The pruned thin-product program at c = 21.2733, θ = 0.111731 solves the thin matrix product on all instances

Procedure $74$ (allInstancesF) of the program programF solves the thin matrix product task thinTask within allInstancesTimeF c₃₀ steps and with need allInstancesNeedF:

for an instance $x$ with parameters $N, D, w, U$ (two integer matrices $X\in\mathbb Z^{N\times D}$, $Y\in\mathbb Z^{D\times N}$ with entries bounded by $U$ and $w$ wanted positions $(i_k,j_k)$), the call writes the $w$ entries $\sum_{t<D}X_{i_k t}Y_{t j_k}$ to the output area and changes no other cell below the free pointer. The time is $30000+t_{\mathrm{off}}(N,D,w)$ when $D^{1249}\le N^{71}$, where $t_{\mathrm{off}}$ is the time tOffline32P of the offline routine of Corollary 32 with pruned encodings at the rational parameters ratParamsF ($c=212733/10000$, $\theta=111731/10^6$), and $30000+40(w+1)(D+1)$ (brute force) otherwise.

programF is program31P ratParamsF (the programs of Section 4 with pruned encodings, Corollaries 31-32) followed by allInstancesF (number 74), which tests $D^{1249}\le N^{71}$ with regimeTestF (number 75).

“Procedure $p$ of program $P$ solves the task within $T$ steps with need $\nu$” (SolvesN) means: in every program that begins with $P$, for every instance and every memory satisfying the task's precondition, and all limits (word size, cells, call depth) allowing $\nu$ of the instance's parameters, the call of $p$ with the instance's arguments ends within $T$ of the parameters steps and leaves a memory satisfying the postcondition.