Long-read next generation amplicon sequencing shows promise for studying complete genes or genomes from complex and diverse populations. Current long-read sequencing technologies have challenging error profiles, hindering data processing and incorporation into downstream analyses. Here we consider the problem of how to reconstruct, free of sequencing error, the true sequence variants and their associated frequencies. Called “amplicon denoising”, this problem has been extensively studied for short-read sequencing technologies, but current solutions do not appear to generalize well to long reads with high indel error rates. We introduce two methods: one that runs nearly instantly and is very accurate for medium length reads (here ~2.6kb) and high template coverage, and another, slower method that is more robust when reads are very long or coverage is lower. On one real dataset with ground truth, and on a number of simulated datasets, we compare our two approaches to each other and to existing algorithms. We outperform all tested methods in accuracy, with competitive run times even for our slower method. Fast Amplicon Denoising (FAD) and Robust Amplicon Denoising (RAD) are implemented purely in the Julia scientific computing language, and are hereby released along with a complete toolkit of functions that allow long-read amplicon sequence analysis pipelines to be constructed in pure Julia. Further, we make available a webserver to dramatically simplify the processing of long-read PacBio sequences.