Floating Point Arithmetic Error
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base 2 (binary) fractions. For example, the decimal fraction 0.125 has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction 0.001 has value 0/2 + 0/4 floating point python + 1/8. These two fractions have identical values, the only real difference floating point arithmetic examples being that the first is written in base 10 fractional notation, and the second in base 2. Unfortunately, most floating point rounding error decimal fractions cannot be represented exactly as binary fractions. A consequence is that, in general, the decimal floating-point numbers you enter are only approximated by the binary floating-point numbers actually stored floating point rounding error example in the machine. The problem is easier to understand at first in base 10. Consider the fraction 1/3. You can approximate that as a base 10 fraction: 0.3 or, better, 0.33 or, better, 0.333 and so on. No matter how many digits you're willing to write down, the result will never be exactly 1/3, but will be an increasingly better approximation of
Floating Point Numbers Explained
1/3. In the same way, no matter how many base 2 digits you're willing to use, the decimal value 0.1 cannot be represented exactly as a base 2 fraction. In base 2, 1/10 is the infinitely repeating fraction 0.0001100110011001100110011001100110011001100110011... Stop at any finite number of bits, and you get an approximation. On a typical machine running Python, there are 53 bits of precision available for a Python float, so the value stored internally when you enter the decimal number 0.1 is the binary fraction 0.00011001100110011001100110011001100110011001100110011010 which is close to, but not exactly equal to, 1/10. It's easy to forget that the stored value is an approximation to the original decimal fraction, because of the way that floats are displayed at the interpreter prompt. Python only prints a decimal approximation to the true decimal value of the binary approximation stored by the machine. If Python were to print the true decimal value of the binary approximation stored for 0.1, it would have to display >>> 0.1 0.1000000000000000055511151231257827021181583404541015625 That is more digits than most people find useful, so Python keeps the number of digits manageab
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Floating Point Ieee
more about Stack Overflow the company Business Learn more about hiring developers or posting what is a float python ads with us Stack Overflow Questions Jobs Documentation Tags Users Badges Ask Question x Dismiss Join the Stack Overflow Community python float decimal places 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 vote 29 down vote favorite https://docs.python.org/2/tutorial/floatingpoint.html 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 22:34 community wiki 4 revs, 3 users 57%David http://stackoverflow.com/questions/2100490/floating-point-inaccuracy-examples 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 smaller one being “eaten” since there is no way to fit it into the larger scale.
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.