Let X1, X2, . . . , Xn be n independent and uniformly distributed random points in a compact region R ⊂ R2 of area 1. Let TSP(X1, . . . , Xn) denote the length of the optimal Euclidean traveling salesman tour that traverses all these points. The classical Beardwood-Halton-Hammersley theorem proves the existence of a universal constant β2 such TSP(X1, . . . , Xn)/ √ n → β2 almost surely, which satisfies 0.625 ≤ β2 ≤ 0.92117. This paper presents a computer-aided proof using numerical quadrature and decision trees that β2 < 0.90304. Although our improvement is still somewhat small, our approach has the advantage that it is primarily limited by computer hardware, and is thus amenable to further improvements over time.