Lecture 4 Arithmetic Techniques §1. More Accurate Machine Arithmetic

Since numerical error is the source of non-robustness, it is natural to try to improve upon the standard floating point arithmetic. Can we improve on the conventional floating point arithmetic, which has already reached a highly optimal design under the IEEE Standard? More generally, what techniques are available in the FP mode of computation? In conventional floating point systems, the number of bits allocated to the mantissa and the exponent are fixed. Matsui and Iri [21] proposed a system where these can vary. But let us look at a remarkable proposal of Clenshaw, Olver and Turner [5] called level-index arithmetic (LI). First, any non-negative real number x is assigned a unique level l(x) ∈ N defined by the property that ln(x) ∈ [0, 1), where ln(x) denotes n applications of the natural logarithm ln. The index of x defined by i(x) := ln(x). The function ψ(x) = l(x) + i(x) is called the generalized logarithm; its inverse φ(x) is the generalized exponential defined by