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/* mpfr_coth - Hyperbolic 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 hyperbolic cotangent is defined by coth(x) = 1/tanh(x)
coth (NaN) = NaN.
coth (+Inf) = 1
coth (-Inf) = -1
coth (+0) = +0.
coth (-0) = -0.
*/
#define FUNCTION mpfr_coth
#define INVERSE mpfr_tanh
#define ACTION_NAN(y) do { MPFR_SET_NAN(y); MPFR_RET_NAN; } while (1)
#define ACTION_INF(y) return mpfr_set_si (y, MPFR_IS_POS(x) ? 1 : -1, rnd_mode)
#define ACTION_ZERO(y,x) do { MPFR_SET_SAME_SIGN(y,x); MPFR_SET_ZERO(y); \
MPFR_RET(0); } while (1)
/* We know |coth(x)| > 1, thus if the approximation z is such that
1 <= z <= 1 + 2^(-p) where p is the target precision, then the
result is either 1 or nextabove(1) = 1 + 2^(1-p). */
#define ACTION_SPECIAL \
if (MPFR_GET_EXP(z) == 1) /* 1 <= |z| < 2 */ \
{ \
/* the following is exact by Sterbenz theorem */ \
mpfr_sub_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, GMP_RNDN); \
if (MPFR_IS_ZERO(z) || MPFR_GET_EXP(z) <= - (mp_exp_t) precy) \
{ \
mpfr_add_si (z, z, MPFR_SIGN(z) > 0 ? 1 : -1, GMP_RNDN); \
break; \
} \
}
/* The analysis is adapted from that for mpfr_csc:
near x=0, coth(x) = 1/x + x/3 + ..., more precisely we have
|coth(x) - 1/x| <= 0.32 for |x| <= 1. Like for csc, the error term has
the same sign as 1/x, thus |coth(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) > 0.32, 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 |coth(x) - 1/x| <= 0.32, if 2^(-2n) ufp(y) >= 0.64, then
|y - coth(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)). */
#define ACTION_TINY(y,x,r) \
if (MPFR_EXP(x) + 1 <= -2 * (mp_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 == GMP_RNDU) \
{ \
if (signx > 0) \
mpfr_nextabove (y); /* 2^k + epsilon */ \
inexact = 1; \
} \
else if (rnd_mode == GMP_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"