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  "body": "\n## Definition\n\nThe *Chudnovsky algorithm* computes approximations of [[pi|0C1VYVNV2N]] from the Chudnovsky formula\n$$\\frac{1}{\\pi} = 12\\sum_{k=0}^{\\infty} \\frac{(-1)^k\\,(6k)!\\,(13591409 + 545140134\\,k)}{(3k)!\\,(k!)^3\\,640320^{3k + 3/2}},$$\nwhere $n!$ is the [[factorial|0XWG0JGDFW]].\nTo obtain $\\pi$ to $D$ decimal digits, it adds the first $N$ terms of this [[series|0AF7X08PCR]] and inverts the sum.\nHere $N$ is slightly more than $D/14$.\nThe partial sum is evaluated by binary splitting.\nThe rational numbers that form the terms are combined pairwise in a balanced tree, so the work is done with large-integer products.\n\n## Remarks\n\nThe ratio of successive terms tends to $-12^3/640320^3 \\approx -6.6 \\times 10^{-15}$, since $(6k)!/((3k)!(k!)^3)$ grows like a constant times $1728^k k^{-3/2}$ by Stirling's formula; this gives about $14.18$ digits per term.\nExample: the $k = 0$ term gives $\\pi \\approx 640320^{3/2}/(12\\cdot 13591409) = 3.14159265358973\\ldots$, correct to $13$ decimal places.\nNon-example: summing the Leibniz series $\\pi/4 = 1 - \\frac13 + \\frac15 - \\cdots$ is not this algorithm and is far slower: its error after $N$ terms is of order $1/N$.\nThe number $640320^3 = 262537412640768000$ is close to $e^{\\pi\\sqrt{163}} - 744$, which reflects the [[j-invariant|FEC2XH822R]] value $j\\big(\\frac{1 + \\sqrt{-163}}{2}\\big) = -640320^3$.\n",
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