Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point number systems. 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 and it has the symbols Greek epsilon ε {\displaystyle \varepsilon } . There are two prevailing definitions, denoted here as rounding machine epsilon or the formal definition and interval machine epsilon or mainstream definition. In the mainstream definition, machine epsilon is independent of rounding method, and is defined simply as the difference between 1 and the next larger floating point number. In the formal definition, machine epsilon is dependent on the type of rounding used and is also called unit roundoff, which has the symbol bold Roman u {\displaystyle \mathbf {u} } . The two terms can generally be considered to differ by simply a factor of two, with the formal definition yielding an epsilon half the size of the mainstream definition, as summarized in the tables in the next section.
Values for standard hardware arithmetics The following table lists machine epsilon values for standard floating-point formats.
Alternative definitions for epsilon The IEEE standard does not define the terms machine epsilon and unit roundoff, so differing definitions of these terms are in use, which can cause some confusion. The two terms differ by simply a factor of two. The more-widely used term (referred to as the mainstream definition in this article), is used in most modern programming languages and is simply defined as machine epsilon is the difference between 1 and the next larger floating point number. The formal definition can generally be considered to yield an epsilon half the size of the mainstream definition, although its definition does vary depending on the form of rounding used. The two terms are described at length in the next two subsections.
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