Tight and Rigorous Error Bounds for Basic Building Blocks of Double-Word Arithmetic

Abstract

We analyze several classical basic building blocks of double-word arithmetic (frequently called “double-double arithmetic” in the literature): the addition of a double-word number and a floating-point number, the addition of two double-word numbers, the multiplication of a double-word number by a floating-point number, the multiplication of two double-word numbers, the division of a double-word number by a floating-point number, and the division of two double-word numbers. For multiplication and division we get better relative error bounds than the ones previously published. For addition of two double-word numbers, we show that the previously published bound was incorrect, and we provide a new relative error bound. We introduce new algorithms for division. We also give examples that illustrate the tightness of our bounds.