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  "body": "\n## Definition\n\nThe *IEEE 754 standard* is the technical standard for [[floating-point arithmetic|RTKRSC3KPH]] of the Institute of Electrical and Electronics Engineers.\nIts main requirements are the following.\n\n1. Formats: a format has a base $\\beta \\in \\set{2, 10}$, a precision $p$ and an exponent range $[e_{\\min}, e_{\\max}]$ with $e_{\\min} = 1 - e_{\\max}$. The formats binary32 and binary64 have $(p, e_{\\max}) = (24, 127)$ and $(53, 1023)$.\n2. Special values: each format contains $+0$ and $-0$, $+\\infty$ and $-\\infty$, NaN (\"not a number\") values, and [[subnormal numbers|DHXXSN12JC]].\n3. Encoding: a binary64 number has $1$ sign bit, $11$ bits of biased exponent (bias $1023$) and $52$ bits of fraction. The leading significand bit of a normalized number is implicit.\n4. Correct rounding: the operations $+$, $-$, $\\times$, $/$, square root and fused multiply-add must return the exact result [[rounded|5NPVX29Y0C]] by the current rounding mode.\n5. Rounding modes: the default is round to nearest with ties to even. The other modes are ties away from zero and directed rounding toward $0$, $+\\infty$ and $-\\infty$.\n\n## Remarks\n\nExample: the decimal number $1$ in binary64 has sign bit $0$, biased exponent $0 + 1023 = 1023$ and fraction bits all $0$.\nNon-example: an arithmetic that always computes $x + y$ by chopping the extra digits does not meet clause 4, since in the default mode the result must be rounded to nearest.\nIn binary64 the largest finite number is $(2 - 2^{-52}) \\times 2^{1023} \\approx 1.8 \\times 10^{308}$, and the smallest positive normalized number is $2^{-1022}$.\nAn overflowing result is rounded to $\\pm\\infty$ under round to nearest, and an invalid operation such as $0/0$ or $\\sqrt{-1}$ returns NaN.\n",
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