Absolute fast converging phylogenetic reconstruction methods are provably guaranteed to recover the true tree with high probability from sequences that grow only polynomially in the number of leaves, once the edge lengths are bounded arbitrarily from above and below. Only a few methods have been determined to be absolute fast converging, these have all been developed in just the last few years, and most are polynomial time. In this paper, we compare pre-existing fast converging methods as well as some new polynomial time methods that we have developed. Our study, based upon simulating evolution under a wide range of model conditions, establishes that our new methods outperform both neighbor joining and the previous fast converging methods, returning very accurate large trees, when these other methods do poorly.