The Rényi divergence is a measure of divergence between distributions. It has recently found several applications in lattice-based cryptography. The contribution of this paper is twofold. First, we give theoretic results which renders it more efficient and easier to use. This is done by providing two lemmas, which give tight bounds in very common situations – for distributions that are tailcut or have a bounded relative error. We then connect the Rényi divergence to the max-log distance. This allows the Rényi divergence to indirectly benefit from all the advantages of a distance. Second, we apply our new results to five practical usecases. It allows us to claim 256 bits of security for a floating-point precision of 53 bits, in cases that until now either required more than 150 bits of precision or were limited to 100 bits of security: rejection sampling, trapdoor sampling (61 bits in this case) and a new sampler by Micciancio and Walter. We also propose a new and compact approach for table-based sampling, and squeeze the standard deviation of trapdoor samplers by a factor that provides a gain of 30 bits of security in practice.