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We investigate the computation of Csisza´r's bounds for the joint source-channel coding (JSCC) error exponent EJ of a communication system consisting of a discrete memoryless source and a discrete memoryless channel. We provide equivalent expressions for these bounds and derive explicit formulas for the rates where the bounds are attained. These equivalent representations can be readily computed for arbitrary source-channel pairs via Arimoto's algorithm. When the channel's distribution satisfies a symmetry property, the bounds admit closed-form parametric expressions. We then use our results to provide a systematic comparison between the JSCC error exponent EJ and the tandem coding error exponent ET, which applies if the source and channel are separately coded. It is shown that ET≤EJ≤2ET. We establish conditions for which EJ>ET and for which EJ=2ET. Numerical examples indicate that EJ is close to 2ET for many source-channel pairs. This gain translates into a power saving larger than 2 dB for a binary source transmitted over additive white Gaussian noise (AWGN) channels and Rayleigh-fading channels with finite output quantization. Finally, we study the computation of the lossy JSCC error exponent under the Hamming distortion measure.