efee5258bc
The MPFR library is a C library for multiple-precision floating-point computations with exact rounding (also called correct rounding). It is based on the GMP multiple-precision library and should replace the MPF class in further releases of GMP. GCC >= 4.2 requires MPFR.
97 lines
5.4 KiB
C
97 lines
5.4 KiB
C
/* mpfr_cot - cotangent function.
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Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
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Contributed by the Arenaire and Cacao projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
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http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
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51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
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/* the cotangent is defined by cot(x) = 1/tan(x) = cos(x)/sin(x).
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cot (NaN) = NaN.
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cot (+Inf) = csc (-Inf) = NaN.
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cot (+0) = +Inf.
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cot (-0) = -Inf.
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*/
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#define FUNCTION mpfr_cot
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#define INVERSE mpfr_tan
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#define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1)
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#define ACTION_INF(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1)
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#define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_INF(y); \
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MPFR_RET(0); } while (1)
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/* (This analysis is adapted from that for mpfr_coth.)
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Near x=0, cot(x) = 1/x - x/3 + ..., more precisely we have
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|cot(x) - 1/x| <= 0.36 for |x| <= 1. The error term has
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the opposite sign as 1/x, thus |cot(x)| <= |1/x|. Then:
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(i) either x is a power of two, then 1/x is exactly representable, and
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as long as 1/2*ulp(1/x) > 0.36, we can conclude;
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(ii) otherwise assume x has <= n bits, and y has <= n+1 bits, then
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|y - 1/x| >= 2^(-2n) ufp(y), where ufp means unit in first place.
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Since |cot(x) - 1/x| <= 0.36, if 2^(-2n) ufp(y) >= 0.72, then
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|y - cot(x)| >= 2^(-2n-1) ufp(y), and rounding 1/x gives the correct
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result. If x < 2^E, then y > 2^(-E), thus ufp(y) > 2^(-E-1).
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A sufficient condition is thus EXP(x) + 1 <= -2 MAX(PREC(x),PREC(Y)).
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The division can be inexact in case of underflow or overflow; but
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an underflow is not possible as emin = - emax. The overflow is a
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real overflow possibly except when |x| = 2^emin. */
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#define ACTION_TINY(y,x,r) \
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if (MPFR_EXP(x) + 1 <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \
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{ \
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int two2emin; \
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int signx = MPFR_SIGN(x); \
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MPFR_ASSERTN (MPFR_EMIN_MIN + MPFR_EMAX_MAX == 0); \
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if ((two2emin = mpfr_get_exp (x) == __gmpfr_emin + 1 && \
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mpfr_powerof2_raw (x))) \
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{ \
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/* Case |x| = 2^emin. 1/x is not representable; so, compute \
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1/(2x) instead (exact), and correct the result later. */ \
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mpfr_set_si_2exp (y, signx, __gmpfr_emax, MPFR_RNDN); \
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inexact = 0; \
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} \
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else \
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inexact = mpfr_ui_div (y, 1, x, r); \
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if (inexact == 0) /* x is a power of two */ \
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{ /* result always 1/x, except when rounding to zero */ \
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if (rnd_mode == MPFR_RNDA) \
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rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \
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if (rnd_mode == MPFR_RNDU || (rnd_mode == MPFR_RNDZ && signx < 0)) \
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{ \
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if (signx < 0) \
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mpfr_nextabove (y); /* -2^k + epsilon */ \
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inexact = 1; \
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} \
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else if (rnd_mode == MPFR_RNDD || rnd_mode == MPFR_RNDZ) \
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{ \
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if (signx > 0) \
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mpfr_nextbelow (y); /* 2^k - epsilon */ \
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inexact = -1; \
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} \
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else /* round to nearest */ \
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inexact = signx; \
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if (two2emin) \
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mpfr_mul_2ui (y, y, 1, r); /* overflow in MPFR_RNDN */ \
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} \
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/* Underflow is not possible with emin = - emax, but we cannot */ \
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/* add an assert as the underflow flag could have already been */ \
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/* set before the call to mpfr_cot. */ \
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MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \
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goto end; \
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}
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#include "gen_inverse.h"
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