IEEE754 Precision- k base-β Arithmetic Inherited by Precision- m Base-β Arithmetic for k < m

Abstract

Suppose an m -digit floating-point arithmetic in base β ≥ 2 following the IEEE754 arithmetic standard is available. We show how a k -digit arithmetic with k < m can be inherited solely using m -digit operations. This includes the rounding into k digits, the four basic operations and the square root, all for even or odd base β. In particular, we characterize the relation between k and m so that no double rounding occurs when computing in m digits and rounding the result into k digits. We discuss rounding to nearest as well as directed rounding, and our approach covers exceptional values including signed zero. For binary arithmetic, a Matlab toolbox based on binary64 including k -bit scalar, vector and matrix operations as well as k -bit interval arithmetic is part of Version 8 of INTLAB, the Matlab toolbox for reliable computing.