Error Using Floating Point Arymetic
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by David Goldberg, published in the March, 1991 issue of Computing Surveys. Copyright 1991, Association for floating point arithmetic error analysis Computing Machinery, Inc., reprinted by permission. Abstract Floating-point arithmetic floating point arithmetic examples is considered an esoteric subject by many people. This is rather surprising because floating-point is floating point arithmetic calculator 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
Floating Point Arithmetic Pdf
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 bash floating point arithmetic 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.0 [Computer Systems Organization]: General -- instruction set design; D.3.4 [Programming Languages]: Processors -- compilers, optimization; G.1.0 [Numerical Analysis]: General -- computer arithmetic, error analysis, numerical algorithms (Secondary) D.2.1 [Software Engineering]: Requirements/Specifications -- languages; D.3.4 Programming Languages]: Formal Definitions and Theory -- semantics; D.4.1 Operating Systems]: Process Management -- synchronization. General Terms: Algorithms, Design, Languages Additional Key Words and Phrases: Denormalized number, exception, floating-point, floating-point standard, gradual underflow, guard digit, NaN, overflow, relative error, rounding error, rounding mode, ulp, underflow. Introduction Builders of computer systems often need information about floating-point arithmetic. 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,
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the Stack Overflow Community Stack Overflow is a community of 4.7 million programmers, just like you, helping each other. Join them; it only takes a minute: Sign up Floating point inaccuracy examples up https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html vote 29 down vote favorite 46 How do you explain floating point inaccuracy to fresh programmers and laymen who still think computers are infinitely wise and accurate? Do you have a favourite example or anecdote which seems to get the idea across much better than an precise, but dry, explanation? How is this taught in Computer Science classes? floating-point floating-accuracy share edited Apr 24 '10 at http://stackoverflow.com/questions/2100490/floating-point-inaccuracy-examples 22:34 community wiki 4 revs, 3 users 57%David Rutten locked by Bill the Lizard May 6 '13 at 12:41 This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. This question and its answers are frozen and cannot be changed. More info: help center. Take a look into this article: What Every Computer Scientist Should Know About Floating-Point Arithmetic –Rubens Farias Jan 20 '10 at 10:17 1 You can comprove this with this simple javascript:alert(0.1*0.1*10); –user216441 Apr 24 '10 at 23:07 comments disabled on deleted / locked posts / reviews| 7 Answers 7 active oldest votes up vote 26 down vote accepted There are basically two major pitfalls people stumble in with floating-point numbers. The problem of scale. Each FP number has an exponent which determines the overall “scale” of the number so you can represent either really small values or really larges ones, though the number of digits you can devote for that is limited. Adding two numbers of different scale will sometimes result in the
asked for help on http://effbot.org/pyfaq/why-are-floating-point-calculations-so-inaccurate.htm some forum and got pointed to a long article with lots of formulas that floating point didn’t seem to help with your problem. Well, this site is here to: Explain concisely why you get that unexpected result Tell you how to deal with floating point arithmetic this problem If you’re interested, provide in-depth explanations of why floating-point numbers have to work like that and what other problems can arise You should look at the Basic Answers first - but don’t stop there! © Published at floating-point-gui.de under the Creative Commons Attribution License (BY) The Floating-Point Guide Home Basic Answers References xkcd Number Formats Binary Fractions Floating-Point Exact Types On Using Integers Errors Rounding Comparison Propagation Languagecheat sheets C# Java JavaScript Perl PHP Python Ruby SQL
Python. It's not. It's a problem caused by the internal representation of floating point numbers, which uses a fixed number of binary digits to represent a decimal number. Some decimal numbers can't be represented exactly in binary, resulting in small roundoff errors.In decimal math, there are many numbers that can't be represented with a fixed number of decimal digits, e.g. 1/3 = 0.3333333333…….In base 2, 1/2 = 0.1, 1/4 = 0.01, 1/8 = 0.001, etc. .2 equals 2/10 equals 1/5, resulting in the binary fractional number 0.001100110011001…Floating point numbers only have 32 or 64 bits of precision, so the digits are cut off at some point, and the resulting number is 0.199999999999999996 in decimal, not 0.2.A floating point's repr function prints as many digits are necessary to make eval(repr(f)) == f true for any float f. The str function prints fewer digits and this often results in the more sensible number that was probably intended:>>> 0.2 0.20000000000000001 >>> print 0.2 0.2Again, this has nothing to do with Python, but with the way the underlying C platform handles floating point numbers, and ultimately with the inaccuracy you'll always have when writing down numbers as a string of a fixed number of digits.One of the consequences of this is that it is dangerous to compare the result of some computation to a float with == ! Tiny inaccuracies may mean that == fails. Instead, you have to check that the difference between the two numbers is less than a certain threshold:epsilon = 0.0000000000001 # Tiny allowed error expected_result = 0.4 if expected_result-epsilon <= computation() <= expected_result+epsilon: ...Please see the chapter on floating point arithmetic (dead link) in the Python tutorial for more information.CATEGORY: general::: effbot.org::: pyfaq ::: rendered by a django application. hosted by webfaction.