Editing Round-off error (section)

== Machine epsilon ==

[[Machine epsilon]] can be used to measure the level of roundoff error in the floating-point number system. Here are two different definitions.<ref name="Forrester_2018"/>

* The machine epsilon, denoted <math>\epsilon_\text{mach}</math>, is the maximum possible [[approximation error|absolute relative error]] in representing a nonzero real number <math>x</math> in a floating-point number system. <math display="block">\epsilon_\text{mach} = \max_{x} \frac{|x-fl(x)|}{|x|}</math>
* The machine epsilon, denoted <math>\epsilon_\text{mach}</math>, is the smallest number <math>\epsilon</math> such that <math>fl(1+\epsilon) > 1</math>. Thus, <math>fl(1+\delta)=fl(1)=1</math> whenever <math>|\delta| < \epsilon_\text{mach}</math>.