int
sin_x(arb_ptr out, const arb_t inp, void * params, long order, long prec)
{
int xlen = FLINT_MIN(2, order);
arb_set(out, inp);
if (xlen > 1)
arb_one(out + 1);
_arb_poly_sin_series(out, out, xlen, order, prec);
eval_count++;
return 0;
}
/* sin((pi/2)x) */
static int
sin_pi2_x(arb_ptr out, const arb_t inp, void * params, slong order, slong prec)
{
arb_ptr x;
x = _arb_vec_init(2);
arb_set(x, inp);
arb_one(x + 1);
arb_const_pi(out, prec);
arb_mul_2exp_si(out, out, -1);
_arb_vec_scalar_mul(x, x, 2, out, prec);
_arb_poly_sin_series(out, x, order, order, prec);
_arb_vec_clear(x, 2);
return 0;
}
int
sin_1x(arb_ptr out, const arb_t inp, void * params, long order, long prec)
{
arb_ptr x;
int xlen = FLINT_MIN(2, order);
x = _arb_vec_init(xlen);
arb_set(x, inp);
if (xlen > 1)
arb_one(x + 1);
_arb_poly_inv_series(out, x, xlen, order, prec);
_arb_poly_sin_series(out, out, order, order, prec);
_arb_vec_clear(x, xlen);
eval_count++;
return 0;
}
void
_arb_poly_rgamma_series(arb_ptr res, arb_srcptr h, long hlen, long len, long prec)
{
int reflect;
long i, rflen, r, n, wp;
arb_ptr t, u, v;
arb_struct f[2];
hlen = FLINT_MIN(hlen, len);
wp = prec + FLINT_BIT_COUNT(prec);
t = _arb_vec_init(len);
u = _arb_vec_init(len);
v = _arb_vec_init(len);
arb_init(f);
arb_init(f + 1);
/* use zeta values at small integers */
if (arb_is_int(h) && (arf_cmpabs_ui(arb_midref(h), prec / 2) < 0))
{
r = arf_get_si(arb_midref(h), ARF_RND_DOWN);
_arb_poly_lgamma_series_at_one(u, len, wp);
_arb_vec_neg(u, u, len);
_arb_poly_exp_series(t, u, len, len, wp);
if (r == 1)
{
_arb_vec_swap(v, t, len);
}
else if (r <= 0)
{
arb_set(f, h);
arb_one(f + 1);
rflen = FLINT_MIN(len, 2 - r);
_arb_poly_rising_ui_series(u, f, FLINT_MIN(2, len), 1 - r, rflen, wp);
_arb_poly_mullow(v, t, len, u, rflen, len, wp);
}
else
{
arb_one(f);
arb_one(f + 1);
rflen = FLINT_MIN(len, r);
_arb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r - 1, rflen, wp);
/* TODO: use div_series? */
_arb_poly_inv_series(u, v, rflen, len, wp);
_arb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* otherwise use Stirling series */
arb_gamma_stirling_choose_param(&reflect, &r, &n, h, 1, 0, wp);
/* rgamma(h) = (gamma(1-h+r) sin(pi h)) / (rf(1-h, r) * pi), h = h0 + t*/
if (reflect)
{
/* u = gamma(r+1-h) */
arb_sub_ui(f, h, r + 1, wp);
arb_neg(f, f);
_arb_poly_gamma_stirling_eval(t, f, n, len, wp);
_arb_poly_exp_series(u, t, len, len, wp);
for (i = 1; i < len; i += 2)
arb_neg(u + i, u + i);
/* v = sin(pi x) */
arb_const_pi(f + 1, wp);
arb_mul(f, h, f + 1, wp);
_arb_poly_sin_series(v, f, 2, len, wp);
_arb_poly_mullow(t, u, len, v, len, len, wp);
/* rf(1-h,r) * pi */
if (r == 0)
{
arb_const_pi(u, wp);
_arb_vec_scalar_div(v, t, len, u, wp);
}
else
{
arb_sub_ui(f, h, 1, wp);
arb_neg(f, f);
arb_set_si(f + 1, -1);
rflen = FLINT_MIN(len, r + 1);
_arb_poly_rising_ui_series(v, f, FLINT_MIN(2, len), r, rflen, wp);
arb_const_pi(u, wp);
_arb_vec_scalar_mul(v, v, rflen, u, wp);
/* divide by rising factorial */
/* TODO: might better to use div_series, when it has a good basecase */
_arb_poly_inv_series(u, v, rflen, len, wp);
_arb_poly_mullow(v, t, len, u, len, len, wp);
}
}
else
{
/* rgamma(h) = rgamma(h+r) rf(h,r) */
if (r == 0)
{
arb_add_ui(f, h, r, wp);
_arb_poly_gamma_stirling_eval(t, f, n, len, wp);
_arb_vec_neg(t, t, len);
_arb_poly_exp_series(v, t, len, len, wp);
}
else
{
arb_set(f, h);
arb_one(f + 1);
rflen = FLINT_MIN(len, r + 1);
_arb_poly_rising_ui_series(t, f, FLINT_MIN(2, len), r, rflen, wp);
arb_add_ui(f, h, r, wp);
_arb_poly_gamma_stirling_eval(v, f, n, len, wp);
_arb_vec_neg(v, v, len);
_arb_poly_exp_series(u, v, len, len, wp);
_arb_poly_mullow(v, u, len, t, rflen, len, wp);
}
}
}
/* compose with nonconstant part */
arb_zero(t);
_arb_vec_set(t + 1, h + 1, hlen - 1);
_arb_poly_compose_series(res, v, len, t, hlen, len, prec);
arb_clear(f);
arb_clear(f + 1);
_arb_vec_clear(t, len);
_arb_vec_clear(u, len);
_arb_vec_clear(v, len);
}
