Unit in the last place

In computer science and numerical analysis, unit in the last place or unit of least precision (ULP) is the spacing between floating-point numbers, i.e., the value the least significant digit represents if it is 1. It is used as a measure of accuracy in numeric calculations.

Definition

In radix b, if x has exponent E, then ULP(x) = machine epsilon · b^E,^[1] but alternative definitions exist in the numerics and computing literature for ULP, exponent and machine epsilon, and they may give different equalities.

Another definition, suggested by John Harrison, is slightly different: ULP(x) is the distance between the two closest straddling floating-point numbers a and b (i.e., those with a ≤ x ≤ b and a ≠ b), assuming that the exponent range is not upper-bounded.^[2][3] These definitions differ only at signed powers of the radix.

The IEEE 754 specification—followed by all modern floating-point hardware—requires that the result of an elementary arithmetic operation (addition, subtraction, multiplication, division, and square root since 1985, and FMA since 2008) be correctly rounded, which implies that in rounding to nearest, the rounded result is within 0.5 ULP of the mathematically exact result, using John Harrison's definition; conversely, this property implies that the distance between the rounded result and the mathematically exact result is minimized (but for the halfway cases, it is satisfied by two consecutive floating-point numbers). Reputable numeric libraries compute the basic transcendental functions to between 0.5 and about 1 ULP. Only a few libraries compute them within 0.5 ULP, this problem being complex due to the Table-Maker's Dilemma.^[4]

Example

Let x be a nonnegative floating-point number and assume that the active rounding attribute is round to nearest, ties to even, denoted RN. If ULP(x) is less than or equal to 1, then RN(x + 1) > x. Otherwise, RN(x + 1) = x or RN(x + 1) = x +  ULP(x), depending on the value of the least significant digit and the exponent of x. 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

Here we start with 0 in 32-bit single-precision and repeatedly add 1 until the operation is idempotent. The result is equal to 2²⁴ since the significand for a single-precision number in this example contains 24 bits.

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

>>> x = 1.0
>>> p = 0
>>> while x != x + 1:
...   x = x * 2
...   p = p + 1
... 
>>> x
9007199254740992.0
>>> p
53
>>> x + 2 + 1
9007199254740996.0

In this case, we start with x = 1 and repeatedly double it until x = x + 1. The result is 2⁵³, because the double-precision floating-point format uses a 53-bit significand.

Language support

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

The C language library provides functions to calculate the next floating-point number in some given direction: nextafterf and nexttowardf for float, nextafter and nexttoward for double, nextafterl and nexttowardl for long double, declared in <math.h>.^[5]

The Boost C++ Libraries offer boost::math::float_next, boost::math::float_prior, boost::math::nextafter and boost::math::float_advance functions to obtain nearby (and distant) floating-point values, ^[6] and boost::math::float_distance(a, b) to calculate the floating-point distance between two doubles. ^[7]

See also

• IEEE 754
• ISO/IEC 10967, part 1 requires an ulp function
• Least significant bit (LSB)
• Machine epsilon

References

Bibliography

• Goldberg, David (1991-03). "Rounding Error" in "What Every Computer Scientist Should Know About Floating-Point Arithmetic". Computing Surveys, ACM, March 1991. Retrieved from http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html#689.