1/* mpfr_coth - Hyperbolic cotangent function. 2 3Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc. 4Contributed by the Arenaire and Cacao projects, INRIA. 5 6This file is part of the GNU MPFR Library. 7 8The GNU MPFR Library is free software; you can redistribute it and/or modify 9it under the terms of the GNU Lesser General Public License as published by 10the Free Software Foundation; either version 3 of the License, or (at your 11option) any later version. 12 13The GNU MPFR Library is distributed in the hope that it will be useful, but 14WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 15or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 16License for more details. 17 18You should have received a copy of the GNU Lesser General Public License 19along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 20http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 2151 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 22 23/* the hyperbolic cotangent is defined by coth(x) = 1/tanh(x) 24 coth (NaN) = NaN. 25 coth (+Inf) = 1 26 coth (-Inf) = -1 27 coth (+0) = +Inf. 28 coth (-0) = -Inf. 29*/ 30 31#define FUNCTION mpfr_coth 32#define INVERSE mpfr_tanh 33#define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) 34#define ACTION_INF(y) return mpfr_set_si (y, MPFR_IS_POS(x) ? 1 : -1, rnd_mode) 35#define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \ 36 MPFR_RET(0); } while (1) 37 38/* We know |coth(x)| > 1, thus if the approximation z is such that 39 1 <= z <= 1 + 2^(-p) where p is the target precision, then the 40 result is either 1 or nextabove(1) = 1 + 2^(1-p). */ 41#define ACTION_SPECIAL \ 42 if (MPFR_GET_EXP(z) == 1) /* 1 <= |z| < 2 */ \ 43 { \ 44 /* the following is exact by Sterbenz theorem */ \ 45 mpfr_sub_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, MPFR_RNDN); \ 46 if (MPFR_IS_ZERO(z) || MPFR_GET_EXP(z) <= - (mpfr_exp_t) precy) \ 47 { \ 48 mpfr_add_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, MPFR_RNDN); \ 49 break; \ 50 } \ 51 } 52 53/* The analysis is adapted from that for mpfr_csc: 54 near x=0, coth(x) = 1/x + x/3 + ..., more precisely we have 55 |coth(x) - 1/x| <= 0.32 for |x| <= 1. Like for csc, the error term has 56 the same sign as 1/x, thus |coth(x)| >= |1/x|. Then: 57 (i) either x is a power of two, then 1/x is exactly representable, and 58 as long as 1/2*ulp(1/x) > 0.32, we can conclude; 59 (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then 60 |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. 61 Since |coth(x) - 1/x| <= 0.32, if 2^(-2n) ufp(y) >= 0.64, then 62 |y - coth(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct 63 result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). 64 A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)). */ 65#define ACTION_TINY(y,x,r) \ 66 if (MPFR_EXP(x) + 1 <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ 67 { \ 68 int signx = MPFR_SIGN(x); \ 69 inexact = mpfr_ui_div (y, 1, x, r); \ 70 if (inexact == 0) /* x is a power of two */ \ 71 { /* result always 1/x, except when rounding away from zero */ \ 72 if (rnd_mode == MPFR_RNDA) \ 73 rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \ 74 if (rnd_mode == MPFR_RNDU) \ 75 { \ 76 if (signx > 0) \ 77 mpfr_nextabove (y); /* 2^k + epsilon */ \ 78 inexact = 1; \ 79 } \ 80 else if (rnd_mode == MPFR_RNDD) \ 81 { \ 82 if (signx < 0) \ 83 mpfr_nextbelow (y); /* -2^k - epsilon */ \ 84 inexact = -1; \ 85 } \ 86 else /* round to zero, or nearest */ \ 87 inexact = -signx; \ 88 } \ 89 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ 90 goto end; \ 91 } 92 93#include "gen_inverse.h" 94