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Floating point arithmetics generally possess many regularity properties in addition to those that are typically used in roundoff error analyses; these properties can be exploited to produce computations that are more accurate and cost effective than many programmers might think possible. Furthermore, many of these properties are quite simple to state and to comprehend, but few programmers seem to be aware of them (or at least willing to rely on them). This dissertation presents some of these properties and explores their consequences for computability, accuracy, cost, and portability. For example, we consider several algorithms for summing a sequence of numbers and show that under very general hypotheses, we can compute a sum to full working precision at only somewhat greater cost than a simple accumulation, which can often produce a sum with no significant figures at all. This example, as well as others we present, can be generalized further by substituting still more complex algorith...

Language Constructs Termination exception mechanisms like in Ada and C++ are supposed to terminate an unsuccessful computation as soon as possible after an exception occurs. However, none of the examples of numeric exception handling presented earlier depends ACM Transactions on Programming Languages and Systems, Vol. 18, No. 2, March 1996. Handling Floating-Point Exceptions 167 on the immediate termination of a calculation signaling an exception. The IEEE exception flags scheme actually takes advantage of the fact that an immediate jump is not necessary; by raising a flag, making a substitution, and continuing, the IEEE Standard supports both an attempted/alternate form and a default substitution with a single, simple reponse to exceptions. A detraction of the IEEE flag solution, though, is its obvious lack of structure. Instead of being forced to set and reset flags, one would ideally have available a language construct that more directly reflected the attempted/alternate algorit...

This paper describes a test of a computer's floating-point arithmetic unit. The test has two goals. The first goal deals with the needs of users of computers, and the second goal deals with manufacturers of computers. The first and major goal is to determine if the machine supports a particular mathematical model of computer arithmetic. This model was developed as an aid in the design, analysis, implementation and testing of portable, high-quality numerical software. If a computer supports the arithmetic model, then software written using the model will perform correctly and to specified accuracy on that machine. The second goal of the test is to check that the basic operations perform as the manufacturer intended. For example, if division ( x x // yy ) is implemented as a composite operation ( x x × (1//yy) ), then the test should detect that fact. Also, the accuracy lost in such a division due to the extra arithmetic operations can tell the manufacturer whether it has been implemente...

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Abstract. This chapter describes our work on formal verification of floating-point algorithms using the HOL Light theorem prover. 1

ing with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 869-0481, or permissions@acm.org. 2 \Delta A. Cuyt, B. Verdonk and D. Verschaeren 1. INTRODUCTION AND MOTIVATION The IEEE standard [IEEE 1985] for floating-point arithmetic, which became official in 1985 and which we shall refer to as IEEE--754, has been adopted by most major microprocessor manufacturers. Whereas guaranteeing 100% correctness of an IEEE floating-point implementation is hardly feasible, as the famous Intel Pentium bug clearly demonstrated, several good but unrelated tools exist to test different aspects of a floating-point implementation for compliance with the IEEE--754 standard. Concurrent with the adoption of IEEE--754 by the microprocessor ind...

Fortran 77 is the most widely used language for scientific programming. Its long-awaited revision is now called Fortran 90. It was finalized (down to the last editorial detail) on 11 April 1991, published as an ISO Standard in August 1991, and the first compiler is now on the market. This seems an appropriate moment to review its history and explain its advantages. Submitted for publication in Computing. Central Computing Department, Atlas Centre, Rutherford Appleton Laboratory, Oxon OX11 0QX. March 1994. CONTENTS 1 Introduction..................................................................... 1 2 History........................................................................... 1 3 Language evolution ......................................................... 2 4 Array features .................................................................. 2 5 Parallel processing ............................................................ 4 6 Derived data types ...................

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