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/* mpfr_csc - cosecant function.
Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012 Free Software Foundation, Inc.
Contributed by the AriC and Caramel 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 3 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.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
/* the cosecant is defined by csc(x) = 1/sin(x).
csc (NaN) = NaN.
csc (+Inf) = csc (-Inf) = NaN.
csc (+0) = +Inf.
csc (-0) = -Inf.
*/
#define FUNCTION mpfr_csc
#define INVERSE mpfr_sin
#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_set_divby0 (); MPFR_RET(0); } while (1)
/* near x=0, we have csc(x) = 1/x + x/6 + ..., more precisely we have
|csc(x) - 1/x| <= 0.2 for |x| <= 1. The analysis is similar to that for
gamma(x) near x=0 (see gamma.c), except here the error term has the same
sign as 1/x, thus |csc(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) > 0.2, 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 |csc(x) - 1/x| <= 0.2, if 2^(-2n) ufp(y) >= 0.4, then
|y - csc(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) <= -2 MAX(PREC(x),PREC(Y)). */
#define ACTION_TINY(y,x,r) \
if (MPFR_EXP(x) <= -2 * (mpfr_exp_t) MAX(MPFR_PREC(x), MPFR_PREC(y))) \
{ \
int signx = MPFR_SIGN(x); \
inexact = mpfr_ui_div (y, 1, x, r); \
if (inexact == 0) /* x is a power of two */ \
{ /* result always 1/x, except when rounding away from zero */ \
if (rnd_mode == MPFR_RNDA) \
rnd_mode = (signx > 0) ? MPFR_RNDU : MPFR_RNDD; \
if (rnd_mode == MPFR_RNDU) \
{ \
if (signx > 0) \
mpfr_nextabove (y); /* 2^k + epsilon */ \
inexact = 1; \
} \
else if (rnd_mode == MPFR_RNDD) \
{ \
if (signx < 0) \
mpfr_nextbelow (y); /* -2^k - epsilon */ \
inexact = -1; \
} \
else /* round to zero, or nearest */ \
inexact = -signx; \
} \
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags); \
goto end; \
}
#include "gen_inverse.h"
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