# \$MinMachineNumber

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

# 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_2024_\$minmachinenumber, author="Wolfram Research", title="{\$MinMachineNumber}", year="2018", howpublished="\url{https://reference.wolfram.com/language/ref/\$MinMachineNumber.html}", note=[Accessed: 25-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_\$minmachinenumber, organization={Wolfram Research}, title={\$MinMachineNumber}, year={2018}, url={https://reference.wolfram.com/language/ref/\$MinMachineNumber.html}, note=[Accessed: 25-June-2024 ]}