# Comparison

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!
```

## 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 *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 *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 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 || (absA + absB < 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.

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!

## Comparing floating-point values as integers

There is an alternative to heaping conceptual complexity onto such an apparently simple task: instead of comparing `a`

and `b`

as real numbers, 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 Comparing floating-point numbers paper for more details.

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