In this paper we suggest a modification of classical Lupanov's method [Lupanov1958]
that allows building circuits over the basis $\{\&,\vee,\neg\}$ for Boolean functions of $n$ variables with size at most
$$
\frac{2^n}{n}\left(1+\frac{3\log n + O(1)}{n}\right),
$$
and with more uniform distribution of outgoing arcs by circuit gates.

For almost all Boolean functions of $n$ variables in the circuits for
these functions, which are built using our method, the fraction of gates
with fan-out 2 is asymptotically at least 1/32. This fact disproves upper bound [Yamamoto2011]
on the number of circuits with exact number of gates with fan-out at least 2.