  Incremental redundancy with ACK/NACK feedback produces a variable-length
stop-feedback (VLSF) code constrained to have $m$ decoding times, with an
ACK/NACK feedback to the transmitter at each decoding time. This paper focuses
on the numerical evaluation of the maximal achievable rate of random VLSF codes
as a function of $m$ for the binary-input additive white Gaussian noise
channel, binary symmetric channel, and binary erasure channel (BEC). Leveraging
Edgeworth and Petrov expansions, we develop tight approximations to the tail
probability of length-$n$ cumulative information density that are accurate for
any blocklength $n$. We reduce Yavas et al.'s non-asymptotic achievability
bound on VLSF codes with $m$ decoding times to an integer program of minimizing
the upper bound on the average blocklength subject to the average error
probability, minimum gap, and integer constraints. We develop two distinct
methods to solve this program. Numerical evaluations show that Polyanskiy's
achievability bound for VLSF codes, which assumes $m = \infty$, can be
approached with a small $m$ for all three channels. For BEC, we consider
systematic transmission followed by random linear fountain coding. This allows
us to obtain a new achievability bound stronger than a previous bound and new
VLSF codes whose rate further outperforms Polyanskiy's bound.
