void
acb_hypgeom_erf_propagated_error(mag_t re, mag_t im, const acb_t z)
{
mag_t x, y;
mag_init(x);
mag_init(y);
/* |exp(-(x+y)^2)| = exp(y^2-x^2) */
arb_get_mag(y, acb_imagref(z));
mag_mul(y, y, y);
arb_get_mag_lower(x, acb_realref(z));
mag_mul_lower(x, x, x);
if (mag_cmp(y, x) >= 0)
{
mag_sub(re, y, x);
mag_exp(re, re);
}
else
{
mag_sub_lower(re, x, y);
mag_expinv(re, re);
}
/* Radius. */
mag_hypot(x, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
mag_mul(re, re, x);
/* 2/sqrt(pi) < 289/256 */
mag_mul_ui(re, re, 289);
mag_mul_2exp_si(re, re, -8);
if (arb_is_zero(acb_imagref(z)))
{
/* todo: could bound magnitude even for complex numbers */
mag_set_ui(y, 2);
mag_min(re, re, y);
mag_zero(im);
}
else if (arb_is_zero(acb_realref(z)))
{
mag_swap(im, re);
mag_zero(re);
}
else
{
mag_set(im, re);
}
mag_clear(x);
mag_clear(y);
}
void
acb_dirichlet_zeta_rs(acb_t res, const acb_t s, slong K, slong prec)
{
if (acb_is_exact(s))
{
acb_dirichlet_zeta_rs_mid(res, s, K, prec);
}
else
{
acb_t t;
mag_t rad, err, err2;
slong acc;
acc = acb_rel_accuracy_bits(s);
acc = FLINT_MAX(acc, 0);
acc = FLINT_MIN(acc, prec);
prec = FLINT_MIN(prec, acc + 20);
acb_init(t);
mag_init(rad);
mag_init(err);
mag_init(err2);
/* rad = rad(s) */
mag_hypot(rad, arb_radref(acb_realref(s)), arb_radref(acb_imagref(s)));
/* bound |zeta'(s)| */
acb_dirichlet_zeta_deriv_bound(err, err2, s);
/* error <= |zeta'(s)| * rad(s) */
mag_mul(err, err, rad);
/* evaluate at midpoint */
acb_get_mid(t, s);
acb_dirichlet_zeta_rs_mid(res, t, K, prec);
acb_add_error_mag(res, err);
acb_clear(t);
mag_clear(rad);
mag_clear(err);
mag_clear(err2);
}
}
void
acb_lambertw_cleared_cut(acb_t res, const acb_t z, const fmpz_t k, int flags, slong prec)
{
acb_t ez1;
acb_init(ez1);
/* compute e*z + 1 */
arb_const_e(acb_realref(ez1), prec);
acb_mul(ez1, ez1, z, prec);
acb_add_ui(ez1, ez1, 1, prec);
if (acb_is_exact(z))
{
acb_lambertw_main(res, z, ez1, k, flags, prec);
}
else
{
acb_t zz;
mag_t err, rad;
mag_init(err);
mag_init(rad);
acb_init(zz);
acb_lambertw_bound_deriv(err, z, ez1, k);
mag_hypot(rad, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
mag_mul(err, err, rad);
acb_set(zz, z);
mag_zero(arb_radref(acb_realref(zz)));
mag_zero(arb_radref(acb_imagref(zz))); /* todo: recompute ez1? */
acb_lambertw_main(res, zz, ez1, k, flags, prec);
acb_add_error_mag(res, err);
mag_clear(err);
mag_clear(rad);
acb_clear(zz);
}
acb_clear(ez1);
}
/* error propagation based on derivatives */
void
acb_hypgeom_airy_direct_prop(acb_t ai, acb_t aip, acb_t bi, acb_t bip,
const acb_t z, slong n, slong prec)
{
mag_t aib, aipb, bib, bipb, zb, rad;
acb_t zz;
int real;
mag_init(aib);
mag_init(aipb);
mag_init(bib);
mag_init(bipb);
mag_init(zb);
mag_init(rad);
acb_init(zz);
real = acb_is_real(z);
arf_set(arb_midref(acb_realref(zz)), arb_midref(acb_realref(z)));
arf_set(arb_midref(acb_imagref(zz)), arb_midref(acb_imagref(z)));
mag_hypot(rad, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
acb_get_mag(zb, z);
acb_hypgeom_airy_bound(aib, aipb, bib, bipb, z);
acb_hypgeom_airy_direct(ai, aip, bi, bip, zz, n, prec);
if (ai != NULL)
{
mag_mul(aipb, aipb, rad);
if (real)
arb_add_error_mag(acb_realref(ai), aipb);
else
acb_add_error_mag(ai, aipb);
}
if (aip != NULL)
{
mag_mul(aib, aib, rad);
mag_mul(aib, aib, zb); /* |Ai''(z)| = |z Ai(z)| */
if (real)
arb_add_error_mag(acb_realref(aip), aib);
else
acb_add_error_mag(aip, aib);
}
if (bi != NULL)
{
mag_mul(bipb, bipb, rad);
if (real)
arb_add_error_mag(acb_realref(bi), bipb);
else
acb_add_error_mag(bi, bipb);
}
if (bip != NULL)
{
mag_mul(bib, bib, rad);
mag_mul(bib, bib, zb); /* |Bi''(z)| = |z Bi(z)| */
if (real)
arb_add_error_mag(acb_realref(bip), bib);
else
acb_add_error_mag(bip, bib);
}
mag_clear(aib);
mag_clear(aipb);
mag_clear(bib);
mag_clear(bipb);
mag_clear(zb);
mag_clear(rad);
acb_clear(zz);
}
/* derivatives: |8/sqrt(pi) sin(2z^2)|, |8/sqrt(pi) cos(2z^2)| <= 5 exp(4|xy|) */
void
acb_hypgeom_fresnel_erf_error(acb_t res1, acb_t res2, const acb_t z, slong prec)
{
mag_t re;
mag_t im;
acb_t zmid;
mag_init(re);
mag_init(im);
acb_init(zmid);
if (arf_cmpabs_ui(arb_midref(acb_realref(z)), 1000) < 0 &&
arf_cmpabs_ui(arb_midref(acb_imagref(z)), 1000) < 0)
{
arb_get_mag(re, acb_realref(z));
arb_get_mag(im, acb_imagref(z));
mag_mul(re, re, im);
mag_mul_2exp_si(re, re, 2);
mag_exp(re, re);
mag_mul_ui(re, re, 5);
}
else
{
arb_t t;
arb_init(t);
arb_mul(t, acb_realref(z), acb_imagref(z), prec);
arb_abs(t, t);
arb_mul_2exp_si(t, t, 2);
arb_exp(t, t, prec);
arb_get_mag(re, t);
mag_mul_ui(re, re, 5);
arb_clear(t);
}
mag_hypot(im, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
mag_mul(re, re, im);
if (arb_is_zero(acb_imagref(z)))
{
mag_set_ui(im, 8); /* For real x, |S(x)| < 4, |C(x)| < 4. */
mag_min(re, re, im);
mag_zero(im);
}
else if (arb_is_zero(acb_realref(z)))
{
mag_set_ui(im, 8);
mag_min(im, re, im);
mag_zero(re);
}
else
{
mag_set(im, re);
}
arf_set(arb_midref(acb_realref(zmid)), arb_midref(acb_realref(z)));
arf_set(arb_midref(acb_imagref(zmid)), arb_midref(acb_imagref(z)));
acb_hypgeom_fresnel_erf(res1, res2, zmid, prec);
if (res1 != NULL)
{
arb_add_error_mag(acb_realref(res1), re);
arb_add_error_mag(acb_imagref(res1), im);
}
if (res2 != NULL)
{
arb_add_error_mag(acb_realref(res2), re);
arb_add_error_mag(acb_imagref(res2), im);
}
mag_clear(re);
mag_clear(im);
acb_clear(zmid);
}
static void
acb_log_sin_pi_half(acb_t res, const acb_t z, slong prec, int upper)
{
acb_t t, u, zmid;
arf_t n;
arb_t pi;
acb_init(t);
acb_init(u);
acb_init(zmid);
arf_init(n);
arb_init(pi);
arf_set(arb_midref(acb_realref(zmid)), arb_midref(acb_realref(z)));
arf_set(arb_midref(acb_imagref(zmid)), arb_midref(acb_imagref(z)));
arf_floor(n, arb_midref(acb_realref(zmid)));
arb_sub_arf(acb_realref(zmid), acb_realref(zmid), n, prec);
arb_const_pi(pi, prec);
if (arf_cmpabs_2exp_si(arb_midref(acb_imagref(zmid)), 2) < 1)
{
acb_sin_pi(t, zmid, prec);
acb_log(t, t, prec);
}
else /* i*pi*(z-0.5) + log((1-exp(-2i*pi*z))/2) */
{
acb_mul_2exp_si(t, zmid, 1);
acb_neg(t, t);
if (upper)
acb_conj(t, t);
acb_exp_pi_i(t, t, prec);
acb_sub_ui(t, t, 1, prec);
acb_neg(t, t);
acb_mul_2exp_si(t, t, -1);
acb_log(t, t, prec);
acb_one(u);
acb_mul_2exp_si(u, u, -1);
acb_sub(u, zmid, u, prec);
if (upper)
acb_conj(u, u);
acb_mul_onei(u, u);
acb_addmul_arb(t, u, pi, prec);
if (upper)
acb_conj(t, t);
}
if (upper)
arb_submul_arf(acb_imagref(t), pi, n, prec);
else
arb_addmul_arf(acb_imagref(t), pi, n, prec);
/* propagated error bound from the derivative pi cot(pi z) */
if (!acb_is_exact(z))
{
mag_t zm, um;
mag_init(zm);
mag_init(um);
acb_cot_pi(u, z, prec);
acb_mul_arb(u, u, pi, prec);
mag_hypot(zm, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
acb_get_mag(um, u);
mag_mul(um, um, zm);
acb_add_error_mag(t, um);
mag_clear(zm);
mag_clear(um);
}
acb_set(res, t);
acb_clear(t);
acb_clear(u);
acb_clear(zmid);
arf_clear(n);
arb_clear(pi);
}
