Give An Example Of A Floating-point Roundoff Error
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Floating Point Rounding Error Example
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Round Off Error In Numerical Method
takes a minute: Sign up What is a simple example of floating point/rounding error? up vote 22 down vote favorite 5 I've heard of "error" when using floating point variables. Now I'm trying to solve this puzzle and I think floating point error I'm getting some rounding/floating point error. So I'm finally going to figure out the basics of floating point error. What is a simple example of floating point/rounding error (preferably in C++) ? Edit: For example say I have an event that has probability p of succeeding. I do this event 10 times (p does not change and all trials are independent). What is the probability of exactly 2 successful trials? I have this coded as: double p_2x_success = pow(1-p, (double)8) * truncation error pow(p, (double)2) * (double)choose(8, 2); Is this an opportunity for floating point error? c++ floating-accuracy share|improve this question edited Oct 30 '08 at 7:21 asked Oct 30 '08 at 7:12 MrDatabase 11.8k3191145 I think what you really need is this: What Every Computer Scientist Should Know About Floating-Point Arithmetic. –Patrick Oct 30 '08 at 7:17 Read this: blog.frama-c.com/index.php?post/2013/05/02/nearbyintf1 –Nicholas Wilson May 3 '13 at 19:09 See simple Java example, shuld be the same in C: stackoverflow.com/a/15790782/254109 –xmedeko Nov 5 '13 at 12:06 add a comment| 5 Answers 5 active oldest votes up vote 20 down vote for(double d = 0; d != 0.3; d += 0.1); // never terminates share|improve this answer edited Nov 1 '08 at 19:07 answered Oct 30 '08 at 10:46 Motti 52.3k27132206 add a comment| up vote 18 down vote Picture is worth a thousand words - try to draw equation f(k) : and you will get such XY graph (X and Y are in logarithmic scale). If computer could represent 32-bit floats without rounding error then for every k we should get zero. But instead error increases with bigger values of k because of floating point error accumulation. hth! share|improve this answer answered Apr 17 '11 at 15:49 Agnius Vasiliauskas 5,25022849 add a comment| up vote 8 down vote Generally, floating point error refers to when a number that cannot be stored in the IEEE floating point representation. I
by David Goldberg, published in the March, 1991 issue of Computing Surveys. Copyright 1991, Association for
Round Off Error In Floating Point Representation
Computing Machinery, Inc., reprinted by permission. Abstract Floating-point arithmetic floating point converter is considered an esoteric subject by many people. This is rather surprising because floating-point is
Floating Point Arithmetic
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 http://stackoverflow.com/questions/249467/what-is-a-simple-example-of-floating-point-rounding-error 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 https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html 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
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss http://programmers.stackexchange.com/questions/101163/what-causes-floating-point-rounding-errors the workings and policies of this site About Us Learn more http://floating-point-gui.de/errors/rounding/ about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Programmers Questions Tags Users Badges Unanswered Ask Question _ Programmers Stack Exchange is a question and answer site for professional programmers interested in conceptual questions about software development. Join them; floating point it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top What causes floating point rounding errors? up vote 40 down vote favorite 22 I am aware that floating point arithmetic has precision problems. I usually overcome them round off error by switching to a fixed decimal representation of the number, or simply by neglecting the error. However, I do not know what are the causes of this inaccuracy. Why are there so many rounding issues with float numbers? floating-point numeric-precision share|improve this question asked Aug 15 '11 at 13:07 nmat 318135 25 To be precise, it's not really the error caused by rounding that most people worry about -- it's the fact that binary floating-point rounding behaves in unintuitive ways. Switching to a decimal representation can make the rounding behave in a more intuitive way, but in exchange you will nearly always increase the relative error (or else have to increase the storage space to compensate). –Daniel Pryden Aug 15 '11 at 16:35 8 My attempt to clear up the most common confusions: floating-point-gui.de –Michael Borgwardt Aug 16 '11 at 11:22 i think what @DanielPryden means is "Switching to a [fixed-point] representation can make the rounding behave in a more intuitive way...". what causes rounding proble
are three reasons why this can be necessary: Large Denominators In any base, the larger the denominator of an (irreducible) fraction, the more digits it needs in positional notation. A sufficiently large denominator will require rounding, no matter what the base or number of available digits is. For example, 1/1000 cannot be accurately represented in less than 3 decimal digits, nor can any multiple of it (that does not allow simplifying the fraction). Periodical digits Any (irreducible) fraction where the denominator has a prime factor that does not occur in the base requires an infinite number of digits that repeat periodically after a certain point. For example, in decimal 1/4, 3/5 and 8/20 are finite, because 2 and 5 are the prime factors of 10. But 1/3 is not finite, nor is 2/3 or 1/7 or 5/6, because 3 and 7 are not factors of 10. Fractions with a prime factor of 5 in the denominator can be finite in base 10, but not in base 2 - the biggest source of confusion for most novice users of floating-point numbers. Non-rational numbers Non-rational numbers cannot be represented as a regular fraction at all, and in positional notation (no matter what base) they require an infinite number of non-recurring digits. Rounding modes There are different methods to do rounding, and this can be very important in programming, because rounding can cause different problems in various contexts that can be addressed by using a better rounding mode. The most common rounding modes are: Rounding towards zero - simply truncate the extra digits. The simplest method, but it introduces larger errors than necessary as well as a bias towards zero when dealing with mainly positive or mainly negative numbers. Rounding half away from zero - if the truncated fraction is greater than or equal to half the base, increase the last remaining digit. This is the method generally taught in school and used by most people. It minimizes errors, but also introduces a bias (away from zero). Rounding half to even also known as banker’s rounding - if the truncated fraction is greater than half the base, increase the last remaining digit. If it is equal to half the base, increase the digit only if that produces an even result. This minimizes errors and bias, and is therefore preferred for bookkeeping. Examples in base 10: Towards zero Half away from zero Half to even 1.4 1 1 1 1.5 1 2 2 -1.6 -1 -2 -