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, called the unit roundoff, 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.
