1/* mpfr_cot - cotangent 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 cotangent is defined by cot(x) = 1/tan(x) = cos(x)/sin(x). 24 cot (NaN) = NaN. 25 cot (+Inf) = csc (-Inf) = NaN. 26 cot (+0) = +Inf. 27 cot (-0) = -Inf. 28*/ 29 30#define FUNCTION mpfr_cot 31#define INVERSE mpfr_tan 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 37/* (This analysis is adapted from that for mpfr_coth.) 38 Near x=0, cot(x) = 1/x - x/3 + ..., more precisely we have 39 |cot(x) - 1/x| <= 0.36 for |x| <= 1. The error term has 40 the opposite sign as 1/x, thus |cot(x)| <= |1/x|. Then: 41 (i) either x is a power of two, then 1/x is exactly representable, and 42 as long as 1/2*ulp(1/x) > 0.36, we can conclude; 43 (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then 44 |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. 45 Since |cot(x) - 1/x| <= 0.36, if 2^(-2n) ufp(y) >= 0.72, then 46 |y - cot(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct 47 result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). 48 A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)). 49 The division can be inexact in case of underflow or overflow; but 50 an underflow is not possible as emin = - emax. The overflow is a 51 real overflow possibly except when |x| = 2^emin. */ 52#define ACTION_TINY(y,x,r) \ 53 if (MPFR_EXP(x) + 1 <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ 54 { \ 55 int two2emin; \ 56 int signx = MPFR_SIGN(x); \ 57 MPFR_ASSERTN (MPFR_EMIN_MIN + MPFR_EMAX_MAX == 0); \ 58 if ((two2emin = mpfr_get_exp (x) == __gmpfr_emin + 1 && \ 59 mpfr_powerof2_raw (x))) \ 60 { \ 61 /* Case |x| = 2^emin. 1/x is not representable; so, compute \ 62 1/(2x) instead (exact), and correct the result later. */ \ 63 mpfr_set_si_2exp (y, signx, __gmpfr_emax, MPFR_RNDN); \ 64 inexact = 0; \ 65 } \ 66 else \ 67 inexact = mpfr_ui_div (y, 1, x, r); \ 68 if (inexact == 0) /* x is a power of two */ \ 69 { /* result always 1/x, except when rounding to zero */ \ 70 if (rnd_mode == MPFR_RNDA) \ 71 rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \ 72 if (rnd_mode == MPFR_RNDU || (rnd_mode == MPFR_RNDZ && signx < 0)) \ 73 { \ 74 if (signx < 0) \ 75 mpfr_nextabove (y); /* -2^k + epsilon */ \ 76 inexact = 1; \ 77 } \ 78 else if (rnd_mode == MPFR_RNDD || rnd_mode == MPFR_RNDZ) \ 79 { \ 80 if (signx > 0) \ 81 mpfr_nextbelow (y); /* 2^k - epsilon */ \ 82 inexact = -1; \ 83 } \ 84 else /* round to nearest */ \ 85 inexact = signx; \ 86 if (two2emin) \ 87 mpfr_mul_2ui (y, y, 1, r); /* overflow in MPFR_RNDN */ \ 88 } \ 89 /* Underflow is not possible with emin = - emax, but we cannot */ \ 90 /* add an assert as the underflow flag could have already been */ \ 91 /* set before the call to mpfr_cot. */ \ 92 MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ 93 goto end; \ 94 } 95 96#include "gen_inverse.h" 97