--- /dev/null
+/* 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 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 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, GMP_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, GMP_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, GMP_RNDN); /* err(ln2) < 1/2 ulp(t) */
+ mpfr_add (t, t, ln2, GMP_RNDN); /* err <= 3/2 ulp(t) */
+ mpfr_clear (ln2);
+ err = 1;
+ }
+ else
+ {
+ exp_te = MPFR_GET_EXP (t);
+ mpfr_sub_ui (t, t, 1, GMP_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, GMP_RNDD); /* t = x - 1 */
+ mpfr_mul_2ui (t, t, 1, GMP_RNDN); /* 2t */
+ mpfr_sqrt (t, t, GMP_RNDN); /* sqrt(2t) */
+ err = 1;
+ }
+ else
+ {
+ d = exp_te - MPFR_GET_EXP (t);
+ mpfr_sqrt (t, t, GMP_RNDN); /* sqrt(x^2-1) */
+ mpfr_add (t, t, x, GMP_RNDN); /* sqrt(x^2-1)+x */
+ mpfr_log (t, t, GMP_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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