  Quantum algorithms for simulation of Hamiltonian evolution are often based on
product formulae. The fractal method of Suzuki gives a systematic way to find
arbitrarily high-order product formulae, but results in a large number of
exponentials. On the other hand, product formulae with fewer exponentials can
be found by numerical solution of simultaneous nonlinear equations. It is also
possible to reduce the cost of long-time simulations by processing, where a
kernel is repeated and a processor need only be applied at the beginning and
end of the simulation. In this work, we found thousands of new product formulae
of both 8th and 10th order, and numerically tested these formulae, together
with many formulae from prior literature. We provide methods to fairly compare
product formulae of different lengths and different orders. We have found a new
8th order processed product formula with exceptional performance, that
outperforms all other tested product formulae for about eight orders of
magnitude in system parameters $T$ (time) and $\epsilon$ (allowable error).
That includes most reasonable combinations of parameters to be used in quantum
algorithms.
