Unit in the last place

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In computer science, Unit in the Last Place or Unit of Least Precision (ULP) is the gap between two very close floating-point numbers. To be exact ulp(x) is the gap between the two floating-point numbers closest to the value x.

The amount of error in the evaluation of a floating-point operation is often expressed in ULP. An average error of 1 ULP is often seen as a tolerable error.

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[edit] Example

If the ULP is less than or equal to 1, then adding 1 to a floating-point number will succeed. Otherwise, adding 1 will have no effect. This is demonstrated in the following Haskell code typed at an interactive prompt:

> until (\x -> x == x+1) (+1) 0 :: Float
1.6777216e7
> it-1
1.6777215e7
> it+1
1.6777216e7
> it+1
1.6777216e7

Here, we start with 32-bit single-precision 0 and keep adding 1 until the operation is idempotent. The result is about 17 million.

Another example, in Python, is (typed at an interactive prompt):

>>> x = 1.0
>>> while (x != x + 1.0):
...   x = x * 10.0
... 
>>> print x
1e+16

using double-precision floating-point.

[edit] Language support

Since Java 1.5, the Java standard library has included Math.ulp(double) and Math.ulp(float) functions.

[edit] References

[edit] See also

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