Prof. Kahan came up with the notion of an ulp. An ulp of a real number value is the distance between the floating-point numbers which bracket the real value. Of course since the only reals we can represent happen to be floating-point numbers, this corresponds to the distance between adjacent floating-point values. To avoid ambiguities at powers of two (where the distance between numbers changes), we can define an ulp of a floating-point value to be the distance to the next floating-point value larger in magnitude.

In Java, all the arithmetic operations {+, -, *, /, sqrt} have error of less than 1/2 ulp assuming their inputs are exact; i.e. assuming you want to take the square root of the floating-point number closest to 0.1 not 0.1 exactly. Much of the math library must be accurate to 1 ulp although a few functions only require 2 ulps. Of course even with correct rounding at each step, the final answer could be widely inaccurate due to accumulated and amplified round-off errors.

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