This paper addresses a fundamental problem in random variate generation: given access to a random source that emits a stream of independent fair bits, what is the most accurate and entropy-efficient algorithm for sampling from a discrete probability distribution (p₁, …, pₙ), where the output distribution (p̂₁, …, p̂ₙ) of the sampling algorithm can be specified with a given level of bit precision? We present a theoretical framework for formulating this problem and provide new techniques for finding sampling algorithms that are optimal both statistically (in the sense of sampling accuracy) and information-theoretically (in the sense of entropy consumption). We leverage these results to build a system that, for a broad family of measures of statistical accuracy, delivers a sampling algorithm whose expected entropy usage is minimal among those that induce the same distribution (i.e., is “entropy-optimal”) and whose output distribution (p̂₁, …, p̂ₙ) is a closest approximation to the target distribution (p₁, …, pₙ) among all entropy-optimal sampling algorithms that operate within the specified precision budget. This optimal approximate sampler is also a closer approximation than any (possibly entropy-suboptimal) sampler that consumes a bounded amount of entropy with the specified precision, a class which includes floating-point implementations of inversion sampling and related methods found in many standard software libraries. We evaluate the accuracy, entropy consumption, precision requirements, and wall-clock runtime of our optimal approximate sampling algorithms on a broad set of probability distributions, demonstrating the ways that they are superior to existing approximate samplers and establishing that they often consume significantly fewer resources than are needed by exact samplers.