Effect of guard digits and normalization options on floating point multiplication

Abstract

The effect of two important computer design parameters on floating point multiplication is studied. The parameters are the number of guard digits, and the decision whether the post-arithmetic normalization shift is done before or after rounding. Our results on the accurary of floating point multiplication depend on a stochastic model for computer numbers. There is an interesting interplay between guard digit possibilities and the normalization option. For example, our own first impression was that normalization before rounding would always be preferable to normalization after rounding. Our results show that this is not the case when there is only one guard digit. In a related vein, our results also give comprehensive justification of the following observation: On an even base machine whose architecture provides for normalization before rounding, one should not merely ask for symmetric rounding (rather than rounding by chopping). Instead one should ask for symmetric roundingplus two guard digits.