Generation and analysis of hard to round cases for binary floating point division

We investigate two sets of hard to round p×p bit fractions arising from division of a normalized p bit floating point dividend by a normalized p bit floating point divisor. These sets are characterized by the p×p bit fraction´s quotient bit string, beginning with or just after the round bit, having the maximum number (p-1) of repeating like bits, specifically 00…01 or 11…10 for the directed rounding “RD-hard” set and 100…01 or 11…10 for the round-to-nearest “RN_-hard” set. We show both the p×p bit RD-hard and RN-hard sets to be of size at least 2^p-2 and at most 2^p-1. Two dimensional quotient vs. divisor plots empirically reveal both the RD-hard and RN-hard sets of p×p bit fractions to be jointly widely distributed. Analysis of patterns and linear sequences of fractions visible in the quotient vs. divisor plots leads to simplified procedures for generating test suites of hard to round fractions. Our strongest computational result is the derivation of formulas that allow 2^(p/2)+O(1) RD-hard and RN-hard p×p bit fractions to be enumerated based on sequential incrementation of respective numerators and denominators