int main()
{
slong iter;
flint_rand_t state;
flint_printf("rsqrt....");
fflush(stdout);
flint_randinit(state);
/* check (a^(-1/2))^(-2) = a */
for (iter = 0; iter < 10000; iter++)
{
acb_t a, b, c;
slong prec;
acb_init(a);
acb_init(b);
acb_init(c);
acb_randtest(a, state, 1 + n_randint(state, 2000), 10);
acb_randtest(b, state, 1 + n_randint(state, 2000), 10);
prec = 2 + n_randint(state, 2000);
acb_rsqrt(b, a, prec);
acb_inv(c, b, prec);
acb_mul(c, c, c, prec);
if (!acb_contains(c, a))
{
flint_printf("FAIL: containment\n\n");
flint_printf("a = "); acb_print(a); flint_printf("\n\n");
flint_printf("b = "); acb_print(b); flint_printf("\n\n");
flint_printf("c = "); acb_print(c); flint_printf("\n\n");
abort();
}
acb_rsqrt(a, a, prec);
if (!acb_equal(a, b))
{
flint_printf("FAIL: aliasing\n\n");
flint_printf("a = "); acb_print(a); flint_printf("\n\n");
flint_printf("b = "); acb_print(b); flint_printf("\n\n");
abort();
}
acb_clear(a);
acb_clear(b);
acb_clear(c);
}
flint_randclear(state);
flint_cleanup();
flint_printf("PASS\n");
return EXIT_SUCCESS;
}
/* REAL: erf(x) = 2x/sqrt(pi) * exp(-x^2) 1F1(1, 3/2, x^2) */
void
acb_hypgeom_erf_1f1b(acb_t res, const acb_t z, slong prec)
{
acb_t a, b, t, w;
acb_init(a);
acb_init(b);
acb_init(t);
acb_init(w);
acb_set_ui(b, 3);
acb_mul_2exp_si(b, b, -1);
acb_mul(w, z, z, prec);
acb_hypgeom_pfq_direct(t, a, 0, b, 1, w, -1, prec);
acb_neg(w, w);
acb_exp(w, w, prec);
acb_mul(t, t, w, prec);
acb_const_pi(w, prec);
acb_rsqrt(w, w, prec);
acb_mul(t, t, w, prec);
acb_mul(t, t, z, prec);
acb_mul_2exp_si(res, t, 1);
acb_clear(a);
acb_clear(b);
acb_clear(t);
acb_clear (w);
}
void
acb_hypgeom_bessel_j_asymp_prefactors(acb_t Ap, acb_t Am, acb_t C,
const acb_t nu, const acb_t z, long prec)
{
if (arb_is_positive(acb_realref(z)))
{
acb_t t, u;
acb_init(t);
acb_init(u);
/* -(2nu+1)/4 * pi + z */
acb_mul_2exp_si(t, nu, 1);
acb_add_ui(t, t, 1, prec);
acb_mul_2exp_si(t, t, -2);
acb_neg(t, t);
acb_const_pi(u, prec);
acb_mul(t, t, u, prec);
acb_add(t, t, z, prec);
acb_mul_onei(t, t);
acb_exp_invexp(Ap, Am, t, prec);
/* (2 pi z)^(-1/2) */
acb_const_pi(C, prec);
acb_mul_2exp_si(C, C, 1);
acb_mul(C, C, z, prec);
acb_rsqrt(C, C, prec);
acb_clear(t);
acb_clear(u);
return;
}
acb_hypgeom_bessel_j_asymp_prefactors_fallback(Ap, Am, C, nu, z, prec);
}
void
acb_hypgeom_bessel_i_asymp_prefactors(acb_t A, acb_t B, acb_t C,
const acb_t nu, const acb_t z, long prec)
{
acb_t t, u;
acb_init(t);
acb_init(u);
/* C = (2 pi z)^(-1/2) */
acb_const_pi(C, prec);
acb_mul_2exp_si(C, C, 1);
acb_mul(C, C, z, prec);
acb_rsqrt(C, C, prec);
if (arb_is_positive(acb_imagref(z)) ||
(arb_is_zero(acb_imagref(z)) && arb_is_negative(acb_realref(z))))
{
acb_exp_pi_i(t, nu, prec);
acb_mul_onei(t, t);
}
else if (arb_is_negative(acb_imagref(z)) ||
(arb_is_zero(acb_imagref(z)) && arb_is_positive(acb_realref(z))))
{
acb_neg(t, nu);
acb_exp_pi_i(t, t, prec);
acb_mul_onei(t, t);
acb_neg(t, t);
}
else
{
acb_exp_pi_i(t, nu, prec);
acb_mul_onei(t, t);
acb_neg(u, nu);
acb_exp_pi_i(u, u, prec);
acb_mul_onei(u, u);
acb_neg(u, u);
arb_union(acb_realref(t), acb_realref(t), acb_realref(u), prec);
arb_union(acb_imagref(t), acb_imagref(t), acb_imagref(u), prec);
}
acb_exp_invexp(B, A, z, prec);
acb_mul(A, A, t, prec);
acb_clear(t);
acb_clear(u);
}
/* (+/- iz)^(-1/2-v) * z^v * exp(+/- iz) */
void
acb_hypgeom_bessel_j_asymp_prefactors_fallback(acb_t Ap, acb_t Am, acb_t C,
const acb_t nu, const acb_t z, long prec)
{
acb_t t, u, v;
acb_init(t);
acb_init(u);
acb_init(v);
/* v = -1/2-nu */
acb_one(v);
acb_mul_2exp_si(v, v, -1);
acb_add(v, v, nu, prec);
acb_neg(v, v);
acb_mul_onei(t, z); /* t = iz */
acb_neg(u, t); /* u = -iz */
/* Ap, Am = (+/- iz)^(-1/2-nu) */
acb_pow(Ap, t, v, prec);
acb_pow(Am, u, v, prec);
/* Ap, Am *= exp(+/- iz) */
acb_exp_invexp(u, v, t, prec);
acb_mul(Ap, Ap, u, prec);
acb_mul(Am, Am, v, prec);
/* z^nu */
acb_pow(t, z, nu, prec);
acb_mul(Ap, Ap, t, prec);
acb_mul(Am, Am, t, prec);
/* (2 pi)^(-1/2) */
acb_const_pi(C, prec);
acb_mul_2exp_si(C, C, 1);
acb_rsqrt(C, C, prec);
acb_clear(t);
acb_clear(u);
acb_clear(v);
}
void
acb_hypgeom_bessel_k_asymp(acb_t res, const acb_t nu, const acb_t z, slong prec)
{
acb_t t, a, b, w;
acb_init(t);
acb_init(a);
acb_init(b);
acb_init(w);
acb_one(a);
acb_mul_2exp_si(a, a, -1);
acb_add(a, a, nu, prec);
acb_mul_2exp_si(b, nu, 1);
acb_add_ui(b, b, 1, prec);
acb_mul_2exp_si(w, z, 1);
acb_hypgeom_u_asymp(t, a, b, w, -1, prec);
acb_neg(w, z);
acb_exp(w, w, prec);
acb_mul(t, t, w, prec);
acb_mul_2exp_si(w, z, 1);
acb_rsqrt(w, w, prec);
acb_mul(res, t, w, prec);
arb_const_sqrt_pi(acb_realref(w), prec);
acb_mul_arb(res, res, acb_realref(w), prec);
acb_clear(t);
acb_clear(a);
acb_clear(b);
acb_clear(w);
}
/* IMAG: erf(z) = 2z/sqrt(pi) * 1F1(1/2, 3/2, -z^2) */
void
acb_hypgeom_erf_1f1a(acb_t res, const acb_t z, slong prec)
{
acb_t a, t, w;
acb_struct b[2];
acb_init(a);
acb_init(b);
acb_init(b + 1);
acb_init(t);
acb_init(w);
acb_one(a);
acb_mul_2exp_si(a, a, -1);
acb_set_ui(b, 3);
acb_mul_2exp_si(b, b, -1);
acb_one(b + 1);
acb_mul(w, z, z, prec);
acb_neg(w, w);
acb_hypgeom_pfq_direct(t, a, 1, b, 2, w, -1, prec);
acb_const_pi(w, prec);
acb_rsqrt(w, w, prec);
acb_mul(t, t, w, prec);
acb_mul(t, t, z, prec);
acb_mul_2exp_si(res, t, 1);
acb_clear(a);
acb_clear(b);
acb_clear(b + 1);
acb_clear(t);
acb_clear(w);
}
void
_acb_hypgeom_legendre_q_double(acb_t res, const acb_t n, const acb_t m,
const acb_t z, slong prec)
{
acb_t t, u, v;
acb_init(t);
acb_init(u);
acb_init(v);
if (acb_is_int(m))
{
acb_sub_ui(t, z, 1, prec);
acb_mul_2exp_si(u, m, -1);
acb_pow(v, t, u, prec);
acb_neg(t, t);
acb_neg(u, u);
acb_pow(t, t, u, prec);
acb_mul(t, t, v, prec);
acb_hypgeom_legendre_q(u, n, m, z, 0, prec);
acb_mul(t, t, u, prec);
acb_mul_2exp_si(u, m, -1);
if (!acb_is_int(u))
acb_neg(t, t);
acb_sub_ui(u, z, 1, prec);
acb_sqrt(u, u, prec);
acb_sub_ui(v, z, 1, prec);
acb_neg(v, v);
acb_rsqrt(v, v, prec);
acb_mul(u, u, v, prec);
acb_hypgeom_legendre_p(v, n, m, z, 1, prec);
acb_mul(u, u, v, prec);
acb_const_pi(v, prec);
acb_mul(u, u, v, prec);
acb_mul_2exp_si(u, u, -1);
acb_sub(res, t, u, prec);
}
else
{
acb_sub(t, n, m, prec);
acb_add_ui(t, t, 1, prec);
acb_mul_2exp_si(u, m, 1);
acb_rising(t, t, u, prec);
acb_neg(u, m);
acb_hypgeom_legendre_p(u, n, u, z, 1, prec);
acb_mul(t, t, u, prec);
acb_hypgeom_legendre_p(u, n, m, z, 1, prec);
acb_sub(t, u, t, prec);
acb_exp_pi_i(u, m, prec);
acb_mul(t, t, u, prec);
acb_sin_pi(u, m, prec);
acb_div(t, t, u, prec);
acb_const_pi(u, prec);
acb_mul(t, t, u, prec);
acb_mul_2exp_si(t, t, -1);
acb_set(res, t);
}
acb_clear(t);
acb_clear(u);
acb_clear(v);
}
