1/* mpfr_csch - Hyperbolic cosecant function. 2 3Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. 4Contributed by the AriC and Caramel 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 cosecant is defined by csch(x) = 1/sinh(x). 24 csch (NaN) = NaN. 25 csch (+Inf) = +0. 26 csch (-Inf) = -0. 27 csch (+0) = +Inf. 28 csch (-0) = -Inf. 29*/ 30 31#define FUNCTION mpfr_csch 32#define INVERSE mpfr_sinh 33#define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) 34#define ACTION_INF(y) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_ZERO (y); \ 35 MPFR_RET(0); } while (1) 36#define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \ 37 mpfr_set_divby0 (); MPFR_RET(0); } while (1) 38 39/* (This analysis is adapted from that for mpfr_csc.) 40 Near x=0, we have csch(x) = 1/x - x/6 + ..., more precisely we have 41 |csch(x) - 1/x| <= 0.2 for |x| <= 1. The error term has the opposite 42 sign as 1/x, thus |csch(x)| <= |1/x|. Then: 43 (i) either x is a power of two, then 1/x is exactly representable, and 44 as long as 1/2*ulp(1/x) > 0.2, we can conclude; 45 (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then 46 |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. 47 Since |csch(x) - 1/x| <= 0.2, if 2^(-2n) ufp(y) >= 0.4, then 48 |y - csch(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct 49 result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). 50 A sufficient condition is thus EXP(x) <= -2 MAX(PREC(x),PREC(Y)). */ 51#define ACTION_TINY(y,x,r) \ 52 if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ 53 { \ 54 int signx = MPFR_SIGN(x); \ 55 inexact = mpfr_ui_div (y, 1, x, r); \ 56 if (inexact == 0) /* x is a power of two */ \ 57 { /* result always 1/x, except when rounding to zero */ \ 58 if (rnd_mode == MPFR_RNDA) \ 59 rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \ 60 if (rnd_mode == MPFR_RNDU || (rnd_mode == MPFR_RNDZ && signx < 0)) \ 61 { \ 62 if (signx < 0) \ 63 mpfr_nextabove (y); /* -2^k + epsilon */ \ 64 inexact = 1; \ 65 } \ 66 else if (rnd_mode == MPFR_RNDD || rnd_mode == MPFR_RNDZ) \ 67 { \ 68 if (signx > 0) \ 69 mpfr_nextbelow (y); /* 2^k - epsilon */ \ 70 inexact = -1; \ 71 } \ 72 else /* round to nearest */ \ 73 inexact = signx; \ 74 } \ 75 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ 76 goto end; \ 77 } 78 79#include "gen_inverse.h" 80