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  "body": "\n## Definition\n\nFix [[integers|0C4JS9RF5M]] $\\beta \\ge 2$ (the *base*), $p \\ge 1$ (the *precision*), and $e_{\\min} \u003c e_{\\max}$.\nA *floating-point number* of this system is $0$ or a real number of the form\n$$x = \\pm\\, (d_0 . d_1 d_2 \\ldots d_{p-1})_\\beta \\times \\beta^{e} = \\pm \\left( \\sum_{i=0}^{p-1} d_i \\beta^{-i} \\right) \\beta^{e}.$$\nHere the digits $d_i$ are integers with $0 \\le d_i \\le \\beta - 1$, and the exponent $e$ is an integer with $e_{\\min} \\le e \\le e_{\\max}$.\nThe number is *normalized* if $d_0 \\ne 0$.\nLet $F$ be the set of these numbers.\nLet $\\mathrm{fl}$ be a [[rounding|5NPVX29Y0C]] to $F$ of the real numbers between $\\min F$ and $\\max F$, usually rounding to nearest.\n*Floating-point arithmetic* replaces each operation $\\circ \\in \\set{+, -, \\times, /}$ on $x, y \\in F$, whenever $x \\circ y$ lies in this range, by\n$$x \\circledast y = \\mathrm{fl}(x \\circ y),$$\nthat is, the exact result rounded to $F$.\n\n## Remarks\n\nExample: with $\\beta = 10$ and $p = 3$, the numbers $x = 1.23$ and $y = 4.56 \\times 10^{-2}$ lie in $F$, and $x \\oplus y = \\mathrm{fl}(1.2756) = 1.28$ under rounding to nearest.\nNon-example: with $\\beta = 10$ and $p = 3$, the number $1.234$ is not in $F$, since it needs four significant digits.\nFloating-point addition is commutative but not associative: with $\\beta = 10$ and $p = 3$, $(1.00 \\oplus 0.004) \\oplus 0.004 = 1.00$, while $1.00 \\oplus (0.004 \\oplus 0.004) = 1.01$.\nIn the range of normalized numbers and with rounding to nearest, $x \\circledast y = (x \\circ y)(1 + \\delta)$ with $\\abs{\\delta} \\le \\tfrac12 \\beta^{1-p}$.\n",
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