is the smallest positive machine‐precision number that can be represented in normalized form on your computer system.
- Machine-precision numbers smaller in magnitude than $MinMachineNumber have less than $MachinePrecision digits of accuracy.
- Accuracy[0.] equals Accuracy[$MinMachineNumber]. »
- In the underlying binary representation, numbers smaller in magnitude than $MinMachineNumber have significands that do not start with a leading 1. »
Examplesopen allclose all
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):
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).
Wolfram Language. 1991. "$MinMachineNumber." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2018. https://reference.wolfram.com/language/ref/$MinMachineNumber.html.
Wolfram Language. (1991). $MinMachineNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/$MinMachineNumber.html