Tight upper bounds on the minimum precision required of the divisor and the partial remainder in high-radix division

Abstract

Digit-recurrence binary dividers are sped up via two complementary methods: keeping the partial remainder in redundant form and selecting the quotient digits in a radix higher than 2. Use of a redundant partial remainder replaces the standard addition in each cycle by a carry-free addition, thus making the cycles shorter. Deriving the quotient in high radix reduces the number of cycles (by a factor of about h for radix 2/sup h/). To make the redundant partial remainder scheme work, quotient digits must be chosen from a redundant set, such as [-2, 2] in radix 4. The redundancy provides some tolerance to imprecision so that the quotient digits can be selected based on examining truncated versions of the partial remainder and divisor. No closed form formula for the required precision in the partial remainder and divisor, as a function of the quotient digit set and the range of the partial remainder, is known. We establish tight upper bounds on the required precision for the partial remainder and divisor. The bounds are tight in the sense that each is only one bit over a well-known simple lower bound. We also discuss the implications of these bounds for the quotient digit selection process.