Copyright 2005, 2006, 2007 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.
This file is part of the MPFR Library.
The 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 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 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. */
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)
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. */ \
MPFR_ASSERTN (! mpfr_underflow_p ()); \
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \
goto end; \
}
#include "gen_inverse.h"
