| /* 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); |
| } |