Machine Epsilon

Updated 15 Sep 2026

•
What it is
- •
  the gap between $1.0$ and the next-largest representable floating-point number — a measure of a floating-point format's precision
•
Formula
- •
  $\epsilon_{mach} = 2^{-N_m}$ , where $N_m$ is the number of mantissa bits
•
Values
- •
  single precision ( $N_m=23$ ): $\epsilon_{mach} = 2^{-23} \approx 1.192 \times 10^{-7}$
- •
  double precision ( $N_m=52$ ): $\epsilon_{mach} = 2^{-52} \approx 2.220 \times 10^{-16}$
•
Significance
- •
  sets the fundamental floor on rounding error for any single floating-point operation — this value reappears throughout error analysis (see Optimal Step Size, Patriot Missile Failure)
•
Related
- •
  IEEE 754 Floating-Point Standard

Linked references 6