Machine epsilon is the gap between 1 and the next larger representable floating-point number in a given format. For a format with base 
and 
significant base-
digits, including the leading digit, it is
 |
For binary double precision, 
and 
, giving 
, approximately 
. The Wolfram
Language supplies this value as $MachineEpsilon.
Machine epsilon measures the spacing of floating-point numbers near 1, not the smallest positive representable number. With rounding to nearest, half this value bounds the relative rounding error for numbers in the normal range. Some numerical-analysis references use "machine epsilon" for this half-sized bound instead, so the convention must be checked when comparing formulas.
