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Lemniscate - Zuniga (2023-i).cfg
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Lemniscate - Zuniga (2023-i).cfg
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// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename".
//
// y-cruncher custom custom:"filename.cfg"
//
// Author: Jorge Zuniga (optimized by Alex Yee)
// Date: September 28, 2023
//
// Value: 5.24411510858423962092967917978223882736550990286324...
//
// _ _
// inf ( (-1/1024)^n P(n) | 3/4 1/8 3/8 5/8 7/8 | )
// 2 omega = Sqrt(2) SUM ( -------------------------------- | | )
// n=1 ( -(4n-1)(8n-1)(8n-3)(8n-5)(8n-7) |_ 1/2 1/16 5/16 9/16 13/16 _|n )
// _
// Or
// inf ( P(n) k=n ( -(4k-1)(8k-1)(8k-3)(8k-5)(8k-7) ) )
// 2 omega = Sqrt(2) SUM (-------------------------------- PROD (------------------------------------------) )
// n=1 ( -(4n-1)(8n-1)(8n-3)(8n-5)(8n-7) k=1 ( 128(2k-1)(16k-3)(16k-7)(16k-11)(16k-15) ) )
//
//
// P(n) = 3358720*n^4 - 5896192*n^3 + 3695360*n^2 - 966176*n + 86016
//
// _ _
// | a b c ... z | (a)_n (b)_n (c)_n ... (z)_n
// | | = -------------------------------
// |_ A B C ... Z _|n (A)_n (B)_n (C)_n ... (Z)_n
//
//
// (w)_n Pochhammer Symbol. (w)_n = w (w + 1) (w + 2) .... (w + n - 1)
//
//
// Cost = 2.8854
//
{
NameShort : "Lemniscate"
NameLong : "Lemniscate Constant"
AlgorithmShort : "Zuniga (2023-i)"
AlgorithmLong : "Zuniga (2023-i)"
Formula : {
Multiply : [
{SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
PolynomialP : [172032 -1932352 7390720 -11792384 6717440]
PolynomialQ : [-443520 5384448 -23609344 48103424 -46137344 16777216]
PolynomialR : [105 -1828 11136 -30208 36864 -16384]
}}
{Invsqrt : 2}
]
}
}