Recently, a fixed compute-resource hardware architecture was proposed to enable the iterative solution of systems of linear equations to arbitrary accuracies. This technique, named ARCHITECT, achieves exact numeric computation by using online arithmetic to allow the refinement of results from earlier iterations over time, eschewing rounding error. ARCHITECT has a key drawback, however: often, many more digits than strictly necessary are generated, with this problem exacerbating the more accurate a solution is sought. In this paper, we infer the locations of these superfluous digits within stationary iterative calculations by exploiting online arithmetic’s digit dependencies and using forward error analysis. We demonstrate that their lack of computation is guaranteed not to affect the ability to reach a solution of any accuracy. Versus ARCHITECT, our illustrative hardware implementation achieves a geometric mean 20.1x speedup in the solution of a set of representative linear systems through the avoidance of redundant digit calculation. For the calculation of high-precision results, we also obtain an up-to 22.4x memory requirement reduction over the same architecture. Finally, we demonstrate that iterative solvers implemented following our proposals show superiority over conventional arithmetic implementations by virtue of their runtime-tunable precisions.