int
f_horror(acb_ptr res, const acb_t z, void * param, slong order, slong prec)
{
acb_t s, t;
if (order > 1)
flint_abort(); /* Would be needed for Taylor method. */
acb_init(s);
acb_init(t);
acb_real_floor(res, z, order != 0, prec);
if (acb_is_finite(res))
{
acb_sub(res, z, res, prec);
acb_set_d(t, 0.5);
acb_sub(res, res, t, prec);
acb_sin_cos(s, t, z, prec);
acb_real_max(s, s, t, order != 0, prec);
acb_mul(res, res, s, prec);
}
acb_clear(s);
acb_clear(t);
return 0;
}
void
_acb_poly_sin_series(acb_ptr g, acb_srcptr h, slong hlen, slong n, slong prec)
{
hlen = FLINT_MIN(hlen, n);
if (hlen == 1)
{
acb_sin(g, h, prec);
_acb_vec_zero(g + 1, n - 1);
}
else if (n == 2)
{
acb_t t;
acb_init(t);
acb_sin_cos(g, t, h, prec);
acb_mul(g + 1, h + 1, t, prec); /* safe since hlen >= 2 */
acb_clear(t);
}
else
{
acb_ptr t = _acb_vec_init(n);
_acb_poly_sin_cos_series(g, t, h, hlen, n, prec);
_acb_vec_clear(t, n);
}
}
void
acb_cot(acb_t r, const acb_t z, slong prec)
{
if (arb_is_zero(acb_imagref(z)))
{
arb_cot(acb_realref(r), acb_realref(z), prec);
arb_zero(acb_imagref(r));
}
else if (arb_is_zero(acb_realref(z)))
{
arb_coth(acb_imagref(r), acb_imagref(z), prec);
arb_neg(acb_imagref(r), acb_imagref(r));
arb_zero(acb_realref(r));
}
else
{
acb_t t;
acb_init(t);
if (arf_cmpabs_2exp_si(arb_midref(acb_imagref(z)), 0) < 0)
{
acb_sin_cos(r, t, z, prec + 4);
acb_div(r, t, r, prec);
}
else
{
acb_mul_2exp_si(t, z, 1);
if (arf_sgn(arb_midref(acb_imagref(z))) > 0)
{
acb_mul_onei(t, t);
acb_exp(t, t, prec + 4);
acb_sub_ui(r, t, 1, prec + 4);
acb_div(r, t, r, prec + 4);
acb_mul_2exp_si(r, r, 1);
acb_sub_ui(r, r, 1, prec);
acb_mul_onei(r, r);
}
else
{
acb_div_onei(t, t);
acb_exp(t, t, prec + 4);
acb_sub_ui(r, t, 1, prec + 4);
acb_div(r, t, r, prec + 4);
acb_mul_2exp_si(r, r, 1);
acb_sub_ui(r, r, 1, prec);
acb_div_onei(r, r);
}
}
acb_clear(t);
}
}
/* f(z) = max(sin(z), cos(z)) */
int
f_max_sin_cos(acb_ptr res, const acb_t z, void * param, slong order, slong prec)
{
acb_t s, c;
if (order > 1)
flint_abort(); /* Would be needed for Taylor method. */
acb_init(s);
acb_init(c);
acb_sin_cos(s, c, z, prec);
acb_real_max(res, s, c, order != 0, prec);
acb_clear(s);
acb_clear(c);
return 0;
}
/* f(z) = z sin(z) / (1 + cos(z)^2) */
int
f_sin_cos_frac(acb_ptr res, const acb_t z, void * param, slong order, slong prec)
{
acb_t s, c;
if (order > 1)
flint_abort(); /* Would be needed for Taylor method. */
acb_init(s);
acb_init(c);
acb_sin_cos(s, c, z, prec);
acb_mul(c, c, c, prec);
acb_add_ui(c, c, 1, prec);
acb_mul(s, s, z, prec);
acb_div(res, s, c, prec);
acb_clear(s);
acb_clear(c);
return 0;
}
void
_acb_poly_sin_cos_series_tangent(acb_ptr s, acb_ptr c,
const acb_srcptr h, slong hlen, slong len, slong prec, int times_pi)
{
acb_ptr t, u, v;
acb_t s0, c0;
hlen = FLINT_MIN(hlen, len);
if (hlen == 1)
{
if (times_pi)
acb_sin_cos_pi(s, c, h, prec);
else
acb_sin_cos(s, c, h, prec);
_acb_vec_zero(s + 1, len - 1);
_acb_vec_zero(c + 1, len - 1);
return;
}
/*
sin(x) = 2*tan(x/2)/(1+tan(x/2)^2)
cos(x) = (1-tan(x/2)^2)/(1+tan(x/2)^2)
*/
acb_init(s0);
acb_init(c0);
t = _acb_vec_init(3 * len);
u = t + len;
v = u + len;
/* sin, cos of h0 */
if (times_pi)
acb_sin_cos_pi(s0, c0, h, prec);
else
acb_sin_cos(s0, c0, h, prec);
/* t = tan((h-h0)/2) */
acb_zero(u);
_acb_vec_scalar_mul_2exp_si(u + 1, h + 1, hlen - 1, -1);
if (times_pi)
{
acb_const_pi(t, prec);
_acb_vec_scalar_mul(u + 1, u + 1, hlen - 1, t, prec);
}
_acb_poly_tan_series(t, u, hlen, len, prec);
/* v = 1 + t^2 */
_acb_poly_mullow(v, t, len, t, len, len, prec);
acb_add_ui(v, v, 1, prec);
/* u = 1/(1+t^2) */
_acb_poly_inv_series(u, v, len, len, prec);
/* sine */
_acb_poly_mullow(s, t, len, u, len, len, prec);
_acb_vec_scalar_mul_2exp_si(s, s, len, 1);
/* cosine */
acb_sub_ui(v, v, 2, prec);
_acb_vec_neg(v, v, len);
_acb_poly_mullow(c, v, len, u, len, len, prec);
/* sin(h0 + h1) = cos(h0) sin(h1) + sin(h0) cos(h1)
cos(h0 + h1) = cos(h0) cos(h1) - sin(h0) sin(h1) */
if (!acb_is_zero(s0))
{
_acb_vec_scalar_mul(t, s, len, c0, prec);
_acb_vec_scalar_mul(u, c, len, s0, prec);
_acb_vec_scalar_mul(v, s, len, s0, prec);
_acb_vec_add(s, t, u, len, prec);
_acb_vec_scalar_mul(t, c, len, c0, prec);
_acb_vec_sub(c, t, v, len, prec);
}
_acb_vec_clear(t, 3 * len);
acb_clear(s0);
acb_clear(c0);
}