  It is shown that the absolute constant in the Berry--Esseen inequality for
i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if
$1\le n\le 500000$, where $n$ is a number of summands. This result is got both
with the help of a supercomputer and an interpolation theorem, which is proved
in the paper as well. In addition, applying the method developed by S. Nagaev
and V. Chebotarev in 2009--2011, an upper bound is obtained for the absolute
constant in the Berry--Esseen inequality in the case under consideration, which
differs from the Esseen constant by no more than 0.06%. As an auxiliary result,
we prove a bound in the local Moivre--Laplace theorem which has a simple and
explicit form. Despite the best possible result, obtained by J. Schulz in 2016,
we propose our approach to the problem of finding the absolute constant in the
Berry--Esseen inequality for two-point distributions since this approach,
combining analytical methods and the use of computers, could be useful in
solving other mathematical problems.
