1 /* mpfr_atanh -- Inverse Hyperbolic Tangente 2 3 Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc. 4 Contributed by the Arenaire and Caramel projects, INRIA. 5 6 This file is part of the GNU MPFR Library. 7 8 The GNU MPFR Library is free software; you can redistribute it and/or modify 9 it under the terms of the GNU Lesser General Public License as published by 10 the Free Software Foundation; either version 3 of the License, or (at your 11 option) any later version. 12 13 The GNU MPFR Library is distributed in the hope that it will be useful, but 14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16 License for more details. 17 18 You should have received a copy of the GNU Lesser General Public License 19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 21 51 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 atanh is done by 27 atanh= 1/2*ln(x+1)-1/2*ln(1-x) */ 28 29 int 30 mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode) 31 { 32 int inexact; 33 mpfr_t x, t, te; 34 mpfr_prec_t Nx, Ny, Nt; 35 mpfr_exp_t err; 36 MPFR_ZIV_DECL (loop); 37 MPFR_SAVE_EXPO_DECL (expo); 38 39 MPFR_LOG_FUNC 40 (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode), 41 ("y[%Pu]=%.*Rg inexact=%d", 42 mpfr_get_prec (y), mpfr_log_prec, y, inexact)); 43 44 /* Special cases */ 45 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt))) 46 { 47 /* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result 48 between -1 and 1 */ 49 if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt)) 50 { 51 MPFR_SET_NAN (y); 52 MPFR_RET_NAN; 53 } 54 else /* necessarily xt is 0 */ 55 { 56 MPFR_ASSERTD (MPFR_IS_ZERO (xt)); 57 MPFR_SET_ZERO (y); /* atanh(0) = 0 */ 58 MPFR_SET_SAME_SIGN (y,xt); 59 MPFR_RET (0); 60 } 61 } 62 63 /* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */ 64 if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0)) 65 { 66 if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt)) 67 { 68 MPFR_SET_INF (y); 69 MPFR_SET_SAME_SIGN (y, xt); 70 mpfr_set_divby0 (); 71 MPFR_RET (0); 72 } 73 MPFR_SET_NAN (y); 74 MPFR_RET_NAN; 75 } 76 77 /* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */ 78 MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1, 79 rnd_mode, {}); 80 81 MPFR_SAVE_EXPO_MARK (expo); 82 83 /* Compute initial precision */ 84 Nx = MPFR_PREC (xt); 85 MPFR_TMP_INIT_ABS (x, xt); 86 Ny = MPFR_PREC (y); 87 Nt = MAX (Nx, Ny); 88 /* the optimal number of bits : see algorithms.ps */ 89 Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4; 90 91 /* initialise of intermediary variable */ 92 mpfr_init2 (t, Nt); 93 mpfr_init2 (te, Nt); 94 95 /* First computation of cosh */ 96 MPFR_ZIV_INIT (loop, Nt); 97 for (;;) 98 { 99 /* compute atanh */ 100 mpfr_ui_sub (te, 1, x, MPFR_RNDU); /* (1-xt)*/ 101 mpfr_add_ui (t, x, 1, MPFR_RNDD); /* (xt+1)*/ 102 mpfr_div (t, t, te, MPFR_RNDN); /* (1+xt)/(1-xt)*/ 103 mpfr_log (t, t, MPFR_RNDN); /* ln((1+xt)/(1-xt))*/ 104 mpfr_div_2ui (t, t, 1, MPFR_RNDN); /* (1/2)*ln((1+xt)/(1-xt))*/ 105 106 /* error estimate: see algorithms.tex */ 107 /* FIXME: this does not correspond to the value in algorithms.tex!!! */ 108 /* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/ 109 err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1); 110 111 if (MPFR_LIKELY (MPFR_IS_ZERO (t) 112 || MPFR_CAN_ROUND (t, err, Ny, rnd_mode))) 113 break; 114 115 /* reactualisation of the precision */ 116 MPFR_ZIV_NEXT (loop, Nt); 117 mpfr_set_prec (t, Nt); 118 mpfr_set_prec (te, Nt); 119 } 120 MPFR_ZIV_FREE (loop); 121 122 inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt)); 123 124 mpfr_clear(t); 125 mpfr_clear(te); 126 127 MPFR_SAVE_EXPO_FREE (expo); 128 return mpfr_check_range (y, inexact, rnd_mode); 129 } 130 131