Measuring the accuracy of ROM reciprocal tables

It is proved that a conventional ROM reciprocal table construction algorithm generates tables that minimize the relative error. The worst case relative errors realized for such optimally computed k-bits-in, m-bits-out ROM reciprocal tables are then determined for all table sizes 3 ⩽ k, m ⩽ 12. It is then proved that the table construction algorithm always generates a k-bits-in, k-bits-out table with relative errors never any greater than 3(2^-k)/4 for any k, and, more generally with g guard bits, that for (k + g)-bits-out the relative error is never any greater than 2^-(k+1)(1 + 1/(2^g+1)). To provide for determining test data without prior construction of a full ROM reciprocal table, a procedure that requires generation and searching of only a small portion of such a table to determine regions containing input data yielding the worst case relative errors is described