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  "body": "\n## Summary\n\nFix a continuous function $f$ on an interval and a degree $n$.\nA minimax approximation of $f$ is a polynomial $p$ of degree at most $n$ that makes the largest error $\\max \\abs{f - p}$ as small as possible.\nIt is the [[best approximation|7N4988HN3P]] for the uniform norm, so it controls the error at every point and not only on average.\nMinimax polynomials are used to build the routines that compute $\\sin$, $\\exp$ and other functions in software libraries.\n\n## Explanations\n\n### L1 human\n\nDraw the graph of $f$ and a thin band around it of the same width $\\varepsilon$ everywhere.\nA polynomial approximates $f$ within $\\varepsilon$ when its graph stays inside the band.\nThe minimax polynomial is the polynomial of the given degree that fits inside the thinnest possible band.\nFor example, the best constant approximation of $f(x) = x$ on $[0, 1]$ is the constant $1/2$, with largest error $1/2$.\nAny other constant $c$ is farther than $1/2$ from $0$ or from $1$.\n",
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