Isolating critical cases for reciprocals using integer factorization

One approach to testing and/or proving correctness of a floating-point algorithm computing a function f is based on finding input floating-point numbers a such that the exact result f(a) is very close to a "rounding boundary", i.e. a floating-point number or a midpoint between them. We show how to do this for the reciprocal function by utilizing prime factorizations. We present the method and show examples, as well as making a fairly detailed study of its expected and worst-case behavior. We point out how this analysis of reciprocals can be useful in analyzing certain reciprocal algorithms, and also show how the approach can be trivially adapted to the reciprocal square root function.