| /* mpfr_cot - cotangent function. |
| |
| Copyright 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao projects, INRIA. |
| |
| This file is part of the GNU MPFR Library. |
| |
| The GNU MPFR Library is free software; you can redistribute it and/or modify |
| it under the terms of the GNU Lesser General Public License as published by |
| the Free Software Foundation; either version 2.1 of the License, or (at your |
| option) any later version. |
| |
| The GNU MPFR Library is distributed in the hope that it will be useful, but |
| WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY |
| or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public |
| License for more details. |
| |
| You should have received a copy of the GNU Lesser General Public License |
| along with the GNU MPFR Library; see the file COPYING.LIB. If not, write to |
| the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, |
| MA 02110-1301, USA. */ |
| |
| /* the cotangent is defined by cot(x) = 1/tan(x) = cos(x)/sin(x). |
| cot (NaN) = NaN. |
| cot (+Inf) = csc (-Inf) = NaN. |
| cot (+0) = +Inf. |
| cot (-0) = -Inf. |
| */ |
| |
| #define FUNCTION mpfr_cot |
| #define INVERSE mpfr_tan |
| #define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) |
| #define ACTION_INF(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1) |
| #define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \ |
| MPFR_RET(0); } while (1) |
| |
| /* (This analysis is adapted from that for mpfr_coth.) |
| Near x=0, cot(x) = 1/x - x/3 + ..., more precisely we have |
| |cot(x) - 1/x| <= 0.36 for |x| <= 1. The error term has |
| the opposite sign as 1/x, thus |cot(x)| <= |1/x|. Then: |
| (i) either x is a power of two, then 1/x is exactly representable, and |
| as long as 1/2*ulp(1/x) > 0.36, we can conclude; |
| (ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then |
| |y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place. |
| Since |cot(x) - 1/x| <= 0.36, if 2^(-2n) ufp(y) >= 0.72, then |
| |y - cot(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct |
| result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1). |
| A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)). |
| The division can be inexact in case of underflow or overflow; but |
| an underflow is not possible as emin = - emax. The overflow is a |
| real overflow possibly except when |x| = 2^emin. */ |
| #define ACTION_TINY(y,x,r) \ |
| if (MPFR_EXP(x) + 1 <= -2 * (mp_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \ |
| { \ |
| int two2emin; \ |
| int signx = MPFR_SIGN(x); \ |
| MPFR_ASSERTN (MPFR_EMIN_MIN + MPFR_EMAX_MAX == 0); \ |
| if ((two2emin = mpfr_get_exp (x) == __gmpfr_emin + 1 && \ |
| mpfr_powerof2_raw (x))) \ |
| { \ |
| /* Case |x| = 2^emin. 1/x is not representable; so, compute \ |
| 1/(2x) instead (exact), and correct the result later. */ \ |
| mpfr_set_si_2exp (y, signx, __gmpfr_emax, GMP_RNDN); \ |
| inexact = 0; \ |
| } \ |
| else \ |
| inexact = mpfr_ui_div (y, 1, x, r); \ |
| if (inexact == 0) /* x is a power of two */ \ |
| { /* result always 1/x, except when rounding to zero */ \ |
| if (rnd_mode == GMP_RNDU || (rnd_mode == GMP_RNDZ && signx < 0)) \ |
| { \ |
| if (signx < 0) \ |
| mpfr_nextabove (y); /* -2^k + epsilon */ \ |
| inexact = 1; \ |
| } \ |
| else if (rnd_mode == GMP_RNDD || rnd_mode == GMP_RNDZ) \ |
| { \ |
| if (signx > 0) \ |
| mpfr_nextbelow (y); /* 2^k - epsilon */ \ |
| inexact = -1; \ |
| } \ |
| else /* round to nearest */ \ |
| inexact = signx; \ |
| if (two2emin) \ |
| mpfr_mul_2ui (y, y, 1, r); /* overflow in GMP_RNDN */ \ |
| } \ |
| /* Underflow is not possible with emin = - emax, but we cannot */ \ |
| /* add an assert as the underflow flag could have already been */ \ |
| /* set before the call to mpfr_cot. */ \ |
| MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \ |
| goto end; \ |
| } |
| |
| #include "gen_inverse.h" |