1/* mpfr_sinh -- hyperbolic sine
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 sinh is done by
27    sinh(x) = 1/2 [e^(x)-e^(-x)]          */
28
29int
30mpfr_sinh (mpfr_ptr y, mpfr_srcptr xt, mpfr_rnd_t rnd_mode)
31{
32  mpfr_t x;
33  int inexact;
34
35  MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", xt, xt, rnd_mode),
36                 ("y[%#R]=%R inexact=%d", y, y, inexact));
37
38  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
39    {
40      if (MPFR_IS_NAN (xt))
41        {
42          MPFR_SET_NAN (y);
43          MPFR_RET_NAN;
44        }
45      else if (MPFR_IS_INF (xt))
46        {
47          MPFR_SET_INF (y);
48          MPFR_SET_SAME_SIGN (y, xt);
49          MPFR_RET (0);
50        }
51      else /* xt is zero */
52        {
53          MPFR_ASSERTD (MPFR_IS_ZERO (xt));
54          MPFR_SET_ZERO (y);   /* sinh(0) = 0 */
55          MPFR_SET_SAME_SIGN (y, xt);
56          MPFR_RET (0);
57        }
58    }
59
60  /* sinh(x) = x + x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
61  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP(xt), 2, 1,
62                                    rnd_mode, {});
63
64  MPFR_TMP_INIT_ABS (x, xt);
65
66  {
67    mpfr_t t, ti;
68    mpfr_exp_t d;
69    mpfr_prec_t Nt;    /* Precision of the intermediary variable */
70    long int err;    /* Precision of error */
71    MPFR_ZIV_DECL (loop);
72    MPFR_SAVE_EXPO_DECL (expo);
73    MPFR_GROUP_DECL (group);
74
75    MPFR_SAVE_EXPO_MARK (expo);
76
77    /* compute the precision of intermediary variable */
78    Nt = MAX (MPFR_PREC (x), MPFR_PREC (y));
79    /* the optimal number of bits : see algorithms.ps */
80    Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
81    /* If x is near 0, exp(x) - 1/exp(x) = 2*x+x^3/3+O(x^5) */
82    if (MPFR_GET_EXP (x) < 0)
83      Nt -= 2*MPFR_GET_EXP (x);
84
85    /* initialise of intermediary variables */
86    MPFR_GROUP_INIT_2 (group, Nt, t, ti);
87
88    /* First computation of sinh */
89    MPFR_ZIV_INIT (loop, Nt);
90    for (;;)
91      {
92        MPFR_BLOCK_DECL (flags);
93
94        /* compute sinh */
95        MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDD));
96        if (MPFR_OVERFLOW (flags))
97          /* exp(x) does overflow */
98          {
99            /* sinh(x) = 2 * sinh(x/2) * cosh(x/2) */
100            mpfr_div_2ui (ti, x, 1, MPFR_RNDD); /* exact */
101
102            /* t <- cosh(x/2): error(t) <= 1 ulp(t) */
103            MPFR_BLOCK (flags, mpfr_cosh (t, ti, MPFR_RNDD));
104            if (MPFR_OVERFLOW (flags))
105              /* when x>1 we have |sinh(x)| >= cosh(x/2), so sinh(x)
106                 overflows too */
107              {
108                inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
109                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
110                break;
111              }
112
113            /* ti <- sinh(x/2): , error(ti) <= 1 ulp(ti)
114               cannot overflow because 0 < sinh(x) < cosh(x) when x > 0 */
115            mpfr_sinh (ti, ti, MPFR_RNDD);
116
117            /* multiplication below, error(t) <= 5 ulp(t) */
118            MPFR_BLOCK (flags, mpfr_mul (t, t, ti, MPFR_RNDD));
119            if (MPFR_OVERFLOW (flags))
120              {
121                inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
122                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
123                break;
124              }
125
126            /* doubling below, exact */
127            MPFR_BLOCK (flags, mpfr_mul_2ui (t, t, 1, MPFR_RNDN));
128            if (MPFR_OVERFLOW (flags))
129              {
130                inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
131                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
132                break;
133              }
134
135            /* we have lost at most 3 bits of precision */
136            err = Nt - 3;
137            if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, MPFR_PREC (y),
138                                             rnd_mode)))
139              {
140                inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
141                break;
142              }
143            err = Nt; /* double the precision */
144          }
145        else
146          {
147            d = MPFR_GET_EXP (t);
148            mpfr_ui_div (ti, 1, t, MPFR_RNDU); /* 1/exp(x) */
149            mpfr_sub (t, t, ti, MPFR_RNDN);    /* exp(x) - 1/exp(x) */
150            mpfr_div_2ui (t, t, 1, MPFR_RNDN);  /* 1/2(exp(x) - 1/exp(x)) */
151
152            /* it may be that t is zero (in fact, it can only occur when te=1,
153               and thus ti=1 too) */
154            if (MPFR_IS_ZERO (t))
155              err = Nt; /* double the precision */
156            else
157              {
158                /* calculation of the error */
159                d = d - MPFR_GET_EXP (t) + 2;
160                /* error estimate: err = Nt-(__gmpfr_ceil_log2(1+pow(2,d)));*/
161                err = Nt - (MAX (d, 0) + 1);
162                if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, MPFR_PREC (y),
163                                                 rnd_mode)))
164                  {
165                    inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
166                    break;
167                  }
168              }
169          }
170
171        /* actualisation of the precision */
172        Nt += err;
173        MPFR_ZIV_NEXT (loop, Nt);
174        MPFR_GROUP_REPREC_2 (group, Nt, t, ti);
175      }
176    MPFR_ZIV_FREE (loop);
177    MPFR_GROUP_CLEAR (group);
178    MPFR_SAVE_EXPO_FREE (expo);
179  }
180
181  return mpfr_check_range (y, inexact, rnd_mode);
182}
183