gives the difference between 1.0 and the next-nearest number representable as a machine-precision number.
- $MachineEpsilon is typically 2-n+1, where n is the number of binary bits used in the internal representation of machine‐precision floating‐point numbers.
- $MachineEpsilon measures the granularity of machine‐precision numbers.
Examplesopen allclose all
Basic Examples (1)
The result of adding 1 to $MachineEpsilon is distinct from 1:
Adding a fraction of $MachineEpsilon effectively results in rounding:
Get the nearest machine number greater than another machine number:
and differ only in the least significant bit:
Horner's method for evaluating a polynomial with a running error bound:
A polynomial with large coefficients:
Properties & Relations (3)
$MachineEpsilon is a power of 2:
$MachineEpsilon is twice 10-MachinePrecision:
This is effectively where is the number of bits of machine precision:
1 and 1+$MachineEpsilon differ only in the least significant bit:
Wolfram Research (1991), $MachineEpsilon, Wolfram Language function, https://reference.wolfram.com/language/ref/$MachineEpsilon.html.
Wolfram Language. 1991. "$MachineEpsilon." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/$MachineEpsilon.html.
Wolfram Language. (1991). $MachineEpsilon. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/$MachineEpsilon.html