Machine Epsilon

Updated 01 Oct 2026

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What it is
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  the gap between $1.0$ and the next-largest representable floating-point number — a measure of a floating-point format's precision
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Formula
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  $\epsilon_{mach} = 2^{-N_m}$ , where $N_m$ is the number of mantissa bits
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Values
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  single precision ( $N_m=23$ ): $\epsilon_{mach} = 2^{-23} \approx 1.192 \times 10^{-7}$
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  double precision ( $N_m=52$ ): $\epsilon_{mach} = 2^{-52} \approx 2.220 \times 10^{-16}$
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Interpretation
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  machine epsilon describes floating-point spacing near $1$ under the usual definition
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  it indicates relative precision, not a positive minimum error in every operation: some operations are exacct
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  absolute spacing varies with magnitude; submornal values require seaprate consderation
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Related
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  IEEE 754 Floating-Point Standard

What it is