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/* mpfr_sinh -- hyperbolic sine
Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 2.1 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* The computation of sinh is done by
sinh(x) = 1/2 [e^(x)-e^(-x)] */
int
mpfr_sinh (mpfr_ptr y, mpfr_srcptr xt, mp_rnd_t rnd_mode)
{
mpfr_t x;
int inexact;
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", xt, xt, rnd_mode),
("y[%#R]=%R inexact=%d", y, y, inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
{
if (MPFR_IS_NAN (xt))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (xt))
{
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, xt);
MPFR_RET (0);
}
else /* xt is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (xt));
MPFR_SET_ZERO (y); /* sinh(0) = 0 */
MPFR_SET_SAME_SIGN (y, xt);
MPFR_RET (0);
}
}
/* sinh(x) = x + x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP(xt), 2, 1,
rnd_mode, {});
MPFR_TMP_INIT_ABS (x, xt);
{
mpfr_t t, ti;
mp_exp_t d;
mp_prec_t Nt; /* Precision of the intermediary variable */
long int err; /* Precision of error */
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_GROUP_DECL (group);
MPFR_SAVE_EXPO_MARK (expo);
/* compute the precision of intermediary variable */
Nt = MAX (MPFR_PREC (x), MPFR_PREC (y));
/* the optimal number of bits : see algorithms.ps */
Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
/* If x is near 0, exp(x) - 1/exp(x) = 2*x+x^3/3+O(x^5) */
if (MPFR_GET_EXP (x) < 0)
Nt -= 2*MPFR_GET_EXP (x);
/* initialise of intermediary variables */
MPFR_GROUP_INIT_2 (group, Nt, t, ti);
/* First computation of sinh */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags);
/* compute sinh */
MPFR_BLOCK (flags, mpfr_exp (t, x, GMP_RNDD));
if (MPFR_OVERFLOW (flags))
/* exp(x) does overflow */
{
/* sinh(x) = 2 * sinh(x/2) * cosh(x/2) */
mpfr_div_2ui (ti, x, 1, GMP_RNDD); /* exact */
/* t <- cosh(x/2): error(t) <= 1 ulp(t) */
MPFR_BLOCK (flags, mpfr_cosh (t, ti, GMP_RNDD));
if (MPFR_OVERFLOW (flags))
/* when x>1 we have |sinh(x)| >= cosh(x/2), so sinh(x)
overflows too */
{
inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
break;
}
/* ti <- sinh(x/2): , error(ti) <= 1 ulp(ti)
cannot overflow because 0 < sinh(x) < cosh(x) when x > 0 */
mpfr_sinh (ti, ti, GMP_RNDD);
/* multiplication below, error(t) <= 5 ulp(t) */
MPFR_BLOCK (flags, mpfr_mul (t, t, ti, GMP_RNDD));
if (MPFR_OVERFLOW (flags))
{
inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
break;
}
/* doubling below, exact */
MPFR_BLOCK (flags, mpfr_mul_2ui (t, t, 1, GMP_RNDN));
if (MPFR_OVERFLOW (flags))
{
inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
break;
}
/* we have lost at most 3 bits of precision */
err = Nt - 3;
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, MPFR_PREC (y),
rnd_mode)))
{
inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
break;
}
err = Nt; /* double the precision */
}
else
{
d = MPFR_GET_EXP (t);
mpfr_ui_div (ti, 1, t, GMP_RNDU); /* 1/exp(x) */
mpfr_sub (t, t, ti, GMP_RNDN); /* exp(x) - 1/exp(x) */
mpfr_div_2ui (t, t, 1, GMP_RNDN); /* 1/2(exp(x) - 1/exp(x)) */
/* it may be that t is zero (in fact, it can only occur when te=1,
and thus ti=1 too) */
if (MPFR_IS_ZERO (t))
err = Nt; /* double the precision */
else
{
/* calculation of the error */
d = d - MPFR_GET_EXP (t) + 2;
/* error estimate: err = Nt-(__gmpfr_ceil_log2(1+pow(2,d)));*/
err = Nt - (MAX (d, 0) + 1);
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, MPFR_PREC (y),
rnd_mode)))
{
inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
break;
}
}
}
/* actualisation of the precision */
Nt += err;
MPFR_ZIV_NEXT (loop, Nt);
MPFR_GROUP_REPREC_2 (group, Nt, t, ti);
}
MPFR_ZIV_FREE (loop);
MPFR_GROUP_CLEAR (group);
MPFR_SAVE_EXPO_FREE (expo);
}
return mpfr_check_range (y, inexact, rnd_mode);
}
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