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/* mpfr_acosh -- inverse hyperbolic cosine
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 3 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, see
http://www.gnu.org/licenses/ or 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 acosh is done by *
* acosh= ln(x + sqrt(x^2-1)) */
int
mpfr_acosh (mpfr_ptr y, mpfr_srcptr x , mp_rnd_t rnd_mode)
{
MPFR_SAVE_EXPO_DECL (expo);
int inexact;
int comp;
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
("y[%#R]=%R inexact=%d", y, y, inexact));
/* Deal with special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
/* Nan, or zero or -Inf */
if (MPFR_IS_INF (x) && MPFR_IS_POS (x))
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else /* Nan, or zero or -Inf */
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
}
comp = mpfr_cmp_ui (x, 1);
if (MPFR_UNLIKELY (comp < 0))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_UNLIKELY (comp == 0))
{
MPFR_SET_ZERO (y); /* acosh(1) = 0 */
MPFR_SET_POS (y);
MPFR_RET (0);
}
MPFR_SAVE_EXPO_MARK (expo);
/* General case */
{
/* Declaration of the intermediary variables */
mpfr_t t;
/* Declaration of the size variables */
mp_prec_t Ny = MPFR_PREC(y); /* Precision of output variable */
mp_prec_t Nt; /* Precision of the intermediary variable */
mp_exp_t err, exp_te, d; /* Precision of error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + 4 + MPFR_INT_CEIL_LOG2 (Ny);
/* initialization of intermediary variables */
mpfr_init2 (t, Nt);
/* First computation of acosh */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags);
/* compute acosh */
MPFR_BLOCK (flags, mpfr_mul (t, x, x, MPFR_RNDD)); /* x^2 */
if (MPFR_OVERFLOW (flags))
{
mpfr_t ln2;
mp_prec_t pln2;
/* As x is very large and the precision is not too large, we
assume that we obtain the same result by evaluating ln(2x).
We need to compute ln(x) + ln(2) as 2x can overflow. TODO:
write a proof and add an MPFR_ASSERTN. */
mpfr_log (t, x, MPFR_RNDN); /* err(log) < 1/2 ulp(t) */
pln2 = Nt - MPFR_PREC_MIN < MPFR_GET_EXP (t) ?
MPFR_PREC_MIN : Nt - MPFR_GET_EXP (t);
mpfr_init2 (ln2, pln2);
mpfr_const_log2 (ln2, MPFR_RNDN); /* err(ln2) < 1/2 ulp(t) */
mpfr_add (t, t, ln2, MPFR_RNDN); /* err <= 3/2 ulp(t) */
mpfr_clear (ln2);
err = 1;
}
else
{
exp_te = MPFR_GET_EXP (t);
mpfr_sub_ui (t, t, 1, MPFR_RNDD); /* x^2-1 */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
{
/* This means that x is very close to 1: x = 1 + t with
t < 2^(-Nt). We have: acosh(x) = sqrt(2t) (1 - eps(t))
with 0 < eps(t) < t / 12. */
mpfr_sub_ui (t, x, 1, MPFR_RNDD); /* t = x - 1 */
mpfr_mul_2ui (t, t, 1, MPFR_RNDN); /* 2t */
mpfr_sqrt (t, t, MPFR_RNDN); /* sqrt(2t) */
err = 1;
}
else
{
d = exp_te - MPFR_GET_EXP (t);
mpfr_sqrt (t, t, MPFR_RNDN); /* sqrt(x^2-1) */
mpfr_add (t, t, x, MPFR_RNDN); /* sqrt(x^2-1)+x */
mpfr_log (t, t, MPFR_RNDN); /* ln(sqrt(x^2-1)+x) */
/* error estimate -- see algorithms.tex */
err = 3 + MAX (1, d) - MPFR_GET_EXP (t);
/* error is bounded by 1/2 + 2^err <= 2^(max(0,1+err)) */
err = MAX (0, 1 + err);
}
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Ny, rnd_mode)))
break;
/* reactualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd_mode);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
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