Formal Definition
Rounding is a procedure for choosing the representation of a real number in a floating point number system. For a number system and a rounding procedure, machine epsilon is the maximum relative error of the chosen rounding procedure.
Some background is needed to determine a value from this definition. A floating point number system is characterized by a radix which is also called the base, and by the precision, i.e. the number of radix digits of the significand (including any leading implicit bit). All the numbers with the same exponent, have the spacing, . The spacing changes at the numbers that are perfect powers of ; the spacing on the side of larger magnitude is times larger than the spacing on the side of smaller magnitude.
Since machine epsilon is a bound for relative error, it suffices to consider numbers with exponent . It also suffices to consider positive numbers. For the usual round-to-nearest kind of rounding, the absolute rounding error is at most half the spacing, or . This value is the biggest possible numerator for the relative error. The denominator in the relative error is the number being rounded, which should be as small as possible to make the relative error large. The worst relative error therefore happens when rounding is applied to numbers of the form where is between and . All these numbers round to with relative error . The maximum occurs when is at the upper end of its range. The in the denominator hardly matters, so it is left off for expediency, and just is taken as machine epsilon. As has been shown here, the relative error is worst for numbers that round to, so machine epsilon also is called unit roundoff meaning roughly "the maximum error that can occur when rounding to the unit value".
Thus, the maximum spacing between a normalised floating point number, and an adjacent normalised number is x .
Read more about this topic: Machine Epsilon
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