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/* mpfr_cos -- cosine of a floating-point number
Copyright 2001, 2002, 2003, 2004, 2005 Free Software Foundation.
This file is part of the MPFR Library.
The 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 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin Place, Fifth Floor, Boston,
MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* s <- 1 - r/2! + r^2/4! + ... + (-1)^l r^l/(2l)! + ...
Assumes |r| < 1.
Returns the index l0 of the last term (-1)^l r^l/(2l)!.
The absolute error on s is at most 2 * l0 * 2^(-m).
*/
static int
mpfr_cos2_aux (mpfr_ptr s, mpfr_srcptr r)
{
unsigned int l, b = 2;
mp_exp_t prec, m = MPFR_PREC (s);
mpfr_t t;
MPFR_ASSERTD (MPFR_GET_EXP (r) <= 0);
mpfr_init2 (t, m);
/* First step for l==1 can be simplified,
futhermore multiply by 1 is not efficient since it is an exact
multiplication (mulhigh failed and we must do a complete mul) */
mpfr_div_2ui (t, r, 1, GMP_RNDN); /* exact */
mpfr_sub (s, __gmpfr_one, t, GMP_RNDD);
MPFR_ASSERTD (MPFR_GET_EXP (s) == 0); /* check 1/2 <= s < 1 */
for (l = 2; MPFR_GET_EXP (t) + m >= 0; l++)
{
mpfr_mul (t, t, r, GMP_RNDU); /* err <= (3l-1) ulp */
mpfr_div_ui (t, t, (unsigned long) (2*l-1)*(2*l), GMP_RNDU);
/* err <= 3l ulp */
MPFR_ASSERTD (MPFR_IS_POS (t));
MPFR_ASSERTD (MPFR_IS_POS (s));
if (l % 2 == 0)
mpfr_add (s, s, t, GMP_RNDD);
else
mpfr_sub (s, s, t, GMP_RNDD);
MPFR_ASSERTD (MPFR_GET_EXP (s) == 0); /* check 1/2 <= s < 1 */
/* err(s) <= l * 2^(-m) */
if (MPFR_UNLIKELY (3 * l > (1U << b)))
b++;
/* now 3l <= 2^b, we want 3l*ulp(t) <= 2^(-m)
i.e. b+EXP(t)-PREC(t) <= -m */
prec = m + MPFR_GET_EXP (t) + b;
if (MPFR_LIKELY (prec >= MPFR_PREC_MIN))
mpfr_prec_round (t, prec, GMP_RNDN);
}
mpfr_clear (t);
return l;
}
int
mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mp_prec_t K0, K, precy, m, k, l;
int inexact;
mpfr_t r, s;
mp_exp_t exps, cancel = 0;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_GROUP_DECL (group);
MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
("y[%#R]=%R inexact=%d", y, y, inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
return mpfr_set_ui (y, 1, GMP_RNDN);
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* cos(x) = 1-x^2/2 + ..., so error < 2^(2*EXP(x)-1) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, __gmpfr_one, 0-2*MPFR_GET_EXP (x)+1,0,
rnd_mode, inexact = _inexact; goto end);
/* Compute initial precision */
precy = MPFR_PREC (y);
/* We can choose everything we want for K0.
This formula has been created by trying many things...
and is far from perfect */
K0 = (MPFR_GET_EXP (x) > 0) ? (MPFR_GET_EXP (x)) : 0 ;
K0 = __gmpfr_isqrt (precy / (2+2*K0+MPFR_INT_CEIL_LOG2 (precy)/4) );
m = precy + 3*K0 + 4;
if (MPFR_GET_EXP (x) >= 0)
m += 5*MPFR_GET_EXP (x);
else
m += -MPFR_GET_EXP (x);
MPFR_GROUP_INIT_2 (group, m, r, s);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_mul (r, x, x, GMP_RNDU); /* err <= 1 ulp */
/* we need that |r| < 1 for mpfr_cos2_aux, i.e. up(x^2)/2^(2K) < 1 */
K = K0 + MAX (MPFR_GET_EXP (r), 0);
/*mpfr_div_2ui (r, r, 2 * K, GMP_RNDN); r = (x/2^K)^2, err <= 1 ulp */
MPFR_SET_EXP (r, MPFR_GET_EXP (r)-2*K); /* Can't overflow! */
/* s <- 1 - r/2! + ... + (-1)^l r^l/(2l)! */
l = mpfr_cos2_aux (s, r);
MPFR_SET_ONE (r);
for (k = 0; k < K; k++)
{
mpfr_mul (s, s, s, GMP_RNDU); /* err <= 2*olderr */
MPFR_SET_EXP (s, MPFR_GET_EXP (s)+1); /* Can't overflow */
mpfr_sub (s, s, r, GMP_RNDN); /* err <= 4*olderr */
MPFR_ASSERTD (MPFR_GET_EXP (s) <= 1);
}
/* absolute error on s is bounded by (2l+1/3)*2^(2K-m)
2l+1/3 <= 2l+1 */
k = MPFR_INT_CEIL_LOG2 (2*l+1) + 2*K;
/* now the error is bounded by 2^(k-m) = 2^(EXP(s)-err) */
exps = MPFR_GET_EXP (s);
if (MPFR_LIKELY (MPFR_CAN_ROUND (s, exps + m - k, precy, rnd_mode)))
break;
if (MPFR_UNLIKELY (exps == 1))
/* s = 1 or -1, and except x=0 which was
already checked above, cos(x) cannot
be 1 or -1, so we can round */
{
if (exps + m - k > precy
/* if round to nearest or away, result is s,
otherwise it is round(nexttoward (s, 0)) */
&& MPFR_IS_LIKE_RNDZ (rnd_mode, MPFR_IS_NEG (s)))
mpfr_nexttozero (s);
break;
}
if (exps < cancel)
{
m += cancel - exps;
cancel = exps;
}
MPFR_ZIV_NEXT (loop, m);
MPFR_GROUP_REPREC_2 (group, m, r, s);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
MPFR_GROUP_CLEAR (group);
end:
MPFR_SAVE_EXPO_FREE (expo);
MPFR_RET (mpfr_check_range (y, inexact, rnd_mode));
}
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