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