14a238c70SJohn Marino /* mpfr_csc - cosecant function. 24a238c70SJohn Marino 3*ab6d115fSJohn Marino Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. 4*ab6d115fSJohn Marino Contributed by the AriC and Caramel projects, INRIA. 54a238c70SJohn Marino 64a238c70SJohn Marino This file is part of the GNU MPFR Library. 74a238c70SJohn Marino 84a238c70SJohn Marino The GNU MPFR Library is free software; you can redistribute it and/or modify 94a238c70SJohn Marino it under the terms of the GNU Lesser General Public License as published by 104a238c70SJohn Marino the Free Software Foundation; either version 3 of the License, or (at your 114a238c70SJohn Marino option) any later version. 124a238c70SJohn Marino 134a238c70SJohn Marino The GNU MPFR Library is distributed in the hope that it will be useful, but 144a238c70SJohn Marino WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 154a238c70SJohn Marino or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public 164a238c70SJohn Marino License for more details. 174a238c70SJohn Marino 184a238c70SJohn Marino You should have received a copy of the GNU Lesser General Public License 194a238c70SJohn Marino along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see 204a238c70SJohn Marino http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc., 214a238c70SJohn Marino 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */ 224a238c70SJohn Marino 234a238c70SJohn Marino /* the cosecant is defined by csc(x) = 1/sin(x). 244a238c70SJohn Marino csc (NaN) = NaN. 254a238c70SJohn Marino csc (+Inf) = csc (-Inf) = NaN. 264a238c70SJohn Marino csc (+0) = +Inf. 274a238c70SJohn Marino csc (-0) = -Inf. 284a238c70SJohn Marino */ 294a238c70SJohn Marino 304a238c70SJohn Marino #define FUNCTION mpfr_csc 314a238c70SJohn Marino #define INVERSE mpfr_sin 324a238c70SJohn Marino #define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) 334a238c70SJohn Marino #define ACTION_INF(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) 344a238c70SJohn Marino #define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \ 354a238c70SJohn Marino mpfr_set_divby0 (); MPFR_RET(0); } while (1) 364a238c70SJohn Marino /* near x=0, we have csc(x) = 1/x + x/6 + ..., more precisely we have 374a238c70SJohn Marino |csc(x) - 1/x| <= 0.2 for |x| <= 1. The analysis is similar to that for 384a238c70SJohn Marino gamma(x) near x=0 (see gamma.c), except here the error term has the same 394a238c70SJohn Marino sign as 1/x, thus |csc(x)| >= |1/x|. Then: 404a238c70SJohn Marino (i) either x is a power of two, then 1/x is exactly representable, and 414a238c70SJohn Marino as long as 1/2*ulp(1/x) > 0.2, we can conclude; 424a238c70SJohn Marino (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then 434a238c70SJohn Marino |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. 444a238c70SJohn Marino Since |csc(x) - 1/x| <= 0.2, if 2^(-2n) ufp(y) >= 0.4, then 454a238c70SJohn Marino |y - csc(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct result. 464a238c70SJohn Marino If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). 474a238c70SJohn Marino A sufficient condition is thus EXP(x) <= -2 MAX(PREC(x),PREC(Y)). */ 484a238c70SJohn Marino #define ACTION_TINY(y,x,r) \ 494a238c70SJohn Marino if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ 504a238c70SJohn Marino { \ 514a238c70SJohn Marino int signx = MPFR_SIGN(x); \ 524a238c70SJohn Marino inexact = mpfr_ui_div (y, 1, x, r); \ 534a238c70SJohn Marino if (inexact == 0) /* x is a power of two */ \ 544a238c70SJohn Marino { /* result always 1/x, except when rounding away from zero */ \ 554a238c70SJohn Marino if (rnd_mode == MPFR_RNDA) \ 564a238c70SJohn Marino rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \ 574a238c70SJohn Marino if (rnd_mode == MPFR_RNDU) \ 584a238c70SJohn Marino { \ 594a238c70SJohn Marino if (signx > 0) \ 604a238c70SJohn Marino mpfr_nextabove (y); /* 2^k + epsilon */ \ 614a238c70SJohn Marino inexact = 1; \ 624a238c70SJohn Marino } \ 634a238c70SJohn Marino else if (rnd_mode == MPFR_RNDD) \ 644a238c70SJohn Marino { \ 654a238c70SJohn Marino if (signx < 0) \ 664a238c70SJohn Marino mpfr_nextbelow (y); /* -2^k - epsilon */ \ 674a238c70SJohn Marino inexact = -1; \ 684a238c70SJohn Marino } \ 694a238c70SJohn Marino else /* round to zero, or nearest */ \ 704a238c70SJohn Marino inexact = -signx; \ 714a238c70SJohn Marino } \ 724a238c70SJohn Marino MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ 734a238c70SJohn Marino goto end; \ 744a238c70SJohn Marino } 754a238c70SJohn Marino 764a238c70SJohn Marino #include "gen_inverse.h" 77