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This paper presents a saddlepoint approximation of the random-coding unionbound of Polyanskiy et al. for i.i.d. random coding over discrete memorylesschannels. The approximation is single-letter, and can thus be computedefficiently. Moreover, it is shown to be asymptotically tight for both fixedand varying rates, unifying existing achievability results in the regimes oferror exponents, second-order coding rates, and moderate deviations. For fixedrates, novel exact-asymptotics expressions are specified to within amultiplicative 1+o(1) term. A numerical example is provided for which theapproximation is remarkably accurate even at short block lengths.

The Saddlepoint Approximation: Unified Random Coding Asymptotics for Fixed and Varying Rates