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Floating point verification in HOL Light: the exponential function
 UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
, 1997
"... Since they often embody compact but mathematically sophisticated algorithms, operations for computing the common transcendental functions in floating point arithmetic seem good targets for formal verification using a mechanical theorem prover. We discuss some of the general issues that arise in veri ..."
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Cited by 31 (6 self)
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Since they often embody compact but mathematically sophisticated algorithms, operations for computing the common transcendental functions in floating point arithmetic seem good targets for formal verification using a mechanical theorem prover. We discuss some of the general issues that arise in verifications of this class, and then present a machinechecked verification of an algorithm for computing the exponential function in IEEE754 standard binary floating point arithmetic. We confirm (indeed strengthen) the main result of a previously published error analysis, though we uncover a minor error in the hand proof and are forced to confront several subtle issues that might easily be overlooked informally. The development described here includes, apart from the proof itself, a formalization of IEEE arithmetic, a mathematical semantics for the programming language in which the algorithm is expressed, and the body of pure mathematics needed. All this is developed logically from first prin...
Handling FloatingPoint Exceptions in Numeric Programs
 ACM Transactions on Programming Languages and Systems
, 1996
"... 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 Languag ..."
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Cited by 21 (0 self)
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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 FloatingPoint 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...
Software Needs in Special Functions
 J. Comput. Appl. Math
, 1996
"... . Currently available software for special functions exhibits gaps and defects in comparison to the needs of modern highperformance scientific computing and also, surprisingly, in comparison to what could be constructed from current algorithms. In this paper we expose some of these deficiencies and ..."
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Cited by 7 (1 self)
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. Currently available software for special functions exhibits gaps and defects in comparison to the needs of modern highperformance scientific computing and also, surprisingly, in comparison to what could be constructed from current algorithms. In this paper we expose some of these deficiencies and identify the related need for useroriented testing software. 1. Introduction A recent article by Lozier and Olver [21] provides a survey of algorithms and software for the numerical evaluation of special functions. Its emphasis is on the generation of function values although selected resources for zeros and integrals are included also. Journals, books, conference proceedings, and software documents were examined and a bibliography of nearly 500 references was constructed. Based on this investigation, the functions were classified and crossreferenced to bibliographic entries and to specific software libraries and systems 1 . The bibliography was prepared using the authors' professional...
Floatingpoint verification using theorem proving
 Formal Methods for Hardware Verification, 6th International School on Formal Methods for the Design of Computer, Communication, and Software Systems, SFM 2006, volume 3965 of Lecture Notes in Computer Science
, 2006
"... Abstract. This chapter describes our work on formal verification of floatingpoint algorithms using the HOL Light theorem prover. 1 ..."
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Cited by 7 (1 self)
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Abstract. This chapter describes our work on formal verification of floatingpoint algorithms using the HOL Light theorem prover. 1
ErrorBounding in LevelIndex Computer Arithmetic
 in Numerical Methods and Error
, 1966
"... . This paper proposes the use of levelindex (LI) and symmetric levelindex (SLI) computer arithmetic for practical computation with error bounds. Comparisons are made with floatingpoint and several advantages are identified. 1 Introduction Any approach to the general problem of assessing the tot ..."
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. This paper proposes the use of levelindex (LI) and symmetric levelindex (SLI) computer arithmetic for practical computation with error bounds. Comparisons are made with floatingpoint and several advantages are identified. 1 Introduction Any approach to the general problem of assessing the total error in the output of computer programs depends on a detailed understanding of the computer arithmetic. The finite precision of the arithmetic gives rise to rounding errors that can be an important component of the total error. Accordingly, much effort has gone into refining the algorithms and circuitry that carry out floatingpoint arithmetic. One goal of this effort has been to minimize rounding errors. Another was to ensure that exceptional conditions, such as underflow and overflow, are detected and reported because their occurrence can completely invalidate the results of a computation. The present state of floatingpoint hardware design [5] is close to optimal, and so the question a...
Basic Linear Algebra Operations In Sli Arithmetic
"... . Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or pro ..."
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. Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or products, and a SIMD software implementation of some of these algorithms is described. The main purpose of this paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations. 1. Introduction This paper reports on a continuing project to develop, implement and apply parallel algorithms for SLI (symmetric levelindex) arithmetic. The algorithms are being developed with a view toward a possible future implementation in hardware but at this stage they are being coded for a particular SIMD (single instruction, multiple data) computer system, a DEC MasPar MP1 1 . The algorithms cover individual arithmetic operations and extensions to the BLAs (basic linear alge...
Basic Linear Algebra Operations in SLI Arithmetic
"... . Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or pro ..."
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. Symmetric levelindex arithmetic was introduced to overcome recognized limitations of floatingpoint systems, most notably overflow and underflow. The original recursive algorithms for arithmetic operations could be parallelized to some extent, particularly when applied to extended sums or products, and a SIMD software implementation of some of these algorithms is described. The main purpose of this paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations. 1. Introduction This paper reports on a continuing project to develop, implement and apply parallel algorithms for SLI (symmetric levelindex) arithmetic. The algorithms are being developed with a view toward a possible future implementation in hardware but at this stage they are being coded for a particular SIMD (single instruction, multiple data) computer system, a DEC MasPar MP1 1 . The algorithms cover individual arithmetic operations and extensions to the BLAs (basic linear alge...