In probabilistic inference, credible intervals constructed from posterior samples provide ranges of likely values for continuous parameters of interest. Intuitively, an inference procedure is optimal if it produces the most precise posterior intervals that cover the true parameter value with the expected frequency in repeated experiments. We present theories and methods for automating posterior interval evaluation of inference performance in probabilistic programming using two metrics: 1.) truth coverage, and 2.) ratio of the empirical over the ideal interval widths. Demonstrating with inference on popular regression and state-space models, we show how the metrics provide effective comparisons between different inference procedures, and capture the effects of collinearity and model misspecification. Overall, we claim such automated interval evaluation can accelerate the robust design and comparison of probabilistic inference programs by directly diagnosing how accurately and precisely they can estimate parameters of interest.