Background: Reticulate events play an important role in determining evolutionary relationships. The problem of computing the minimum number of such events to explain discordance between two phylogenetic trees is a hard computational problem. Even for binary trees, exact solvers struggle to solve instances with reticulation number larger than 40-50. Results: Here we present CYCLEKILLER and NONBINARYCYCLEKILLER, the first methods to produce solutions verifiably close to optimality for instances with hundreds or even thousands of reticulations. Conclusions: Using simulations, we demonstrate that these algorithms run quickly for large and difficult instances, producing solutions that are very close to optimality. As a spin-off from our simulations we also present TERMINUSEST, which is the fastest exact method currently available that can handle nonbinary trees: this is used to measure the accuracy of the NONBINARYCYCLEKILLER algorithm. All three methods are based on extensions of previous theoretical work (SIDMA 26(4): 1635-1656, TCBB 10(1): 18-25, SIDMA 28(1): 49-66) and are publicly available. We also apply our methods to real data.