int
mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
int K0, K, precy, m, k, l, inexact;
mpfr_t r, s;
if (MPFR_IS_NAN(x) || MPFR_IS_INF(x))
{
MPFR_SET_NAN(y);
MPFR_RET_NAN;
}
if (MPFR_IS_ZERO(x))
{
mpfr_set_ui (y, 1, GMP_RNDN);
return 0;
}
precy = MPFR_PREC(y);
K0 = _mpfr_isqrt(precy / 2);
/* we need at least K + log2(precy/K) extra bits */
m = precy + 3 * K0 + 3;
mpfr_init2 (r, m);
mpfr_init2 (s, m);
do
{
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_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);
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_ui (s, s, 1, 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; k++, l = (l + 1) >> 1);
/* now the error is bounded by 2^(k-m) = 2^(EXP(s)-err) */
l = mpfr_can_round (s, MPFR_EXP(s) + m - k, GMP_RNDN, rnd_mode, precy);
if (l == 0)
{
m += BITS_PER_MP_LIMB;
mpfr_set_prec (r, m);
mpfr_set_prec (s, m);
}
}
while (l == 0);
inexact = mpfr_set (y, s, rnd_mode);
mpfr_clear (r);
mpfr_clear (s);
return inexact;
}
int
mpfr_cos (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_prec_t K0, K, precy, m, k, l;
int inexact, reduce = 0;
mpfr_t r, s, xr, c;
mpfr_exp_t exps, cancel = 0, expx;
MPFR_ZIV_DECL (loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_GROUP_DECL (group);
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 (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, rnd_mode);
}
}
MPFR_SAVE_EXPO_MARK (expo);
/* cos(x) = 1-x^2/2 + ..., so error < 2^(2*EXP(x)-1) */
expx = MPFR_GET_EXP (x);
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, -2 * expx,
1, 0, rnd_mode, expo, {});
/* Compute initial precision */
precy = MPFR_PREC (y);
if (precy >= MPFR_SINCOS_THRESHOLD)
{
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_cos_fast (y, x, rnd_mode);
}
K0 = __gmpfr_isqrt (precy / 3);
m = precy + 2 * MPFR_INT_CEIL_LOG2 (precy) + 2 * K0;
if (expx >= 3)
{
reduce = 1;
/* As expx + m - 1 will silently be converted into mpfr_prec_t
in the mpfr_init2 call, the assert below may be useful to
avoid undefined behavior. */
MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
mpfr_init2 (c, expx + m - 1);
mpfr_init2 (xr, m);
}
MPFR_GROUP_INIT_2 (group, m, r, s);
MPFR_ZIV_INIT (loop, m);
for (;;)
{
/* If |x| >= 4, first reduce x cmod (2*Pi) into xr, using mpfr_remainder:
let e = EXP(x) >= 3, and m the target precision:
(1) c <- 2*Pi [precision e+m-1, nearest]
(2) xr <- remainder (x, c) [precision m, nearest]
We have |c - 2*Pi| <= 1/2ulp(c) = 2^(3-e-m)
|xr - x - k c| <= 1/2ulp(xr) <= 2^(1-m)
|k| <= |x|/(2*Pi) <= 2^(e-2)
Thus |xr - x - 2kPi| <= |k| |c - 2Pi| + 2^(1-m) <= 2^(2-m).
It follows |cos(xr) - cos(x)| <= 2^(2-m). */
if (reduce)
{
mpfr_const_pi (c, MPFR_RNDN);
mpfr_mul_2ui (c, c, 1, MPFR_RNDN); /* 2Pi */
mpfr_remainder (xr, x, c, MPFR_RNDN);
if (MPFR_IS_ZERO(xr))
goto ziv_next;
/* now |xr| <= 4, thus r <= 16 below */
mpfr_mul (r, xr, xr, MPFR_RNDU); /* err <= 1 ulp */
}
else
mpfr_mul (r, x, x, MPFR_RNDU); /* err <= 1 ulp */
/* now |x| < 4 (or xr if reduce = 1), thus |r| <= 16 */
/* we need |r| < 1/2 for mpfr_cos2_aux, i.e., EXP(r) - 2K <= -1 */
K = K0 + 1 + MAX(0, MPFR_GET_EXP(r)) / 2;
/* since K0 >= 0, if EXP(r) < 0, then K >= 1, thus EXP(r) - 2K <= -3;
otherwise if EXP(r) >= 0, then K >= 1/2 + EXP(r)/2, thus
EXP(r) - 2K <= -1 */
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);
/* l is the error bound in ulps on s */
MPFR_SET_ONE (r);
for (k = 0; k < K; k++)
{
mpfr_sqr (s, s, MPFR_RNDU); /* err <= 2*olderr */
MPFR_SET_EXP (s, MPFR_GET_EXP (s) + 1); /* Can't overflow */
mpfr_sub (s, s, r, MPFR_RNDN); /* err <= 4*olderr */
if (MPFR_IS_ZERO(s))
goto ziv_next;
MPFR_ASSERTD (MPFR_GET_EXP (s) <= 1);
}
/* The absolute error on s is bounded by (2l+1/3)*2^(2K-m)
2l+1/3 <= 2l+1.
If |x| >= 4, we need to add 2^(2-m) for the argument reduction
by 2Pi: if K = 0, this amounts to add 4 to 2l+1/3, i.e., to add
2 to l; if K >= 1, this amounts to add 1 to 2*l+1/3. */
l = 2 * l + 1;
if (reduce)
l += (K == 0) ? 4 : 1;
k = MPFR_INT_CEIL_LOG2 (l) + 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 the error is less
than 2^(-precy) for directed rounding, or 2^(-precy-1) for rounding
to nearest. */
{
if (m > k && (m - k >= precy + (rnd_mode == MPFR_RNDN)))
{
/* If round to nearest or away, result is s = 1 or -1,
otherwise it is round(nexttoward (s, 0)). However in order to
have the inexact flag correctly set below, we set |s| to
1 - 2^(-m) in all cases. */
mpfr_nexttozero (s);
break;
}
}
if (exps < cancel)
{
m += cancel - exps;
cancel = exps;
}
ziv_next:
MPFR_ZIV_NEXT (loop, m);
MPFR_GROUP_REPREC_2 (group, m, r, s);
if (reduce)
{
mpfr_set_prec (xr, m);
mpfr_set_prec (c, expx + m - 1);
}
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, s, rnd_mode);
MPFR_GROUP_CLEAR (group);
if (reduce)
{
mpfr_clear (xr);
mpfr_clear (c);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
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;
}
