Faculty of Engineering and Information Technologies, School of Electrical and Information Engineering
Abstract
Computer arithmetic is one of the more important topics within computer science and engineering. The earliest implementations of computer systems were designed to perform arithmetic operations and cost if not all digital systems will be required to perform some sort of arithmetic as part of their normal operations. This reliance on the arithmetic operations of computers means the accurate representation of real numbers within digital systems is vital, and an understanding of how these systems are implemented and their possible drawbacks is essential in order to design and implement modern high performance systems. At present the most widely implemented system for computer arithmetic is the IEEE754 Floating Point system, while this system is deemed to the be the best available implementation it has several features that can result in serious errors of computation if not implemented correctly. Lack of understanding of these errors and their effects has led to real world disasters in the past on several occasions. Systems for the detection of these errors are highly important and fast, efficient and easy to use implementations of these detection systems is a high priority. Detection of floating point rounding errors normally requires run-time analysis in order to be effective. Several systems have been proposed for the analysis of floating point arithmetic including Interval Arithmetic, Affine Arithmetic and Monte Carlo Arithmetic. While these systems have been well studied using theoretical and software based approaches, implementation of systems that can be applied to real world situations has been limited due to issues with implementation, performance and scalability. The majority of implementations have been software based and have not taken advantage of the performance gains associated with hardware accelerated computer arithmetic systems. This is especially problematic when it is considered that systems requiring high accuracy will often require high performance. The aim of this thesis and associated research is to increase understanding of error and error analysis methods through the development of easy to use and easy to understand implementations of these techniques
