/* 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_t, m = MPFR_PREC(s);
mpfr_t t;
MPFR_ASSERTN (MPFR_EXP(r) <= 0);
mpfr_init2 (t, m);
mpfr_set_ui (t, 1, GMP_RNDN);
mpfr_set_ui(s, 1, GMP_RNDN);
for (l = 1; MPFR_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_ASSERTN (MPFR_EXP(s) == 0); /* check 1/2 <= s < 1 */
/* err(s) <= l * 2^(-m) */
if (3 * l > (1 << b))
b++;
/* now 3l <= 2^b, we want 3l*ulp(t) <= 2^(-m)
i.e. b+EXP(t)-PREC(t) <= -m */
prec_t = m + MPFR_EXP(t) + b;
if (prec_t >= MPFR_PREC_MIN)
mpfr_round_prec (t, GMP_RNDN, prec_t);
}
mpfr_clear (t);
return l;
}
// remove trailing zeros (only by limbs)
void mpfr_remove_trailing_zeros(mpfr_t x) {
unsigned int xn = MPFR_LIMB_SIZE(x);
mp_limb_t* xp = MPFR_MANT(x);
unsigned int i = 0;
while (i < xn && xp[i] == 0) i++;
if (i > 0 && i < xn) mpfr_round_prec(x, GMP_RNDN, (xn-i)*BITS_PER_MP_LIMB);
}
