Constructing High Radix Quotient Digit Selection Tables for SRT Division and Square Root

Abstract

High radix SRT division plays an important role in contemporary microprocessors as the quotient digit selection tables effectively reduce the computation complexity of the quotient digits. The quotient digit selection table is constructed according to the rounded lower and upper bounds of the overlapping regions in the traditional method. The table construction process lacks in mathematical rigor and consequently is susceptible to error. This paper proposes an algebraic method for computing the quotient digit selection tables. We characterize the quotient digit selection functions to construct quotient digit selection tables required for SRT division and SRT square root with any valid redundancy. The functions include the maximum and minimum legal quotient digit selections. We compute the truncations of the remainder <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p$</tex-math></inline-formula> and divisor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d \in [1,2)$</tex-math></inline-formula> . We implement procedures to compute quotient digit selection tables by our functions. The computation of the quotient digit selection table for the case radix-4 with the quotient digit set <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[-2,2]$</tex-math></inline-formula> is presented by using the minimum quotient digit selection function. Our functions can compute quotient digit selection tables in the design phase of SRT division and square root by given radix and redundancy.