Floating point numbers are real numbers

TL;DR

This paper explores how continuous mathematics can provide new insights and results into the evaluation of floating point operations like sums, square roots, and dot products on digital computers.

Contribution

It demonstrates how continuous mathematical analysis can yield sharp, simple, and novel results about floating point arithmetic operations.

Findings

New mathematical results on floating point sum evaluation

Insights into floating point square root computations

Analysis of dot product accuracy in floating point arithmetic

Abstract

Floating point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which continuous mathematics leads to sharp, simple and new results about the evaluation of sums, square roots and dot products in floating point arithmetic.