BUILT-IN MATHEMATICA SYMBOL

$MachineEpsilon

$MachineEpsilon
gives the difference between

and the next-nearest number representable as a machine-precision number.

$MachineEpsilon is typically , where n is the number of binary bits used in the internal representation of machine-precision floating-point numbers.

The result of adding 1 to $MachineEpsilon is distinct from 1:

Adding a fraction of $MachineEpsilon effectively results in rounding:

Out[1]=

The result of adding 1 to $MachineEpsilon is distinct from 1:

Out[2]=

Out[3]=

Adding a fraction of $MachineEpsilon effectively results in rounding:

Out[4]=

Out[5]=

The result of subtracting

from 1 is distinct from 1:

Find machine epsilon algorithmically:

Get the nearest machine number greater than another machine number:

and

are distinct:

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:

Evaluate at x=10; the error is large, but within the bound:

$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:

The resolution of machine numbers is twice as fine just below 1 versus just above 1: