Artcorusse » Artistes contemporains. Tissu cr. Tout d’abord, dans le sens pratique de production couleur et la reproduction, le deuxi. Il est une illustration de la collaboration des artistes et couturiers sur les harmonies des associations de couleurs afin de cr. Artistes ne voient pas la diff. Il ya beaucoup d’illustrations vives de l’histoire d’amour s. Jeud 2. 1h. 30; Lun. Merc. 8h. 00 et 2. ADSL Alice. Box : Cha. Le Pilori 2009-08 Vade retro Satan! Devenez exorciste en faisant un don au Pilori et, pour une somme d Toutes Les Partitions de Musique (Affichage 1401 Serge Hutin (1927-1997) est l'un des meilleurs vulgarisateurs de l'occultisme et du paranormal. Exposition du peintre Nikola. Liste des films: UNE GOURMANDE Caramella-1995 FERMO POSTA TINTO BRASS Fermo posta Tinto Brass-1995 KREOLA Kreola-1993 COSI FAN TUTTE Cosi fan tutte-1992.
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What Every Computer Scientist Should Know About Floating- Point Arithmetic. Appendix D What Every Computer Scientist Should Know About Floating- Point Arithmetic. Note – This appendix is an edited reprint of the paper What Every Computer Scientist Should Know About Floating- Point Arithmetic, by David Goldberg, published in the March, 1. Computing Surveys. Copyright 1. 99. 1, Association for Computing Machinery, Inc., reprinted by permission. Abstract. Floating- point arithmetic is considered an esoteric subject by many people. This is rather surprising because floating- point is ubiquitous in computer systems. Almost every language has a floating- point datatype; computers from PCs to supercomputers have floating- point accelerators; most compilers will be called upon to compile floating- point algorithms from time to time; and virtually every operating system must respond to floating- point exceptions such as overflow. This paper presents a tutorial on those aspects of floating- point that have a direct impact on designers of computer systems. It begins with background on floating- point representation and rounding error, continues with a discussion of the IEEE floating- point standard, and concludes with numerous examples of how computer builders can better support floating- point. Categories and Subject Descriptors: (Primary) C. There are, however, remarkably few sources of detailed information about it. One of the few books on the subject, Floating- Point Computation by Pat Sterbenz, is long out of print. This paper is a tutorial on those aspects of floating- point arithmetic (floating- point hereafter) that have a direct connection to systems building. It consists of three loosely connected parts. The first section, Rounding Error, discusses the implications of using different rounding strategies for the basic operations of addition, subtraction, multiplication and division. It also contains background information on the two methods of measuring rounding error, ulps and relativeerror. The second part discusses the IEEE floating- point standard, which is becoming rapidly accepted by commercial hardware manufacturers. Included in the IEEE standard is the rounding method for basic operations. The discussion of the standard draws on the material in the section Rounding Error. The third part discusses the connections between floating- point and the design of various aspects of computer systems. Information, news, entertainment, shopping, homes, classifieds, restaurants, hotels and weather for Richmond, Virginia. Print yourself your onward or return flight. Print your own onward or return flight ticket with this free service. An onward ticket can be required, based on the countries entry requirements. Roundcube, Hostedemail, Exchange.Topics include instruction set design, optimizing compilers and exception handling. 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Eliminate the stress of searching for the right science project supplies—try a kit from the Science Buddies Store. Get the latest NFL news, video updates, scores, schedules, standings and more on Sporting News. We have a number of resources ranging from video discussions with our scientists, virtual tours through google hangouts to live streaming of our public lectu. In particular, the proofs of many of the theorems appear in this section. The end of each proof is marked with the z symbol. When a proof is not included, the z appears immediately following the statement of the theorem. Rounding Error Squeezing infinitely many real numbers into a finite number of bits requires an approximate representation. Although there are infinitely many integers, in most programs the result of integer computations can be stored in 3. In contrast, given any fixed number of bits, most calculations with real numbers will produce quantities that cannot be exactly represented using that many bits. Therefore the result of a floating- point calculation must often be rounded in order to fit back into its finite representation. This rounding error is the characteristic feature of floating- point computation. The section Relative Error and Ulps describes how it is measured. Since most floating- point calculations have rounding error anyway, does it matter if the basic arithmetic operations introduce a little bit more rounding error than necessary? That question is a main theme throughout this section. The section Guard Digits discusses guard digits, a means of reducing the error when subtracting two nearby numbers. Guard digits were considered sufficiently important by IBM that in 1. System/3. 60 architecture (single precision already had a guard digit), and retrofitted all existing machines in the field. Two examples are given to illustrate the utility of guard digits. The IEEE standard goes further than just requiring the use of a guard digit. It gives an algorithm for addition, subtraction, multiplication, division and square root, and requires that implementations produce the same result as that algorithm. Thus, when a program is moved from one machine to another, the results of the basic operations will be the same in every bit if both machines support the IEEE standard. This greatly simplifies the porting of programs. Other uses of this precise specification are given in Exactly Rounded Operations. Floating- point Formats Several different representations of real numbers have been proposed, but by far the most widely used is the floating- point representation. Floating- point representations have a base (which is always assumed to be even) and a precision p. If = 1. 0 and p = 3, then the number 0. If = 2 and p = 2. In general, a floating- point number will be represented as . Two other parameters associated with floating- point representations are the largest and smallest allowable exponents, emax and emin. Since there are p possible significands, and emax - emin + 1 possible exponents, a floating- point number can be encoded inbits, where the final +1 is for the sign bit. The precise encoding is not important for now. There are two reasons why a real number might not be exactly representable as a floating- point number. The most common situation is illustrated by the decimal number 0. Although it has a finite decimal representation, in binary it has an infinite repeating representation. Thus when = 2, the number 0. A less common situation is that a real number is out of range, that is, its absolute value is larger than . Most of this paper discusses issues due to the first reason. However, numbers that are out of range will be discussed in the sections Infinity and Denormalized Numbers. Floating- point representations are not necessarily unique. For example, both 0. If the leading digit is nonzero (d. The floating- point number 1. When = 2, p = 3, emin = - 1 and emax = 2 there are 1. FIGURE D- 1. The bold hash marks correspond to numbers whose significand is 1. Requiring that a floating- point representation be normalized makes the representation unique. Unfortunately, this restriction makes it impossible to represent zero! A natural way to represent 0 is with 1. For example, the expression (2. Consider the floating- point format with = 1. If the result of a floating- point computation is 3. Similarly, if the real number . In general, if the floating- point number d. Another way to measure the difference between a floating- point number and the real number it is approximating is relative error, which is simply the difference between the two numbers divided by the real number. For example the relative error committed when approximating 3. To compute the relative error that corresponds to . This error is ((/2)- p) . Since numbers of the form d. That is, (2) In particular, the relative error corresponding to . This factor is called the wobble. Setting = (/2)- p to the largest of the bounds in (2) above, we can say that when a real number is rounded to the closest floating- point number, the relative error is always bounded by e, which is referred to as machine epsilon. In the example above, the relative error was . In order to avoid such small numbers, the relative error is normally written as a factor times , which in this case is = (/2)- p = 5(1. Thus the relative error would be expressed as (. To illustrate the difference between ulps and relative error, consider the real number x = 1. It is approximated by. The error is 0. 5 ulps, the relative error is 0. Next consider the computation 8. The exact value is 8x = 9. The error is now 4. The error measured in ulps is 8 times larger, even though the relative error is the same. In general, when the base is , a fixed relative error expressed in ulps can wobble by a factor of up to . And conversely, as equation (2) above shows, a fixed error of . The most natural way to measure rounding error is in ulps. For example rounding to the nearest floating- point number corresponds to an error of less than or equal to . However, when analyzing the rounding error caused by various formulas, relative error is a better measure. A good illustration of this is the analysis in the section Theorem 9. Since can overestimate the effect of rounding to the nearest floating- point number by the wobble factor of , error estimates of formulas will be tighter on machines with a small . When only the order of magnitude of rounding error is of interest, ulps and may be used interchangeably, since they differ by at most a factor of . For example, when a floating- point number is in error by n ulps, that means that the number of contaminated digits is log n. If the relative error in a computation is n, then (3) contaminated digits log n. Guard Digits One method of computing the difference between two floating- point numbers is to compute the difference exactly and then round it to the nearest floating- point number. This is very expensive if the operands differ greatly in size. Assuming p = 3, 2. Rather than using all these digits, floating- point hardware normally operates on a fixed number of digits. Suppose that the number of digits kept is p, and that when the smaller operand is shifted right, digits are simply discarded (as opposed to rounding). Take another example: 1. This becomes x = 1. How bad can the error be? Theorem 1 Using a floating- point format with parameters andp, and computing differences using pdigits, the relative error of the result can be as large as- 1. Proof. A relative error of - 1 in the expression x - y occurs when x = 1. Here y has p digits (all equal to ). Strip. Os blog. Gavran opet leti! Na zatvaranju drugog izdanja viroviti. Iako se broj izdanja stavljen u odnosu na godine postojanja . 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