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acb_atan: improve error bounds

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This commit is contained in:
Fredrik Johansson 2017-09-18 00:06:02 +02:00
parent b84da69711
commit aa30e792ae
1 changed files with 34 additions and 3 deletions
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37
acb/atan.c
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@ -61,18 +61,49 @@ acb_atan(acb_t r, const acb_t z, slong prec)
acb_log1p(t, t, prec); acb_log1p(t, t, prec);
acb_log1p(u, u, prec); acb_log1p(u, u, prec);
acb_sub(t, t, u, prec); acb_sub(t, t, u, prec);
acb_mul_onei(t, t);
acb_mul_2exp_si(r, t, -1);
} }
else else if (acb_is_exact(z))
{ {
acb_onei(t); acb_onei(t);
acb_sub(t, t, z, prec); acb_sub(t, t, z, prec);
acb_div(t, z, t, prec); acb_div(t, z, t, prec);
acb_mul_2exp_si(t, t, 1); acb_mul_2exp_si(t, t, 1);
acb_log1p(t, t, prec); acb_log1p(t, t, prec);
acb_mul_onei(t, t);
acb_mul_2exp_si(r, t, -1);
} }
else
{
mag_t err, err2;
mag_init(err);
mag_init(err2);
acb_mul_onei(t, t); /* atan'(z) = 1/(1+z^2) = 1/((z+i)(z-i)) */
acb_mul_2exp_si(r, t, -1); acb_onei(t);
acb_add(t, z, t, prec);
acb_get_mag_lower(err, t);
acb_onei(t);
acb_sub(t, z, t, prec);
acb_get_mag_lower(err2, t);
mag_mul_lower(err, err, err2);
mag_hypot(err2, arb_radref(acb_realref(z)), arb_radref(acb_imagref(z)));
mag_div(err, err2, err);
arf_set(arb_midref(acb_realref(t)), arb_midref(acb_realref(z)));
arf_set(arb_midref(acb_imagref(t)), arb_midref(acb_imagref(z)));
mag_zero(arb_radref(acb_realref(t)));
mag_zero(arb_radref(acb_imagref(t)));
acb_atan(r, t, prec);
acb_add_error_mag(r, err);
mag_clear(err);
mag_clear(err2);
}
acb_clear(t); acb_clear(t);
acb_clear(u); acb_clear(u);