPFR_RNDD);   /* t = x - 1 */
122                mpfr_mul_2ui (t, t, 1, MPFR_RNDN);  /* 2t */
123                mpfr_sqrt (t, t, MPFR_RNDN);        /* sqrt(2t) */
124                err = 1;
125              }
126            else
127              {
128                d = exp_te - MPFR_GET_EXP (t);
129                mpfr_sqrt (t, t, MPFR_RNDN);        /* sqrt(x^2-1) */
130                mpfr_add (t, t, x, MPFR_RNDN);      /* sqrt(x^2-1)+x */
131                mpfr_log (t, t, MPFR_RNDN);         /* ln(sqrt(x^2-1)+x) */
132
133                /* error estimate -- see algorithms.tex */
134                err = 3 + MAX (1, d) - MPFR_GET_EXP (t);
135                /* error is bounded by 1/2 + 2^err <= 2^(max(0,1+err)) */
136                err = MAX (0, 1 + err);
137              }
138          }
139
140        if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Ny, rnd_mode)))
141          break;
142
143        /* reactualisation of the precision */
144        MPFR_ZIV_NEXT (loop, Nt);
145        mpfr_set_prec (t, Nt);
146      }
147    MPFR_ZIV_FREE (loop);
148
149    inexact = mpfr_set (y, t, rnd_mode);
150
151    mpfr_clear (t);
152  }
153
154  MPFR_SAVE_EXPO_FREE (expo);
155  return mpfr_check_range (y, inexact, rnd_mode);
156}
157