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Concept
Unit in the last place
Summary
In computer science and numerical analysis, unit in the last place or unit of least precision (ulp) is the spacing between two consecutive floating-point numbers, i.e., the value the least significant digit (rightmost digit) represents if it is 1. It is used as a measure of accuracy in numeric calculations.
One definition is: In radix with precision , if , then .
Another definition, suggested by John Harrison, is slightly different: is the distance between the two closest straddling floating-point numbers and (i.e., those with and ), assuming that the exponent range is not upper-bounded. 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.
Let be a positive floating-point number and assume that the active rounding mode is round to nearest, ties to even, denoted . If , then . Otherwise, or , depending on the value of the least significant digit and the exponent of . 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 single precision and repeatedly add 1 until the operation does not change the value.
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