$MinMachineNumber

$MinMachineNumber

is the smallest positive machine‐precision number that can be represented in normalized form on your computer system.

Details

Examples

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Basic Examples  (1)

The smallest hardware floating-point number that can be put in normalized form:

Scope  (3)

Machine numbers smaller than $MinMachineNumber are represented as subnormal machine numbers:

This is still a machine number:

However, x has not gained accuracy relative to $MinMachineNumber:

Find the smallest positive normalized machine number algorithmically:

Find the smallest positive subnormal machine number algorithmically:

Properties & Relations  (4)

Compute the minimum exponent in binary for machine arithmetic:

$MinMachineNumber has that smallest exponent and all bits but the first set to 0 in the significand:

Subnormal machine numbers have the minimum exponent and a leading 0 bit in the significand:

$MinMachineNumber/252 produces that smallest positive subnormal number:

Further division produces a machine zero:

$MaxMachineNumber×$MinMachineNumber is 4.×(1.-$MachineEpsilon/2):

Accuracy[$MinMachineNumber] equals Accuracy[0.]:

Possible Issues  (2)

Computations with machine numbers smaller than $MinMachineNumber can lose all significant digits:

Use SetPrecision to convert a machine number to arbitrary precision and avoid underflow:

The reciprocal of $MaxMachineNumber is smaller than $MinMachineNumber:

Wolfram Research (1991), $MinMachineNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/$MinMachineNumber.html (updated 2018).

Text

Wolfram Research (1991), $MinMachineNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/$MinMachineNumber.html (updated 2018).

CMS

Wolfram Language. 1991. "$MinMachineNumber." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2018. https://reference.wolfram.com/language/ref/$MinMachineNumber.html.

APA

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

BibTeX

@misc{reference.wolfram_2025_$minmachinenumber, author="Wolfram Research", title="{$MinMachineNumber}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/$MinMachineNumber.html}", note=[Accessed: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_$minmachinenumber, organization={Wolfram Research}, title={$MinMachineNumber}, year={2018}, url={https://reference.wolfram.com/language/ref/$MinMachineNumber.html}, note=[Accessed: 22-February-2025 ]}