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  "body": "\n## Definition\n\nLet $F$ be a set of [[floating-point numbers|RTKRSC3KPH]] and let $\\mathrm{fl} \\colon \\R \\to F$ be a [[rounding|5NPVX29Y0C]].\nThe *round-off error* of representing $x \\in \\R$ is $\\mathrm{fl}(x) - x$, and for $x \\ne 0$ its *relative round-off error* is $(\\mathrm{fl}(x) - x)/x$.\nLet an algorithm compute $y = f(x)$ in exact real arithmetic, and let it return $\\hat y$ when each operation is done in floating-point arithmetic.\nThen the round-off error of the computation is $\\hat y - y$.\n\n## Remarks\n\nExample: with base $10$, precision $3$ and rounding to nearest, $\\mathrm{fl}(2/3) = 0.667$, so the round-off error is $0.667 - 2/3 = 1/3000$ and the relative round-off error is $1/2000 = 0.0005$.\nThis is within the unit roundoff $u = \\tfrac12 \\cdot 10^{-2}$ of the system.\nNon-example: replacing $e = \\sum_{k \\ge 0} 1/k!$ by $\\sum_{k=0}^{5} 1/k!$ in exact arithmetic gives an error, but it is a truncation error, not a round-off error.\nIf $x$ lies in the range of normalized numbers and $\\mathrm{fl}$ rounds to nearest, then $\\abs{\\mathrm{fl}(x) - x} \\le u \\abs{x}$, where $u = \\tfrac12 \\beta^{1-p}$ is the unit roundoff, see [[machine epsilon|JA3QXVF2B8]].\n",
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