The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé 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.