The explicit bound of the first form is stated for the words that leave room for the table. Fixed-parameter tractability is stated for every word whose entries fit, and there the table may not be addressable: a machine with $2 ^ w$ cells cannot hold $(p_max + 1) ^ m$ entries when that exceeds $2 ^ w$, although every entry of the word is a word.

The program first computes $min ((p_max + 1) ^ m) (|x| + 1)$ by repeated multiplication, testing each step by a division so that no intermediate value exceeds $|x| + 1$, and compares it with $|x|$.

If the table has at most $|x|$ entries, the sweep of the first form runs unchanged, within $1000 · (p_max+1)^m · (m+1) · (|x|+1)$ instructions.

Otherwise $|x| < (p_max + 1) ^ m ≤ (k + 1) ^ k$, where $k = m + p_max$ is the parameter, so the word, and with it the number of jobs $n$, is bounded by a function of the parameter alone. The program then tries every schedule. An odometer of $n$ digits in base $m + 1$ runs through them, each digit naming a machine or rejecting the job. Each string is checked against the word: every job on an eligible machine, no two overlapping jobs on one machine, and the weight added up with a cap at the threshold. There are $(m + 1) ^ n$ strings, each tested in time polynomial in the length, and $n ≤ |x|$.

The function of the parameter is $g k = 4000 · (k+1) · (k+1)^k · (k+1)^((k+1)^k)$, much larger than the $(m · p_max)^{2m}$ of the first form, which is the cost where the table fits.