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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...
A proven correctly rounded logarithm in doubleprecision
 In Real Numbers and Computers, Schloss Dagstuhl
, 2004
"... Abstract. This article is a case study in the implementation of a portable, proven and efficient correctly rounded elementary function in doubleprecision. We describe the methodology used to achieve these goals in the crlibm library. There are two novel aspects to this approach. The first is the pr ..."
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Cited by 19 (9 self)
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Abstract. This article is a case study in the implementation of a portable, proven and efficient correctly rounded elementary function in doubleprecision. We describe the methodology used to achieve these goals in the crlibm library. There are two novel aspects to this approach. The first is the proof framework, and in general the techniques used to balance performance and provability. The second is the introduction of processorspecific optimization to get performance equivalent to the best current mathematical libraries, while trying to minimize the proof work. The implementation of the natural logarithm is detailed to illustrate these questions. Mathematics Subject Classification. 2604, 65D15, 65Y99. 1.
Towards the postultimate libm
, 2005
"... This article presents advances on the subject of correctly rounded elementary functions since the publication of the libultim mathematical library developed by Ziv at IBM. This library showed that the average performance and memory overhead of correct rounding could be made negligible. However, the ..."
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Cited by 13 (8 self)
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This article presents advances on the subject of correctly rounded elementary functions since the publication of the libultim mathematical library developed by Ziv at IBM. This library showed that the average performance and memory overhead of correct rounding could be made negligible. However, the worstcase overhead was still a factor 1000 or more. It is shown here that, with current processor technology, this worstcase overhead can be kept within a factor of 2 to 10 of current best libms. This low overhead has very positive consequences on the techniques for implementing and proving correctly rounded functions, which are also studied. These results lift the last technical obstacles to a generalisation of (at least some) correctly rounded double precision elementary functions.
Certifying the floatingpoint implementation of an elementary function using Gappa
 IEEE TRANSACTIONS ON COMPUTERS, 2010. 9 HTTP://DX.DOI.ORG/10.1145/1772954.1772987 10 HTTP://DX.DOI.ORG/10.1145/1838599.1838622 11 HTTP://SHEMESH.LARC.NASA.GOV/NFM2010/PAPERS/NFM2010_14_23.PDF 12 HTTP://DX.DOI.ORG/10.1007/9783642142031_11 13 HTTP://DX.
, 2011
"... High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. This certification may require a timeconsuming proof fo ..."
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Cited by 8 (3 self)
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High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. This certification may require a timeconsuming proof for each line of code, and it is usually broken by the smallest change to the code, e.g., for maintenance or optimization purpose. Certifying floatingpoint programs by hand is, therefore, very tedious and errorprone. The Gappa proof assistant is designed to make this task both easier and more secure, due to the following novel features: It automates the evaluation and propagation of rounding errors using interval arithmetic. Its input format is very close to the actual code to validate. It can be used incrementally to prove complex mathematical properties pertaining to the code. It generates a formal proof of the results, which can be checked independently by a lower level proof assistant like Coq. Yet it does not require any specific knowledge about automatic theorem proving, and thus, is accessible to a wide community. This paper demonstrates the practical use of this tool for a widely used class of floatingpoint programs: implementations of elementary functions in a mathematical library.
Assisted verification of elementary functions
, 2005
"... The implementation of a correctly rounded or interval elementary function needs to be proven carefully in the very last details. The proof requires a tight bound on the overall error of the implementation with respect to the mathematical function. Such work is function specific, concerns tens of lin ..."
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Cited by 5 (2 self)
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The implementation of a correctly rounded or interval elementary function needs to be proven carefully in the very last details. The proof requires a tight bound on the overall error of the implementation with respect to the mathematical function. Such work is function specific, concerns tens of lines of code for each function, and will usually be broken by the smallest change to the code (e.g. for maintenance or optimization purpose). Therefore, it is very tedious and errorprone if done by hand. This article discusses the use of the Gappa proof assistant in this context. Gappa has two main advantages over previous approaches: Its input format is very close to the actual C code to validate, and it automates error evaluation and propagation using interval arithmetic. Besides, it can be used to incrementally prove complex mathematical properties pertaining to the C code. Yet it does not require any specific knowledge about automatic theorem proving, and thus is accessible to a wider community. Moreover, Gappa may generate a formal proof of the results that can be checked independently by a lowerlevel proof assistant like Coq, hence providing an even higher confidence in the certification of the numerical code.
Computing Correctly Rounded Integer Powers in FloatingPoint Arithmetic
"... We introduce several algorithms for accurately evaluating powers to a positive integer in floatingpoint arithmetic, assuming a fused multiplyadd (fma) instruction is available. For bounded, yet very large values of the exponent, we aim at obtaining correctlyrounded results in roundtonearest mod ..."
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Cited by 4 (0 self)
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We introduce several algorithms for accurately evaluating powers to a positive integer in floatingpoint arithmetic, assuming a fused multiplyadd (fma) instruction is available. For bounded, yet very large values of the exponent, we aim at obtaining correctlyrounded results in roundtonearest mode, that is, our algorithms return the floatingpoint number that is nearest the exact value.
Software techniques for perfect elementary functions in floatingpoint interval arithmetic
 IN REAL NUMBERS AND COMPUTERS
, 2006
"... ..."
Fast, Sound and Precise Narrowing of the Exponential Function
, 1996
"... In this paper we present an algorithm for narrowing the constraint y = e x . The algorithm has been designed to be fast by using only IEEE multiplication. The main difficulty is to design algorithms which soundly, rapidly, and precisely compute upper and lower bounds on e x and ln(y). We prove ..."
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Cited by 1 (0 self)
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In this paper we present an algorithm for narrowing the constraint y = e x . The algorithm has been designed to be fast by using only IEEE multiplication. The main difficulty is to design algorithms which soundly, rapidly, and precisely compute upper and lower bounds on e x and ln(y). We prove that our algorithms are correct and produce upper and lower bounds which differ by at most 2 ULP. Experiments indicate that the bounds differ by 1 ULP 99.8% of the time. The method we describe is tabledriven as is Gal's accurate tables method, but the table is easier to construct and hence this method may be applicable to wider classes of functions. 1 Introduction One of the most exciting advances in constraint logic programming over the last few years has been the incorporation of the relational interval arithmetic constraint solving algorithm in CLP systems (see CLP(BNR) [16] and CLP(F) [11]). Relational interval arithmetic is an extensions grows out of earlier work by Moore [15] on inte...
Certifying floatingpoint implementations using Gappa
, 2008
"... High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. Such work may require several lines of proof for each lin ..."
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Cited by 1 (0 self)
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High confidence in floatingpoint programs requires proving numerical properties of final and intermediate values. One may need to guarantee that a value stays within some range, or that the error relative to some ideal value is well bounded. Such work may require several lines of proof for each line of code, and will usually be broken by the smallest change to the code (e.g. for maintenance or optimization purpose). Certifying these programs by hand is therefore very tedious and errorprone. This article discusses the use of the Gappa proof assistant in this context. Gappa has two main advantages over previous approaches: Its input format is very close to the actual C code to validate, and it automates error evaluation and propagation using interval arithmetic. Besides, it can be used to incrementally prove complex mathematical properties pertaining to the C code. Yet it does not require any specific knowledge about automatic theorem proving, and thus is accessible to a wide community. Moreover, Gappa may generate a formal proof of the results that can be checked independently by a lowerlevel proof assistant like Coq, hence providing an even higher confidence in the certification of the numerical code. The article demonstrates the use of this tool on a realsize example, an elementary function with correctly rounded output. 1
Programming Manual, Order Number 241430.
"... patents issued by the U.S. Patent and Trademark Office. Intel has patents covering the use or implementation of processors in combination with other products, e.g., certain computer systems. System and method patents or pending patents, of Intel and others, may apply to these systems. A separate lic ..."
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patents issued by the U.S. Patent and Trademark Office. Intel has patents covering the use or implementation of processors in combination with other products, e.g., certain computer systems. System and method patents or pending patents, of Intel and others, may apply to these systems. A separate license may be required for their use (see Intel Terms and Conditions for details). Specific Intel patents include U.S. patent 4,972,338. Information in this document is provided solely to enable use of Intel products. Intel assumes no liability whatsoever, including infringement of any patent or copyright, for sale and use of Intel products except as provided in Intel’s Terms and Conditions of Sale for such products. Intel Corporation makes no warranty for the use of its products and assumes no responsibility for any errors which may appear in this document nor does it make a commitment to update the information contained herein. Intel retains the right to make changes to these specifications at any time, without notice. Contact your local Intel sales office or your distributor to obtain the latest specifications before placing your product order. MDS is an ordering code only and is not used as a product name or trademark of Intel Corporation. Intel Corporation and Intel's FASTPATH are not affiliated with Kinetics, a division of Excelan, Inc. or its FASTPATH trademark or products. *Other brands and names are the property of their respective owners.