  In this work, we present a new way to compute the Taylor polynomial of the
matrix exponential which reduces the number of matrix multiplications in
comparison with the de-facto standard Patterson-Stockmeyer method. This
reduction is sufficient to make the method superior in performance to Pad\'e
approximants by 10-30% over a range of values for the matrix norms and thus we
propose its replacement in standard software kits. Numerical experiments show
the performance of the method and illustrate its stability.
