The fast multipole method (FMM) of Greengard evaluates Coulomb interactions of point charges with computational requirements that increase linearly, with the number of particles. In this work, the central transformations of the FMM are obtained in a very compact manner from simple algebraic manipulations of two addition theorems. The intermediate multipole and Taylor expansions are defined differently from previous work to yield simplified and more efficient transformations. Error estimates are obtained due to the effect of multipole truncation and the use of the multipole to Taylor transformation operator. Efficient implementation of the FMM for potential and forces is discussed, and calculations are presented that probe the accuracy and performance of the method.