We present AQUA, a new probabilistic inference algorithm that operates on probabilistic programs with continuous posterior distributions. AQUA approximates programs via an efficient quantization of the continuous distributions. It represents the distributions of random variables using quantized value intervals (Interval Cube) and corresponding probability densities (Density Cube). AQUA’s analysis transforms Interval and Density Cubes to compute the posterior distribution with bounded error. We also present an adaptive algorithm for selecting the size and the granularity of the Interval and Density Cubes. We evaluate AQUA on 24 programs from the literature. AQUA solved all of 24 benchmarks in less than 43 s (median 1.35 s) with a high-level of accuracy. We show that AQUA is more accurate than state-of-the-art approximate algorithms (Stan’s NUTS and ADVI) and supports programs that are out of reach of exact inference tools, such as PSI and SPPL.