Machine epsilon

Machine epsilon gives an upper bound on the relative error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subject of computational science. The quantity is also called macheps or unit roundoff, and it has the symbols Greek epsilon ${\displaystyle \epsilon }$ or bold Roman u, respectively.

The following values of machine epsilon apply to standard floating point formats:

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, ${\displaystyle b}$, and by the precision ${\displaystyle p}$, i.e. the number of radix ${\displaystyle b}$ digits of the significand (including any leading implicit bit). All the numbers with the same exponent, ${\displaystyle e}$, have the spacing, ${\displaystyle b^{e-(p-1)}}$. The spacing changes at the numbers that are perfect powers of ${\displaystyle b}$; the spacing on the side of larger magnitude is ${\displaystyle b}$ times larger than the spacing on the side of smaller magnitude.

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