Floating Point Relative Error
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problems, integer overflow, and an attempt to extend the ULPs based technique further than really makes sense. The series of articles listed above comparing floating point numbers in c covers the whole topic, but the key article that demonstrates good techniques
Comparing Floating Point Numbers Java
for floating-point comparisons can be found here. This article also includes a cool demonstration, using sin(double(pi)), of why floating point numbers should not be tested for equality the ULPs technique and other relative error techniques breaks down around zero. In short, stop reading. Click this link. Okay, you've been warned. The remainder of this article exists purely
Comparison Of Floating Point Numbers With Equality Operator
for historical reasons. Comparing for equality Floating point math is not exact. Simple values like 0.2 cannot be precisely represented using binary floating point numbers, and the limited precision of floating point numbers means that slight changes in the order of operations can change the result. Different compilers and CPU architectures store temporary results at different precisions, so results will differ float compare depending on the details of your environment. If you do a calculation and then compare the results against some expected value it is highly unlikely that you will get exactly the result you intended. In other words, if you do a calculation and then do this comparison: if (result == expectedResult) then it is unlikely that the comparison will be true. If the comparison is true then it is probably unstable tiny changes in the input values, compiler, or CPU may change the result and make the comparison be false. Comparing with epsilon absolute error Since floating point calculations involve a bit of uncertainty we can try to allow for this by seeing if two numbers are close to each other. If you decide based on error analysis, testing, or a wild guess that the result should always be within 0.00001 of the expected result then you can change your comparison to this: if (fabs(result - expectedResult) < 0.00001) The maximum error value is typically called epsilon. Absolute error calculations have their plac
the same result through different correct methods) often differ slightly, and a simple equality test fails. For example: float a = 0.15 + 0.15 float b = 0.1 + 0.2 if(a == b) // can be false! if(a >= b) // can
Compare Float To 0 Java
also be false! Don’t use absolute error margins The solution is to check not whether the
C++ Float Epsilon
numbers are exactly the same, but whether their difference is very small. The error margin that the difference is compared to is often called floating point equality epsilon. The most simple form: if( Math.abs(a-b) < 0.00001) // wrong - don't do this This is a bad way to do it because a fixed epsilon chosen because it “looks small” could actually be way too large when the http://www.cygnus-software.com/papers/comparingfloats/Comparing%20floating%20point%20numbers.htm numbers being compared are very small as well. The comparison would return “true” for numbers that are quite different. And when the numbers are very large, the epsilon could end up being smaller than the smallest rounding error, so that the comparison always returns “false”. Therefore, it is necessary to see whether the relative error is smaller than epsilon: if( Math.abs((a-b)/b) < 0.00001 ) // still not right! Look out for edge cases There are some important special cases http://floating-point-gui.de/errors/comparison/ where this will fail: When both a and b are zero. 0.0/0.0 is “not a number”, which causes an exception on some platforms, or returns false for all comparisons. When only b is zero, the division yields “infinity”, which may also cause an exception, or is greater than epsilon even when a is smaller. It returns false when both a and b are very small but on opposite sides of zero, even when they’re the smallest possible non-zero numbers. Also, the result is not commutative (nearlyEquals(a,b) is not always the same as nearlyEquals(b,a)). To fix these problems, the code has to get a lot more complex, so we really need to put it into a function of its own: public static boolean nearlyEqual(float a, float b, float epsilon) { final float absA = Math.abs(a); final float absB = Math.abs(b); final float diff = Math.abs(a - b); if (a == b) { // shortcut, handles infinities return true; } else if (a == 0 || b == 0 || diff < Float.MIN_NORMAL) { // a or b is zero or both are extremely close to it // relative error is less meaningful here return diff < (epsilon * Float.MIN_NORMAL); } else { // use relative error return diff / Math.min((absA + absB), Float.MAX_VALUE) < epsilon; } } This method passes tests for many important special cases, but as you can see, it uses some quite non-obvious logic. In
the field of numerical analysis, and by extension in the subject of computational science. The quantity is also called macheps or unit roundoff, and it has the symbols Greek https://en.wikipedia.org/wiki/Machine_epsilon epsilon ϵ {\displaystyle \epsilon } or bold Roman u, respectively. Contents 1 Values http://www.dcs.ed.ac.uk/home/mhe/plume/node10.html for standard hardware floating point arithmetics 2 Formal definition 3 Arithmetic model 4 Variant definitions 5 How to determine machine epsilon 5.1 Approximation 6 See also 7 Notes and references 8 External links Values for standard hardware floating point arithmetics[edit] The following values of machine epsilon apply to standard floating point floating point formats: IEEE 754 - 2008 Common name C++ data type Base b {\displaystyle b} Precision p {\displaystyle p} Machine epsilon[a] b − ( p − 1 ) / 2 {\displaystyle b^{-(p-1)}/2} Machine epsilon[b] b − ( p − 1 ) {\displaystyle b^{-(p-1)}} binary16 half precision short 2 11 (one bit is implicit) 2−11 ≈ 4.88e-04 2−10 ≈ 9.77e-04 binary32 single precision floating point numbers float 2 24 (one bit is implicit) 2−24 ≈ 5.96e-08 2−23 ≈ 1.19e-07 binary64 double precision double 2 53 (one bit is implicit) 2−53 ≈ 1.11e-16 2−52 ≈ 2.22e-16 extended precision _float80[1] 2 64 2−64 ≈ 5.42e-20 2−63 ≈ 1.08e-19 binary128 quad(ruple) precision _float128[1] 2 113 (one bit is implicit) 2−113 ≈ 9.63e-35 2−112 ≈ 1.93e-34 decimal32 single precision decimal _Decimal32[2] 10 7 5 × 10−7 10−6 decimal64 double precision decimal _Decimal64[2] 10 16 5 × 10−16 10−15 decimal128 quad(ruple) precision decimal _Decimal128[2] 10 34 5 × 10−34 10−33 a according to Prof. Demmel, LAPACK, Scilab b according to Prof. Higham; ISO C standard; C, C++ and Python language constants; Mathematica, MATLAB and Octave; various textbooks - see below for the latter definition Formal definition[edit] Rounding is a procedure for choosing the representation of a real number in a floating point number system. For a number system and a rounding procedure, machine epsilon is the maximum relative error of the chosen rounding procedure. Some background is needed to determine a value from this definition. A floating point number system is characterized by a radix whi
to perform error analysis. This involves doing calculations to obtain a bound on the error of a particular expression. One approach is to use the assumption that a real number x is approximated by the number , where . In this equation, is the relative error in the representation. Hence: It is now possible to calculate the effect that certain operations will have on the relative error of a floating point computation. Operations such as floating point multiplication will affect the relative error, but not significantly. Let where the desired result is : Operations such as addition or subtraction, however, can have a much more significant effect on the relative error in certain cases. Consider the subtraction : It is now clear that if x2 is nearly equal to x1, the relative error will be greatly magnified. The use of simple methods like this, or more sophisticated approaches, allow the accuracy of a given computation to be examined. This may allow the user to have faith in the results of a floating point computation. Knuth [17] describes the approach illustrated here and gives a more detailed discussion of the problem. The major problems with such methods are that firstly the error analysis may simply tell the user that he or she should have no faith whatsoever in the correctness of the result produced, and secondly that the error analysis must be performed for every computation and is not general. Next: Interval Arithmetic Up: Approaches to Real Arithmetic Previous: Floating Point Arithmetic Martin Escardo 5/11/2000