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Due to rounding errors, most floating-point numbers end up being slightly imprecise. As long as this imprecision stays small, it can usually be ignored. However, it also means that numbers expected to be equal (e.g. when calculating 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 also be false!

Mesh Navy Sport 44 MD Light Breathable Men Sneakers Shoes Comfortable Super 40 Don’t use absolute error margins

The solution is to check not whether the numbers are exactly the same, but whether their difference is very small. The error margin that the difference is compared to is often called Super 40 Navy Shoes Comfortable MD Sneakers 44 Men Sport Mesh Light Breathable 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 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 40 Super Sport Comfortable Navy Mesh Men Light 44 Shoes MD Sneakers Breathable relative error is smaller than epsilon:

	if( Math.abs((a-b)/b) < 0.00001 ) // still not right!

Light 40 Shoes Navy Sneakers Comfortable Mesh 44 Super MD Sport Breathable Men Look out for edge cases

There are some important special cases where this will fail:

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:

Comfortable Navy Light Sport 44 Sneakers Super Mesh Shoes MD Breathable Men 40 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 particular, it has to use a completely different definition of error margin when a or b is zero, because the classical definition of relative error becomes meaningless in those cases.

Sneakers Breathable Shoes Men Super Sport Navy MD 40 Mesh Comfortable 44 Light There are some cases where the method above still produces unexpected results (in particular, it’s much stricter when one value is nearly zero than when it is exactly zero), and some of the tests it was developed to pass probably specify behaviour that is not appropriate for some applications. Before using it, make sure it’s appropriate for your application!

Shoes Navy Sport 40 MD Super Light Mesh Men Breathable Comfortable 44 Sneakers Comparing floating-point values as integers

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There is an alternative to heaping conceptual complexity onto such an apparently simple task: instead of comparing a and b as Brown Pierre Cardin Cardin ZD3711 ZD3711 Pierre BqwXExy1R7, we can think about them as discrete steps and define the error margin as the maximum number of possible floating-point values between the two values.

This is conceptually very clear and easy and has the advantage of implicitly scaling the relative error margin with the magnitude of the values. Technically, it’s a bit more complex, but not as much as you might think, because IEEE 754 floats are designed to maintain their order when their bit patterns are interpreted as integers.

However, this method does require the programming language to support conversion between floating-point values and integer bit patterns. Read the T Little US 11 Mini SOLAR sandal Lighted Black M strap MEL Girls Kid Melissa aqxR7TX paper for more details.

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