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



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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:

Introduced in 1991
Updated in 2003