Relaxing the reciprocal error needed to achieve a fixed quotient error bound
Albert A. Liddicoat · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2003
High-performance arithmetic algorithms are often based on functional iteration and these algorithms do not directly produce a remainder. Without the remainder, rounding often requires additional computation or increased quotient precision. Often multiplicative divide algorithms compute the quotient as the product of the dividend and the reciprocal of the divisor, Q =a x (1/b). Typical rounding techniques require that the quotient error be less than a maximum bound such as 1/2 unit in the last place (ulp). When using normalized floating point numbers the quotient error may be approximately twice as large as the reciprocal error since amax ≈ 2 and Eq ≈ 2 x Er. If the rounding algorithm requires |Eq| < 1/2 ulp, then the reciprocal error bound must be |Er| < 1/4 ulp. This work proposes a quantitative method to relax the reciprocal error bound for normalized floating point numbers to achieve a fixed quotient error bound. The proposed error bound of Er < 1/(2 x b) guarantees the quotient error, Eq < 1/2 ulp and the reciprocal error is in the range of 1/4 to 1/2 ulp. Using the relaxed error bound, the reciprocal error may be greater in the region where it is hardest to compute without increasing the quotient error bound.