A fast and highly accurate computational scheme to compute a product of Bessel functions with (large) complex order and argument is of utmost importance in some problems in physics involving cylindrical configurations. Since these products can occur in the kernel of multiple integrals, and are subject to mutual cancellations, speed and accuracy of their numerical computation are key. Moreover, the order and argument can vary over a vast range of several orders of magnitude. To deliver a high relative accuracy (10-11) over the entire range, we use the integral representations of the Bessel functions, and introduce a deformation into steepest descent paths. The numerical computation is accelerated by three orders of magnitude, if these paths are replaced by well-chosen piecewise linear segments, while maintaining the set relative accuracy.