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/* mpfr_sin -- sine of a floating-point number
Copyright 2001, 2002, 2003, 2004, 2005, 2006 Free Software Foundation, Inc.
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"
/* determine the sign of sin(x) using argument reduction.
Assumes x is not an exact multiple of Pi (this excludes x=0). */
static int
mpfr_sin_sign (mpfr_srcptr x)
{
mpfr_t c, k;
mp_exp_t K;
int sign;
mp_prec_t m;
mpfr_srcptr y;
MPFR_ZIV_DECL (loop);
K = MPFR_GET_EXP(x);
if (K < 0) /* Trivial case if ABS(x) < 1 */
return MPFR_SIGN (x);
m = K + BITS_PER_MP_LIMB;
mpfr_init2 (c, m);
mpfr_init2 (k, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
/* first determine round(x/Pi): does not have to be exact since
the result is an integer */
mpfr_const_pi (c, GMP_RNDN); /* err <= 1/2*ulp(c) = 2^(1-m) */
/* we need that k is not-to-badly rounded to an integer,
i.e. ulp(k) <= 1, so m >= EXP(k). */
mpfr_div (k, x, c, GMP_RNDN);
mpfr_round (k, k);
sign = 1;
if (!MPFR_IS_ZERO (k)) /* subtract k*approx(Pi) */
{
/* determine parity of k for sign */
if (MPFR_GET_EXP (k) <= 0 || (mpfr_uexp_t) MPFR_GET_EXP (k) <= m)
{
mp_size_t j = BITS_PER_MP_LIMB * MPFR_LIMB_SIZE(k)
- MPFR_GET_EXP(k);
mp_size_t l = j / BITS_PER_MP_LIMB;
/* parity bit is j-th bit starting from least significant bits */
if ((MPFR_MANT(k)[l] >> (j % BITS_PER_MP_LIMB)) & 1)
sign = -1; /* k is odd */
}
K = MPFR_GET_EXP (k); /* k is an integer, thus K >= 1, k < 2^K */
mpfr_mul (k, k, c, GMP_RNDN); /* err <= oldk*err(c) + 1/2*ulp(k)
<= 2^(K+2-m) */
mpfr_sub (k, x, k, GMP_RNDN);
/* assuming |k| <= Pi, err <= 2^(1-m)+2^(K+2-m) < 2^(K+3-m) */
MPFR_ASSERTN (MPFR_IS_ZERO (k) || MPFR_GET_EXP (k) <= 2);
y = k;
}
else
{
K = 1;
y = x;
}
/* sign of sign(y) is uncertain if |y| <= err < 2^(K+3-m),
thus EXP(y) < K+4-m */
if (MPFR_LIKELY (!MPFR_IS_ZERO (y)
&& MPFR_GET_EXP (y) >= K + 4 - (mp_exp_t) m))
break;
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (c, m);
mpfr_set_prec (k, m);
}
if (MPFR_IS_NEG (y))
sign = -sign;
mpfr_clear (k);
mpfr_clear (c);
return sign;
}
int
mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
mpfr_t c;
mp_exp_t e;
mp_prec_t precy, m;
int inexact, sign;
MPFR_ZIV_DECL (loop);
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 /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
/* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -2*MPFR_GET_EXP (x)+2,0,rnd_mode, );
/* Compute initial precision */
precy = MPFR_PREC (y);
m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
e = MPFR_GET_EXP (x);
m += (e < 0) ? -2 * e : e;
sign = mpfr_sin_sign (x);
mpfr_init2 (c, m);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
mpfr_cos (c, x, GMP_RNDZ); /* can't be exact */
mpfr_nexttoinf (c); /* now c = cos(x) rounded away */
mpfr_mul (c, c, c, GMP_RNDU); /* away */
mpfr_ui_sub (c, 1, c, GMP_RNDZ);
mpfr_sqrt (c, c, GMP_RNDZ);
if (MPFR_IS_NEG_SIGN(sign))
MPFR_CHANGE_SIGN(c);
/* Warning: c may be 0! */
if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
{
/* Huge cancellation: increase prec a lot! */
m = MAX (m, MPFR_PREC (x));
m = 2 * m;
}
else
{
/* the absolute error on c is at most 2^(3-m-EXP(c)) */
e = 2 * MPFR_GET_EXP (c) + m - 3;
if (mpfr_can_round (c, e, GMP_RNDN, GMP_RNDZ,
precy + (rnd_mode == GMP_RNDN)))
/* WARNING: even if we know c <= sin(x), don't give GMP_RNDZ
as 3rd argument to mpfr_can_round, since if c is exactly
representable to the target precision (inexact = 0 below),
we would have to add one ulp when rounding away from 0. */
break;
/* check for huge cancellation (Near 0) */
if (e < (mp_exp_t) MPFR_PREC (y))
m += MPFR_PREC (y) - e;
/* Check if near 1 */
if (MPFR_GET_EXP (c) == 1)
m += m;
}
/* Else generic increase */
MPFR_ZIV_NEXT (loop, m);
mpfr_set_prec (c, m);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, c, rnd_mode);
/* inexact cannot be 0, since this would mean that c was representable
within the target precision, but in that case mpfr_can_round will fail */
mpfr_clear (c);
return inexact; /* inexact */
}
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