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  "body": "\n## Definition\n\nConsider a [[floating-point|RTKRSC3KPH]] system with base $\\beta$ and precision $p$.\nThe *machine epsilon* is\n$$\\varepsilon_{\\mathrm{mach}} = \\beta^{1-p},$$\nthe distance from $1$ to the next larger floating-point number.\nThe *unit roundoff* of the system with rounding to nearest is\n$$u = \\tfrac12 \\beta^{1-p}.$$\nSome authors call $u$ the machine epsilon instead.\n\n## Remarks\n\nExample: in IEEE binary64, $\\beta = 2$ and $p = 53$, so $\\varepsilon_{\\mathrm{mach}} = 2^{-52} \\approx 2.22 \\times 10^{-16}$ and $u = 2^{-53} \\approx 1.11 \\times 10^{-16}$.\nCheck: the floating-point numbers in $[1, 2)$ are $1 + k \\, 2^{-52}$ for $k = 0, 1, \\ldots, 2^{52} - 1$, so the successor of $1$ is $1 + 2^{-52}$.\nNon-example: the smallest positive floating-point number, about $4.9 \\times 10^{-324}$ in binary64, is not the machine epsilon; it measures the range, not the relative precision.\nIf $x$ lies in the range of normalized numbers and $\\mathrm{fl}(x)$ is the nearest floating-point number, then $\\mathrm{fl}(x) = x(1 + \\delta)$ with $\\abs{\\delta} \\le u$.\n",
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