1 /* mpfr_csch - Hyperbolic cosecant function. 2 3 Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. 4 Contributed by the AriC 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 /* 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