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/* mpfr_atanu -- atanu(x) = atan(x)*u/(2*pi)
mpfr_atanpi -- atanpi(x) = atan(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 atan(x)*u/(2*pi) */
int
mpfr_atanu (mpfr_ptr y, mpfr_srcptr x, unsigned long u, mpfr_rnd_t rnd_mode)
{
mpfr_t tmp, pi;
mpfr_prec_t prec;
mpfr_exp_t expx;
int inex;
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,
inex));
/* Singular cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x))
{
/* atanu(+Inf,u) = u/4, atanu(-Inf,u) = -u/4 */
if (MPFR_IS_POS (x))
return mpfr_set_ui_2exp (y, u, -2, rnd_mode);
else
{
inex = mpfr_set_ui_2exp (y, u, -2, MPFR_INVERT_RND (rnd_mode));
MPFR_CHANGE_SIGN (y);
return -inex;
}
}
else /* necessarily x=0 */
{
MPFR_ASSERTD(MPFR_IS_ZERO(x));
/* atan(0)=0 with same sign, even when u=0 to ensure
atanu(-x,u) = -atanu(x,u) */
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0); /* exact result */
}
}
if (u == 0) /* return 0 with sign of x, which is coherent with case x=0 */
{
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
if (mpfr_cmpabs_ui (x, 1) == 0)
{
/* |x| = 1: atanu(1,u) = u/8, atanu(-1,u)=-u/8 */
/* 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, -3, rnd_mode);
else
{
inex = mpfr_set_ui_2exp (y, u, -3, MPFR_INVERT_RND(rnd_mode));
MPFR_CHANGE_SIGN(y);
return -inex;
}
}
/* For x>=1, we have pi/2-1/x < atan(x) < pi/2, thus
u/4-u/(2*pi*x) < atanu(x,u) < u/4, and the relative difference between
atanu(x,u) and u/4 is less than 2/(pi*x) < 1/x <= 2^(1-EXP(x)).
If the relative difference is <= 2^(-prec-2), then the difference
between atanu(x,u) and u/4 is <= 1/4*ulp(u/4) <= 1/2*ulp(RN(u/4)).
We also require x >= 2^64, which implies x > 2*u/pi, so that
(u-1)/4 < u/4-u/(2*pi*x) < u/4. */
expx = MPFR_GET_EXP(x);
if (expx >= 65 && expx - 1 >= MPFR_PREC(y) + 2)
{
prec = (MPFR_PREC(y) <= 63) ? 65 : MPFR_PREC(y) + 2;
/* now prec > 64 and prec > MPFR_PREC(y)+1 */
mpfr_init2 (tmp, prec);
/* since expx >= 65, we have emax >= 65, thus u is representable here,
and we don't need to work in an extended exponent range */
inex = mpfr_set_ui (tmp, u, MPFR_RNDN); /* exact since prec >= 64 */
MPFR_ASSERTD(inex == 0);
mpfr_nextbelow (tmp);
/* Since prec >= 65, the last significant bit of tmp is 1, and since
prec > PREC(y), tmp is not representable in the target precision,
which ensures we will get a correct ternary value below. */
MPFR_ASSERTD(mpfr_min_prec(tmp) > MPFR_PREC(y));
if (MPFR_SIGN(x) < 0)
MPFR_CHANGE_SIGN(tmp);
/* since prec >= PREC(y)+2, the rounding of tmp is correct */
inex = mpfr_div_2ui (y, tmp, 2, rnd_mode);
mpfr_clear (tmp);
return inex;
}
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_atan (tmp, x, MPFR_RNDA);
/* tmp = atan(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 = atan(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 = atan(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
|atanu(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 = atan(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);
inex = mpfr_set (y, tmp, rnd_mode);
mpfr_clear (tmp);
mpfr_clear (pi);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd_mode);
}
int
mpfr_atanpi (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
return mpfr_atanu (y, x, 2, rnd_mode);
}
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