Number-theoretic test generation for directed rounding

We present methods to generate systematically the hardest test cases for multiplication, division, and square root subject to directed rounding, essentially extending previous work on number-theoretic floating point testing to rounding modes other than to-nearest. The algorithms focus upon the rounding boundaries of the modes truncate, to-minus infinity, and to-infinity and programs based on them require little beyond exact arithmetic in the working precision to create billions of edge cases. We show that the amount of work required to calculate trial multiplicands pays off in the form of free extra tests due to an interconnection among the operations considered herein. Although these tests do not replace proofs of correctness, they can be used to gain a high degree of confidence that the accuracy requirements as mandated by IEEE Standard 754 have been satisfied.