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/* mpfr_asinu -- asinu(x) = asin(x)*u/(2*pi)
mpfr_asinpi -- asinpi(x) = asin(x)/pi
Copyright 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. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* put in y the correctly rounded value of asin(x)*u/(2*pi) */
int
mpfr_asinu (mpfr_ptr y, mpfr_srcptr x, unsigned long u, mpfr_rnd_t rnd_mode)
{
mpfr_t tmp, pi;
mpfr_prec_t prec;
int compared, inexact;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg u=%lu rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, u,
rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
inexact));
/* Singular cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* necessarily x=0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
/* asin(0)=0 with same sign, even when u=0 to ensure
asinu(-x,u) = -asinu(x,u) */
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0); /* exact result */
}
}
compared = mpfr_cmpabs_ui (x, 1);
if (compared > 0)
{
/* asinu(x) = NaN for |x| > 1, included for u=0, since NaN*0 = NaN */
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
if (u == 0) /* return 0 with sign of x */
{
MPFR_SET_ZERO (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
if (compared == 0)
{
/* |x| = 1: asinu(1,u) = u/4, asinu(-1,u)=-u/4 */
/* we can't use mpfr_set_si_2exp with -u since -u might not be
representable as long */
if (MPFR_SIGN(x) > 0)
return mpfr_set_ui_2exp (y, u, -2, rnd_mode);
else
{
inexact = mpfr_set_ui_2exp (y, u, -2, MPFR_INVERT_RND(rnd_mode));
MPFR_CHANGE_SIGN(y);
return -inexact;
}
}
/* asin(+/-1/2) = +/-pi/6, thus asin(+/-1/2,u) = +/-u/12 is exact when u is
a multiple of 3 */
if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), -1) == 0 && (u % 3) == 0)
{
long v = u / 3;
if (MPFR_IS_NEG (x))
v = -v;
return mpfr_set_si_2exp (y, v, -2, rnd_mode);
}
prec = MPFR_PREC (y);
MPFR_SAVE_EXPO_MARK (expo);
prec += MPFR_INT_CEIL_LOG2(prec) + 10;
mpfr_init2 (tmp, prec);
mpfr_init2 (pi, prec);
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
/* In the error analysis below, each thetax denotes a variable such that
|thetax| <= 2^(1-prec) */
mpfr_asin (tmp, x, MPFR_RNDA);
/* tmp = asin(x) * (1 + theta1), and tmp cannot be zero since we rounded
away from zero, and the case x=0 was treated before */
/* first multiply by u to avoid underflow issues */
mpfr_mul_ui (tmp, tmp, u, MPFR_RNDA);
/* tmp = asin(x)*u * (1 + theta2)^2, and |tmp| >= 0.5*2^emin */
mpfr_const_pi (pi, MPFR_RNDZ); /* round toward zero since we we will
divide by pi, to round tmp away */
/* pi = Pi * (1 + theta3) */
mpfr_div (tmp, tmp, pi, MPFR_RNDA);
/* tmp = asin(x)*u/Pi * (1 + theta4)^4, with |tmp| > 0 */
/* since we rounded away from 0, if we get 0.5*2^emin here, it means
|asinu(x,u)| < 0.25*2^emin (pi is not exact) thus we have underflow */
if (MPFR_EXP(tmp) == __gmpfr_emin)
{
/* mpfr_underflow rounds away for RNDN */
mpfr_clear (tmp);
mpfr_clear (pi);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (y,
(rnd_mode == MPFR_RNDN) ? MPFR_RNDZ : rnd_mode, 1);
}
mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDA); /* exact */
/* tmp = asin(x)*u/(2*Pi) * (1 + theta4)^4 */
/* since |(1 + theta4)^4 - 1| <= 8*|theta4| for prec >= 3,
the relative error is less than 2^(4-prec) */
MPFR_ASSERTD(!MPFR_IS_ZERO(tmp));
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 4,
MPFR_PREC (y), rnd_mode)))
break;
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (tmp, prec);
mpfr_set_prec (pi, prec);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, tmp, rnd_mode);
mpfr_clear (tmp);
mpfr_clear (pi);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
int
mpfr_asinpi (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
return mpfr_asinu (y, x, 2, rnd_mode);
}
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