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/*
Copyright (C) 2017 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "mag.h"
#include "arb.h" /* for atan table */
static double
mag_atan_d(double x)
{
double t, u, v;
int p, q, flip;
t = x;
flip = 0;
if (t > 1.0)
{
t = 1.0 / t;
flip = 1;
}
q = ldexp(1.0, ARB_ATAN_TAB1_BITS);
p = t * (1 << ARB_ATAN_TAB1_BITS);
if (p == (1 << ARB_ATAN_TAB1_BITS))
p--;
#if FLINT_BITS == 64
v = arb_atan_tab1[p][ARB_ATAN_TAB1_LIMBS - 1] * ldexp(1.0, -FLINT_BITS);
#else
v = arb_atan_tab1[p][ARB_ATAN_TAB1_LIMBS - 1] * ldexp(1.0, -FLINT_BITS)
+ arb_atan_tab1[p][ARB_ATAN_TAB1_LIMBS - 2] * ldexp(1.0, -2 * FLINT_BITS);
#endif
t = (q * t - p) / (p * t + q);
/* t - t^3/3 + t^5/5 */
u = t * t;
t = (t * (1.0 / 15.0)) * (15.0 - 5.0 * u + 3.0 * (u * u));
v += t;
if (flip)
v = 1.5707963267948966192 - v;
return v;
}
void
mag_atan(mag_t res, const mag_t x)
{
if (mag_is_zero(x))
{
mag_zero(res);
}
else if (mag_cmp_2exp_si(x, MAG_BITS) > 0)
{
mag_const_pi(res);
mag_mul_2exp_si(res, res, -1);
}
else if (mag_cmp_2exp_si(x, -(MAG_BITS / 2)) < 0)
{
mag_set(res, x);
}
else
{
double t;
t = ldexp(MAG_MAN(x), MAG_EXP(x) - MAG_BITS);
t = mag_atan_d(t);
t = t * (1.0 + 1e-12);
mag_set_d(res, t);
}
}
void
mag_atan_lower(mag_t res, const mag_t x)
{
if (mag_is_zero(x))
{
mag_zero(res);
}
else if (mag_is_inf(x))
{
mag_const_pi_lower(res);
mag_mul_2exp_si(res, res, -1);
}
else if (mag_cmp_2exp_si(x, MAG_BITS) > 0)
{
/* atan(x) > pi/2 - 1/x */
mag_t t;
mag_init(t);
mag_one(t);
mag_div(t, t, x);
mag_const_pi_lower(res);
mag_mul_2exp_si(res, res, -1);
mag_sub_lower(res, res, t);
mag_clear(t);
}
else if (mag_cmp_2exp_si(x, -MAG_BITS) < 0)
{
/* atan(x) > x - x^2 */
mag_t t;
mag_init(t);
mag_mul(t, x, x);
mag_sub_lower(res, x, t);
mag_clear(t);
}
else
{
double t;
t = ldexp(MAG_MAN(x), MAG_EXP(x) - MAG_BITS);
t = mag_atan_d(t);
t = t * (1.0 - 1e-12);
mag_set_d_lower(res, t);
}
}
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