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  "body": "\n## Summary\n\nRound-off error, or rounding error, is the difference between the exact result of a computation and the result computed with [[rounded|5NPVX29Y0C]] numbers.\nIn [[floating-point arithmetic|RTKRSC3KPH]] each single rounding has relative error at most the unit roundoff, about $10^{-16}$ in double precision.\nMany roundings can add up, and a subtraction of nearly equal numbers can magnify them, as in [[catastrophic cancellation|TJPNYW7J8E]].\nIt is distinct from [[truncation error|4AKCQERM89]], which comes from the method even in exact arithmetic.\n\n## Explanations\n\n### L1 human\n\nA computer keeps only a fixed number of digits, so it must round almost every result.\nEach rounding error is tiny, but a long computation does millions of operations.\nSometimes the errors cancel each other, and sometimes they pile up.\nA classic example: adding $0.1$ ten times in double precision does not give exactly $1$, because $0.1$ is rounded when stored and again after each addition.\n",
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