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/* mpfr_tanu -- tanu(x) = tan(2*pi*x/u)
Copyright 2020 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"
/* FIXME[VL]: Implement the range reduction in this function.
That's the whole point of tanu compared to tan. */
/* put in y the corrected-rounded value of tan(2*pi*x/u) */
int
mpfr_tanu (mpfr_ptr y, mpfr_srcptr x, unsigned long u, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t precy, prec;
mpfr_exp_t expx, expt, err;
mpfr_t t;
int inexact = 0, nloops = 0, underflow = 0;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC (
("x[%Pu]=%*.Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%*.Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
inexact));
if (u == 0 || MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
/* for u=0, return NaN */
if (u == 0 || MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
MPFR_SAVE_EXPO_MARK (expo);
precy = MPFR_PREC (y);
expx = MPFR_GET_EXP (x);
/* for x large, since argument reduction is expensive, we want to avoid
any failure in Ziv's strategy, thus we take into account expx too */
prec = precy + MPFR_INT_CEIL_LOG2 (MAX(precy,expx)) + 8;
MPFR_ASSERTD(prec >= 2);
mpfr_init2 (t, prec);
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
int inex;
nloops ++;
/* We first compute an approximation t of 2*pi*x/u, then call tan(t).
If t = 2*pi*x/u + s, then
|tan(t) - tan(2*pi*x/u)| = |s| * (1 + tan(v)^2) where v is in the
interval [t, t+s]. If we ensure that |t| >= |2*pi*x/u|, since tan() is
increasing, we can bound tan(v)^2 by tan(t)^2. */
mpfr_set_prec (t, prec);
mpfr_const_pi (t, MPFR_RNDU); /* t = pi * (1 + theta1) where
|theta1| <= 2^(1-prec) */
mpfr_mul_2ui (t, t, 1, MPFR_RNDN); /* t = 2*pi * (1 + theta1) */
mpfr_mul (t, t, x, MPFR_RNDA); /* t = 2*pi*x * (1 + theta2)^2 where
|theta2| <= 2^(1-prec) */
inex = mpfr_div_ui (t, t, u, MPFR_RNDN);
/* t = 2*pi*x/u * (1 + theta3)^3 where |theta3| <= 2^(1-prec) */
/* if t is zero here, it means the division by u underflows, then
tan(t) also underflows, since |tan(x)| <= |x|. */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
{
inexact = mpfr_underflow (y, rnd_mode, MPFR_SIGN(t));
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_INEXACT
| MPFR_FLAGS_UNDERFLOW);
underflow = 1;
goto end;
}
/* emulate mpfr_div_ui (t, t, u, MPFR_RNDA) above, so that t is rounded
away from zero */
if (MPFR_SIGN(t) > 0 && inex < 0)
mpfr_nextabove (t);
else if (MPFR_SIGN(t) < 0 && inex > 0)
mpfr_nextbelow (t);
expt = MPFR_GET_EXP (t);
/* since prec >= 3, |(1 + theta3)^3 - 1| <= 4*theta3 <= 2^(3-prec)
thus |s| = |t - 2*pi*x/u| <= |t| * 2^(3-prec) */
mpfr_tan (t, t, MPFR_RNDA);
{
/* compute an upper bound for 1+tan(t)^2 */
mpfr_t z;
mpfr_init2 (z, 64);
mpfr_sqr (z, t, MPFR_RNDU);
mpfr_add_ui (z, z, 1, MPFR_RNDU);
expt += MPFR_GET_EXP (z);
/* now |t - tan(2*pi*x/u)| <= ulp(t) + 2^(expt + 3 - prec) */
mpfr_clear (z);
}
/* t cannot be zero here, since we excluded t=0 before, which is the
only exact case where tan(t)=0, and we round away from zero */
err = expt + 3 - prec;
expt = MPFR_GET_EXP (t); /* new exponent of t */
/* the total error is bounded by 2^err + ulp(t) = 2^err + 2^(expt-prec)
thus if err <= expt-prec, it is bounded by 2^(expt-prec+1),
otherwise it is bounded by 2^(err+1). */
err = (err <= expt - prec) ? expt - prec + 1 : err + 1;
/* normalize err for mpfr_can_round */
err = expt - err;
if (MPFR_CAN_ROUND (t, err, precy, rnd_mode))
break;
/* Check exact cases only after the first level of Ziv' strategy, to
avoid slowing down the average case. Exact cases are when 2*pi*x/u
is a multiple of pi/4, i.e., x/u a multiple of 1/8:
(a) x/u = {0,1/2} mod 1: return +0 or -0
(b) x/u = {1/4,3/4} mod 1: return +Inf or -Inf
(c) x/u = {1/8,3/8,5/8,7/8} mod 1: return 1 or -1 */
if (nloops == 1)
{
inexact = mpfr_div_ui (t, x, u, MPFR_RNDA);
mpfr_mul_2ui (t, t, 3, MPFR_RNDA);
if (inexact == 0 && mpfr_integer_p (t))
{
mpz_t z;
unsigned long mod8;
mpz_init (z);
inexact = mpfr_get_z (z, t, MPFR_RNDZ);
MPFR_ASSERTN(inexact == 0);
mod8 = mpz_fdiv_ui (z, 8);
mpz_clear (z);
if (mod8 == 0 || mod8 == 4) /* case (a) */
mpfr_set_zero (y, ((mod8 == 0) ? +1 : -1) * MPFR_SIGN (x));
else if (mod8 == 2 || mod8 == 6) /* case (b) */
{
mpfr_set_inf (y, (mod8 == 2) ? +1 : -1);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_DIVBY0);
}
else /* case (c) */
{
if (mod8 == 1 || mod8 == 5)
mpfr_set_ui (y, 1, rnd_mode);
else
mpfr_set_si (y, -1, rnd_mode);
}
goto end;
}
}
MPFR_ZIV_NEXT (loop, prec);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd_mode);
end:
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return underflow ? inexact : mpfr_check_range (y, inexact, rnd_mode);
}
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