Large-scale testing in modern applications such as genomics often entails a trade-off between accuracy and speed: multiplicity corrections push cutoffs deep into the tails, where normal approximations can fail, while resampling is accurate but computationally expensive for large datasets. To resolve this impasse in the context of conditional independence testing, we introduce spaCRT, a closed-form saddlepoint approximation (SPA) for the distilled conditional randomization test (dCRT) that retains the statistical accuracy of dCRT's resampling while avoiding its computational cost. We prove that spaCRT's relative approximation error vanishes asymptotically by establishing a general theorem on the relative error of conditional SPAs. Because dCRT uses a plug-in nuisance regression, we specialize our guarantees to common choices: low-dimensional generalized linear model (GLM), high-dimensional GLM lasso, and kernel ridge regression. Our general theorem is, to our knowledge, the first rigorous technical tool for analyzing SPAs for resampling tests, which had previously been justified only heuristically. It extends beyond spaCRT, as we exemplify by justifying an SPA for the classical sign-flipping location test. Empirically, spaCRT matches dCRT's statistical performance by approximating its p-values with median error 1-12% across settings while delivering a 250x speedup on a single-cell CRISPR screen dataset with 85,000 hypotheses. Building on dCRT's versatility, spaCRT and its open-source R package enable fast and accurate large-scale testing across diverse applications.

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