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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"
void
mag_bin_uiui(mag_t res, ulong n, ulong k)
{
mag_t t;
if (k > n)
{
mag_zero(res);
return;
}
if (k > n / 2)
k = n - k;
if (k == 0)
{
mag_one(res);
return;
}
if (k == 1)
{
mag_set_ui(res, n);
return;
}
mag_init(t);
if (n < 256 && k < 256)
{
/* using accurate (lookup table) factorials */
mag_fac_ui(res, n);
mag_rfac_ui(t, k);
mag_mul(res, res, t);
mag_rfac_ui(t, n - k);
mag_mul(res, res, t);
}
else
{
/* binary entropy bound (n/(n-k))^(n-k) (n/k)^k = n^n / (k^k (n-k)^(n-k)) */
mag_set_ui(res, n);
mag_div_ui(res, res, n - k);
mag_pow_ui(res, res, n - k);
/* (n/(n-k))^(n-k) is also bounded by exp(k) */
mag_set_ui(t, k);
mag_exp(t, t);
mag_min(res, res, t);
mag_set_ui(t, n);
mag_div_ui(t, t, k);
mag_pow_ui(t, t, k);
mag_mul(res, res, t);
}
mag_clear(t);
}
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