  Let $\crs(K_n)$ be the minimum number of crossings over all rectilinear
drawings of the complete graph on $n$ vertices on the plane. In this paper we
prove that $\crs(K_n) < 0.380473\binom{n}{4}+\Theta(n^3)$; improving thus on
the previous best known upper bound. This is done by obtaining new rectilinear
drawings of $K_n$ for small values of $n$, and then using known constructions
to obtain arbitrarily large good drawings from smaller ones. The "small" sets
where found using a simple heuristic detailed in this paper.
