Pipelined multiplicative division with IEEE rounding

We propose optimized pipelined implementations for Goldschmidt´s division algorithm with IEEE rounding based on Booth radix-8 multiplication. Compared to other FP-division algorithms, our implementations require fewer clock cycles and admit shorter clock periods. The considered optimizations for the quotient approximation are based on a careful general analysis of tight error bounds for the implementation and are accompanied by the utilization of redundant representations, partial compressions, injection-based rounding, and rectangular multipliers for the internal computations. To efficiently achieve IEEE compliant rounding, we introduce the concept of dew-point rounding that allows efficient implementation and reduced requirements for the quotient approximation. On this basis, we propose the implementation of different versions of Goldschmidt´s division algorithm with different pipeline depths. None of these implementations requires a full-sized multiplier at any stage of the computations. In this way we reduce latency, cost, and enable increased throughput at a reasonable cost. We suggest a full range of pipelining depths: On one extreme is a 3-stage pipeline with a restart time that simply equals the latency minus the number of pipeline stages. On the other extreme is a fully pipelined design.