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Binding free energies of bromodomain inhibitors are calculated with recently formulated approaches, namely ESMACS (enhanced sampling of molecular dynamics with approximation of continuum solvent) and TIES (thermodynamic integration with enhanced sampling). A set of compounds is provided by GlaxoSmithKline, which represents a range of chemical functionality and binding affinities. The predicted binding free energies exhibit a good Spearman correlation of 0.78 with the experimental data from the 3-trajectory ESMACS, and an excellent correlation of 0.92 from the TIES approach where applicable. Given access to suitable high end computing resources and a high degree of automation, we can compute individual binding affinities in a few hours with precisions no greater than 0.2 kcal/mol for TIES, and no larger than 0.34 and 1.71 kcal/mol for the 1- and 3-trajectory ESMACS approaches.