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

摘要

Suppose an m-digit floating-point arithmetic in base beta >= 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 beta. 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.