  We propose a numerically efficient method for evaluating the random-coding
union bound with parameter $s$ on the error probability achievable in the
finite-blocklength regime by a pilot-assisted transmission scheme employing
Gaussian codebooks and operating over a memoryless block-fading channel. Our
method relies on the saddlepoint approximation, which, differently from
previous results reported for similar scenarios, is performed with respect to
the number of fading blocks (a.k.a. diversity branches) spanned by each
codeword, instead of the number of channel uses per block. This different
approach avoids a costly numerical averaging of the error probability over the
realizations of the fading process and of its pilot-based estimate at the
receiver and results in a significant reduction of the number of channel
realizations required to estimate the error probability accurately. Our
numerical experiments for both single-antenna communication links and massive
multiple-input multiple-output (MIMO) networks show that, when two or more
diversity branches are available, the error probability can be estimated
accurately with the saddlepoint approximation with respect to the number of
fading blocks using a numerical method that requires about two orders of
magnitude fewer Monte-Carlo samples than with the saddlepoint approximation
with respect to the number of channel uses per block.
