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$MachineEpsilon

gives the difference between 1.0 and the next-nearest number representable as a machine-precision number.

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Basic Examples  
Scope  
Applications  
Properties & Relations  
Neat Examples  
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$MachineEpsilon

$MachineEpsilon

gives the difference between 1.0 and the next-nearest number representable as a machine-precision number.

Details

  • $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.

Examples

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Basic Examples  (1)

Wolfram Language code: $MachineEpsilon

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

Wolfram Language code: x = 1. + $MachineEpsilon
Wolfram Language code: x - 1.

Adding a fraction of $MachineEpsilon effectively results in rounding:

Wolfram Language code: y = 1. + {0.3, 0.5, 0.7}$MachineEpsilon
Wolfram Language code: y - 1.

Scope  (2)

The result of subtracting $MachineEpsilon/2 from 1 is distinct from 1:

Wolfram Language code: x = 1. - $MachineEpsilon / 2
Wolfram Language code: 1. - x

Find machine epsilon algorithmically:

Wolfram Language code: Block[{ϵ = 1.}, While[Not[PossibleZeroQ[(1. + ϵ) - 1.]], ϵ /= 2];ϵ * 2]
Wolfram Language code: % == $MachineEpsilon

Applications  (2)

Get the nearest machine number greater than another machine number:

Wolfram Language code: x = RandomReal[{1, 100}]
Wolfram Language code: y = x * (1. + $MachineEpsilon);

and are distinct:

Wolfram Language code: x - y

and differ only in the least significant bit:

Wolfram Language code: RealDigits[x, 2]
Wolfram Language code: RealDigits[y, 2]

Horner's method for evaluating a polynomial with a running error bound:

Wolfram Language code: horner[p_List, x_] := Module[{u, y, mu}, y = Last[p]; mu = Abs[y] / 2; Do[y = x * y + c;mu = Abs[x] * mu + Abs[y], {c, Take[p, {-2, 1, -1}]}]; (* u is "unit roundoff"*) u = $MachineEpsilon / 2; mu = u * (2 * mu - Abs[y]); {y, mu}]

A polynomial with large coefficients:

Wolfram Language code: poly = N[Expand[Product[(x - i), {i, 20}]]]

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

Wolfram Language code: horner[CoefficientList[poly, x], 10.]

Properties & Relations  (3)

$MachineEpsilon is a power of 2:

Wolfram Language code: Log[2., $MachineEpsilon]

$MachineEpsilon is twice 10-MachinePrecision:

Wolfram Language code: $MachineEpsilon == 2 / 10 ^ MachinePrecision

This is effectively where is the number of bits of machine precision:

Wolfram Language code: b = MachinePrecision * Log[2., 10.]
Wolfram Language code: 2 ^ (1 - b) == $MachineEpsilon

1 and 1+$MachineEpsilon differ only in the least significant bit:

Wolfram Language code: RealDigits[1., 2]
Wolfram Language code: RealDigits[1. + $MachineEpsilon, 2]

Neat Examples  (1)

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

Wolfram Language code: Plot[(1 + x) - 1, {x, -2$MachineEpsilon, 2 $MachineEpsilon}, PlotRange -> All]
Wolfram Research (1991), $MachineEpsilon, Wolfram Language function, https://reference.wolfram.com/language/ref/$MachineEpsilon.html.

Text

Wolfram Research (1991), $MachineEpsilon, Wolfram Language function, https://reference.wolfram.com/language/ref/$MachineEpsilon.html.

CMS

Wolfram Language. 1991. "$MachineEpsilon." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/$MachineEpsilon.html.

APA

Wolfram Language. (1991). $MachineEpsilon. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/$MachineEpsilon.html

BibTeX

@misc{reference.wolfram_2026_$machineepsilon, author="Wolfram Research", title="{$MachineEpsilon}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/$MachineEpsilon.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_$machineepsilon, organization={Wolfram Research}, title={$MachineEpsilon}, year={1991}, url={https://reference.wolfram.com/language/ref/$MachineEpsilon.html}, note=[Accessed: 15-June-2026]}