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A Lucid Interval
 American Scientist
, 2003
"... Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardwa ..."
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Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardware is unreliable. The problem is simply that computers are discrete and finite machines, and they cannot cope with some of the continuous and infinite aspects of mathematics. Even an innocentlooking number like 1 ⁄10 can cause no end of trouble: In most cases, the computer cannot even read it in or print it out exactly, much less perform exact calculations with it. Errors caused by these limitations of digital machines
Taylor Forms  Use and Limits
 Reliable Computing
, 2002
"... This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 19 ..."
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This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.
Rigorous and portable standard functions
 BIT
"... Abstract. Today’s floating point implementations of elementary transcendental functions are usually very accurate. However, with few exceptions, the actual accuracy is not known. In the present paper we describe a rigorous, accurate, fast and portable implementation of the elementary standard functi ..."
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Abstract. Today’s floating point implementations of elementary transcendental functions are usually very accurate. However, with few exceptions, the actual accuracy is not known. In the present paper we describe a rigorous, accurate, fast and portable implementation of the elementary standard functions based on some existing approximate standard functions. The scheme is outlined for IEEE 754, but not difficult to adapt to other floating point formats. A Matlab implementation is available on the net. Accuracy of the proposed algorithms can be rigorously estimated. As an example we prove that the relative accuracy of the exponential function is better than 2.07eps in a slightly reduced argument range (eps denoting the relative rounding error unit). Otherwise, extensive computational tests suggest for all elementary functions and all suitable arguments an accuracy better than about 3eps. 1. A general approach for rigorous standard functions. Todays libraries for the approximation of elementary transcendental functions are very fast and the results are mostly of very high accuracy. For a good introduction and summary of stateoftheart methods cf. [19]. The achieved accuracy does not exceed one or two ulp for almost all input arguments; however, there is no proof for that. Today computers are more and more used for socalled computerassisted proofs, where assumptions of mathematical theorems are verified on the computer in order to draw anticipated conclusions. Famous examples are the celebrated Kepler conjecture [9], the enclosure of the Feigenbaum constant [8], bounds for
COMPUTING SCIENCE A LUCID INTERVAL
"... Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardwa ..."
Abstract
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Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardware is unreliable. The problem is simply that computers are discrete and finite machines, and they cannot cope with some of the continuous and infinite aspects of mathematics. Even an innocentlooking number like 1 ⁄10 can cause no end of trouble: In most cases, the computer cannot even read it in or print it out exactly, much less perform exact calculations with it. Errors caused by these limitations of digital machines are usually small and inconsequential, but sometimes every bit counts. On February 25, 1991, a Patriot missile battery assigned to protect a military installation at Dahrahn, Saudi Arabia, failed to intercept a Scud missile, and the malfunction was blamed on an error in computer arithmetic. The Patriot’s control system kept track of time by counting tenths of a second; to convert the count into full seconds, the computer multiplied by 1 ⁄10. Mathematically, the procedure is unassailable, but computationally it was disastrous. Because the decimal fraction 1 ⁄10 has no exact finite representation in binary notation, the computer had to approximate. Apparently, the conversion constant stored in the program was the 24bit binary fraction 0.00011001100110011001100, which is too small by a factor of about one tenmillionth. The discrepancy sounds tiny, but over four days it built up to about a third of a second. In combination with other peculiarities of the control software, the inaccuracy caused a miscalculation of almost 700 meters in the predicted position of the incoming missile. Twentyeight soldiers died. Of course it is not to be taken for granted that better arithmetic would have saved those 28 lives. (Many other Patriots failed for unrelated reasons; some analysts doubt whether any Scuds were stopped by Patriots.) And surely the underlying problem was not the slight drift in the clock but a design vulnerable to such minor timing