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  "body": "\n## Summary\n\nValidated numerics, also called verified or rigorous computation, is numerical computation whose output comes with a mathematical guarantee.\nInstead of a single approximate number, it returns a set, usually an interval or a box, that provably contains the exact answer.\nAll [[round-off errors|CPN6SD3BTV]] and [[truncation errors|4AKCQERM89]] are bounded, typically with [[interval arithmetic|QPKKTE1ZY0]] or [[affine arithmetic|YPE8E9CVXP]] and directed rounding.\nIts results can serve as steps of a proof; an example is Warwick Tucker's computer-assisted proof that the Lorenz attractor exists.\n\n## Explanations\n\n### L1 human\n\nAn ordinary numerical program says \"the answer is about $1.41421$\" and hopes the digits are right.\nA validated program says \"the answer lies between $1.414213$ and $1.414214$\", and this claim is certain, rounding errors included.\nIt does this by computing with intervals and by rounding lower bounds down and upper bounds up.\nSuch enclosures can turn a computer calculation into part of a mathematical proof.\n",
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