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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'', 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.〔Muller, Jean-Michel (2005-11). "On the definition of ulp(x)". INRIA Technical Report 5504. ACM Transactions on Mathematical Software, Vol. V, No. N, November 2005. Retrieved in 2012-03 from http://ljk.imag.fr/membres/Carine.Lucas/TPScilab/JMMuller/ulp-toms.pdf.〕 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.
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