  The most popular method for computing the matrix logarithm is a combination
of the inverse scaling and squaring method in conjunction with a Pad\'e
approximation, sometimes accompanied by the Schur decomposition. The main
computational effort lies in matrix-matrix multiplications and left matrix
division. In this work we illustrate that the number of such operations can be
substantially reduced, by using a graph based representation of an efficient
polynomial evaluation scheme. A technique to analyze the rounding error is
proposed, and backward error analysis is adapted. We provide substantial
simulations illustrating competitiveness both in terms of computation time and
rounding errors.
