  In this paper, we are interested in the performance of a variable-length
stop-feedback (VLSF) code with $m$ optimal decoding times for the binary-input
additive white Gaussian noise channel. We first develop tight approximations on
the tail probability of length-$n$ cumulative information density. Building on
the work of Yavas \emph{et al.}, for a given information density threshold, we
formulate the integer program of minimizing the upper bound on average
blocklength over all decoding times subject to the average error probability,
minimum gap and integer constraints. Eventually, minimization of locally
minimum upper bounds over all thresholds will yield the globally minimum upper
bound and this is called the two-step minimization. For the integer program, we
present a greedy algorithm that yields possibly suboptimal integer decoding
times. By allowing a positive real-valued decoding time, we develop the
gap-constrained sequential differential optimization (SDO) procedure that
sequentially produces the optimal, real-valued decoding times. We identify the
error regime in which Polyanskiy's scheme of stopping at zero does not improve
the achievability bound. In this error regime, the two-step minimization with
the gap-constrained SDO shows that a finite $m$ suffices to attain Polyanskiy's
bound for VLSF codes with $m = \infty$.
