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/* mpfr_cos -- cosine of a floating-point number
Copyright 2001, 2002, 2003, 2004 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., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include <stdio.h>
#include "mpfr-impl.h"
static int mpfr_cos2_aux _MPFR_PROTO ((mpfr_ptr, mpfr_srcptr));
int
mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
int K0, K, precy, m, k, l;
int inexact;
mpfr_t r, s;
mp_limb_t *rp, *sp;
mp_size_t sm;
mp_exp_t exps, cancel = 0;
TMP_DECL (marker);
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_emin_emax ();
precy = MPFR_PREC(y);
K0 = __gmpfr_isqrt(precy / 2); /* Need K + log2(precy/K) extra bits */
m = precy + 3 * (K0 + 2 * MAX(MPFR_GET_EXP (x), 0)) + 3;
TMP_MARK(marker);
sm = (m + BITS_PER_MP_LIMB - 1) / BITS_PER_MP_LIMB;
MPFR_TMP_INIT(rp, r, m, sm);
MPFR_TMP_INIT(sp, s, m, sm);
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 */
/* 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_mul_2ui (s, s, 1, GMP_RNDU); /* err <= 4*olderr */
mpfr_sub (s, s, r, GMP_RNDN);
}
/* absolute error on s is bounded by (2l+1/3)*2^(2K-m) */
for (k = 2 * K, l = 2 * l + 1; l > 1; l = (l + 1) >> 1)
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, GMP_RNDN, GMP_RNDZ,
precy + (rnd_mode == GMP_RNDN))))
break;
m += BITS_PER_MP_LIMB;
if (exps < cancel)
{
m += cancel - exps;
cancel = exps;
}
sm = (m + BITS_PER_MP_LIMB - 1) / BITS_PER_MP_LIMB;
MPFR_TMP_INIT(rp, r, m, sm);
MPFR_TMP_INIT(sp, s, m, sm);
}
mpfr_restore_emin_emax ();
inexact = mpfr_set (y, s, rnd_mode); /* FIXME: Dont' need check range? */
TMP_FREE(marker);
return inexact;
}
/* 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;
long int prec, m = MPFR_PREC(s);
mpfr_t t;
MPFR_ASSERTD (MPFR_GET_EXP (r) <= 0);
mpfr_init2 (t, m);
MPFR_SET_ONE (t);
mpfr_set (s, t, GMP_RNDN);
for (l = 1; MPFR_GET_EXP (t) + m >= 0; l++)
{
mpfr_mul (t, t, r, GMP_RNDU); /* err <= (3l-1) ulp */
mpfr_div_ui (t, t, (2*l-1)*(2*l), GMP_RNDU); /* err <= 3l ulp */
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;
}
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