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  "body": "\n## Definition\n\nConsider a [[floating-point|RTKRSC3KPH]] system with base $\\beta$, precision $p$ and minimal exponent $e_{\\min}$.\nLet $x \\ne 0$ be a floating-point number written as $x = \\pm (d_0 . d_1 \\ldots d_{p-1})_\\beta \\times \\beta^{e}$, with $d_0 \\ne 0$ if $e \u003e e_{\\min}$.\nThe *unit in the last place* of $x$ is\n$$\\mathrm{ulp}(x) = \\beta^{\\,e - p + 1},$$\nthe value of the digit position $d_{p-1}$.\n\n## Remarks\n\nExample: with $\\beta = 10$ and $p = 3$, the number $x = 3.14 = 3.14 \\times 10^{0}$ has $e = 0$, so $\\mathrm{ulp}(x) = 10^{-2}$; its neighbours are $3.13$ and $3.15$.\nNon-example: in the same system, $0.001$ is not the ulp of $3.14$, since $3.141$ is not a floating-point number of precision $3$.\nThe ulp of $1$ is $\\beta^{1-p}$, the [[machine epsilon|JA3QXVF2B8]].\nSeveral variants of the definition exist; some define $\\mathrm{ulp}(x)$ for every real $x$ as the gap between the two floating-point numbers closest to $x$.\nThey agree away from powers of $\\beta$.\nRounding to nearest gives an error of at most $\\tfrac12 \\mathrm{ulp}$ of the result.\n",
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    "name": "Unit in the last place",
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  "name": "Unit in the last place",
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