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/* mpfr_coth - Hyperbolic cotangent function.
Copyright 2005-2021 Free Software Foundation, Inc.
Contributed by the AriC and Caramba 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
https://www.gnu.org/licenses/ or 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) = +Inf.
coth (-0) = -Inf.
*/
#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_INF(y); \
MPFR_SET_DIVBY0 (); 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), MPFR_RNDN); \
if (MPFR_IS_ZERO(z) || MPFR_GET_EXP(z) <= - (mpfr_exp_t) precy) \
{ \
mpfr_add_si (z, z, MPFR_SIGN (z), MPFR_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 * (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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