In this talk, we present a saddlepoint approximation of the the random-coding union bound of Polyanskiy et al. for i.i.d. random coding over discrete memoryless channels. The approximation can be computed efficiently, and is shown to be asymptotically tight and recover several known asymptotic results at both fixed and varying rates, unifying the regimes of error exponents, second-order coding rates, and moderate deviations. For fixed rates, novel exact-asymptotics expressions are specified to within a multiplicative 1+o(1) term. Using numerical examples, we show that the saddlepoint approximation is remarkably accurate even at short block lengths.