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Custom Formulas/#README.txt
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This folder contains a set of custom formulas you can play with.
Formulas can be loaded directly from the Custom Compute menu or entered from the
command line as "custom:filename".
y-cruncher custom custom:"constant - algorithm"
More formulas can be found here on the official GitHub repo:
https://github.com/Mysticial/y-cruncher-Formulas
Documentation for writing custom formulas can be found here:
http://www.numberworld.org/y-cruncher/guides/custom_formulas.html
--------------------------------------------------------------------------------
The rest of this file documents which pairs of formulas are approved for
compute/verify pairs for the purpose of setting a world record or simply having
verified digits.
Legend:
"Fastest Pair" indicates the fastest approved pair for formulas.
"Dependent Set" indicates a set of similar formula. You cannot do a
compute/verify pair with 2 formulas in the same dependent set.
The term "same relative dependency" will be used several times here. This means
that two formulas are dependent on the same constant in such a way that if the
constant were changed, both formulas will evaluate to the same (incorrect) value.
Formula pairs that fall into these category will use different algorithms for
that constant and are rearranged in a way to avoid any computational dependencies.
--------------------------------------------------------------------------------
2^(1/5):
Fastest Pair:
2^(1d5) - Native.cfg
2^(1d5) - Series.cfg
The series formula is very slow. Instead, just compute: 200000^(1/5).
The digits will be the same, but shifted over by 1.
--------------------------------------------------------------------------------
2^(1/7):
Fastest Pair:
2^(1d7) - Native.cfg
2^(1d7) - Series.cfg
The series formula is very slow. Instead, just compute: 20000000^(1/7).
The digits will be the same, but shifted over by 1.
--------------------------------------------------------------------------------
Catalan's Constant:
Fastest Pair:
Catalan - Pilehrood (short).cfg
Catalan - Guillera (2019).cfg
This constant is supported natively in y-cruncher. So it's faster to use the
built-in implementations instead.
Dependent Set:
Catalan - Huvent (combined).cfg
Catalan - Huvent (optimized).cfg
Catalan - Huvent (original).cfg
Dependent Set:
Catalan - Ramanujan (unoptimized).cfg
Catalan - Ramanujan.cfg
--------------------------------------------------------------------------------
Cbrt(2):
Fastest Pair:
- Cbrt(2) - Native.cfg
- Cbrt(2) - Series2.cfg
The series formula is very slow. Instead, just compute: 2000^(1/3).
The digits will be the same, but shifted over by 1.
--------------------------------------------------------------------------------
Cbrt(3):
Fastest Pair:
- Cbrt(3) - Native.cfg
- Cbrt(3) - Series.cfg
The series formula is very slow. Instead, just compute: 3000^(1/3).
The digits will be the same, but shifted over by 1.
--------------------------------------------------------------------------------
Cos(1):
Fastest Pair:
Cos(1) - Series.cfg
Cos(1) - Half Angle Formula.cfg
--------------------------------------------------------------------------------
e:
Fastest Pair:
e - exp(1).cfg
e - exp(-1).cfg
This constant is supported natively in y-cruncher. So it's faster to use the
built-in implementations instead.
Dependent Set:
e^-1 - Native.cfg
e - exp(-1).cfg
--------------------------------------------------------------------------------
Gamma(1/3):
Fastest Pair:
Gamma(1d3) - Series-Pi
Gamma(1d3) - AGM-Pi
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Gamma(1/4):
Fastest Pair:
Gamma(1d4) - AGM-Pi.cfg
Gamma(1d4) - Series-Pi.cfg
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Gamma(1/6):
Fastest Pair:
Gamma(1d6) - Series-Pi.cfg
Gamma(1d6) - AGM-Pi.cfg
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Gamma(2/3):
Fastest Pair:
Gamma(2d3) - Series-Pi
Gamma(2d3) - AGM-Pi
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Gamma(3/4):
Fastest Pair:
Gamma(3d4) - AGM-Pi.cfg
Gamma(3d4) - Series-Pi.cfg
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Gamma(5/6):
Fastest Pair:
Gamma(2d3) - Series-Pi.cfg
Gamma(2d3) - AGM-Pi.cfg
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Gauss's Constant:
Fastest Pair:
Gauss - AGM.cfg
Gauss - Series.cfg
--------------------------------------------------------------------------------
Golden Ratio:
Fastest Pair:
GoldenRatio - Native Invsqrt.cfg
GoldenRatio - Series.cfg
Note that you really don't want to use the series formula here. The fastest way
to compute+verify is to run the built-in function for this constant and
sqrt(125). The digits will be the same except for the 2nd digit and all the
digits will be shifted over by one.
--------------------------------------------------------------------------------
Lemniscate:
Fastest Pair:
Lemniscate - AGM-Pi.cfg
Lemniscate - Series-Pi.cfg
This is a built-in constant. But the 2nd formula here (Series-Pi) is actually
faster than both the built-in ArcSinlemn formulas.
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Log(2):
Fastest Pair:
Log(2) - Machin (3 terms).cfg
Log(2) - Machin (4 terms).cfg
This constant is supported natively in y-cruncher. So it's faster to use the
built-in implementations instead.
--------------------------------------------------------------------------------
Log(Pi):
Fastest Pair: none
Dependent Set:
Log(Pi) (unoptimized).cfg
Log(Pi).cfg
--------------------------------------------------------------------------------
Pi:
Fastest Pair:
Pi - Chudnovsky.cfg
Pi - Ramanujan.cfg
This constant is supported natively in y-cruncher. So it's faster to use the
built-in implementations instead.
--------------------------------------------------------------------------------
Sin(1):
Fastest Pair:
Sin(1) - Half Angle Formula.cfg
Sin(1) - Series.cfg
--------------------------------------------------------------------------------
Sqrt(2):
Fastest Pair:
Sqrt(2) - Native Invsqrt.cfg
Sqrt(2) - Series.cfg
The series formula is very slow. Instead, just compute: Sqrt(200).
The digits will be the same, but shifted over by 1.
--------------------------------------------------------------------------------
Universal Parabolic Constant:
Fastest Pair: none
Dependent Set:
Universal Parabolic Constant (unoptimized).cfg
Universal Parabolic Constant.cfg
--------------------------------------------------------------------------------
Zeta(2):
Fastest Pair:
Zeta(2) - Chudnovsky.cfg
Zeta(2) - Direct.cfg
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Zeta(3):
Fastest Pair:
Zeta(3) - Wedeniwski.cfg
Zeta(3) - Amdeberhan-Zeilberger.cfg
This constant is supported natively in y-cruncher. So it's faster to use the
built-in implementations instead.
--------------------------------------------------------------------------------
Zeta(4):
Fastest Pair:
Zeta(4) - Chudnovsky.cfg
Zeta(4) - Direct.cfg
These two formulas have the same relative dependency on Pi. Therefore Pi is set
to use two different algorithms.
--------------------------------------------------------------------------------
Zeta(5):
Fastest Pair:
Zeta(5) - BBP-Kruse.cfg
Zeta(5) - Broadhurst (optimized).cfg
Dependent Set:
Zeta(5) - Broadhurst.cfg
Zeta(5) - Broadhurst (Huvent 2006).cfg
Zeta(5) - Broadhurst (optimized).cfg
Custom Formulas/2^(1d5) - Native.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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// 5th root of 2.
//
// Value: 1.14869835499703500679862694677792758944385088909779...
//
{
NameShort : "2^(⅕)"
NameLong : "5th root of 2"
AlgorithmShort : "Native"
AlgorithmLong : "Native InvNthRoot"
Formula : {
Shift : [
{InvNthRoot : [5 16]}
1
]
}
}
Custom Formulas/2^(1d5) - Series.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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 11, 2018
//
// 5th root of 2.
//
// Value: 1.14869835499703500679862694677792758944385088909779...
//
{
NameShort : "2^(⅕)"
NameLong : "5th root of 2"
AlgorithmShort : "Series"
AlgorithmLong : "Hypergeometric Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [4 -5]
PolynomialQ : [0 -10]
PolynomialR : [4 -5]
}
}
}
Custom Formulas/2^(1d7) - Native.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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// 7th root of 2.
//
// Value: 1.10408951367381233764950538762334472132532660078012...
//
{
NameShort : "2^(⅐)"
NameLong : "7th root of 2"
AlgorithmShort : "Native"
AlgorithmLong : "Native InvNthRoot"
Formula : {
Shift : [
{InvNthRoot : [7 64]}
1
]
}
}
Custom Formulas/2^(1d7) - Series.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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 11, 2018
//
// 7th root of 2.
//
// Value: 1.10408951367381233764950538762334472132532660078012...
//
{
NameShort : "2^(⅐)"
NameLong : "7th root of 2"
AlgorithmShort : "Series"
AlgorithmLong : "Hypergeometric Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [6 -7]
PolynomialQ : [0 -14]
PolynomialR : [6 -7]
}
}
}
Custom Formulas/ArcCosh(e).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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 30, 2018
//
// Value: 1.65745445415307727259382874228053473915839276203367...
//
{
NameShort : "ArcCosh(e)"
NameLong : "ArcCosh(e)"
AlgorithmShort : "Log-Formula"
AlgorithmLong : "Log Expansion of ArcCosh(x)"
Formula : {
Log : {Scope : {
Locals : [
{arg : {E : {}}}
]
Formula : {
LinearCombination : [
[1 "arg"]
[1 {Sqrt : {
LinearCombination : [
[1 {Power : ["arg" 2]}]
[1 -1]
]
}}]
]
}
}}
}
}
Custom Formulas/ArcCoth(GoldenRatio).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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 30, 2018
//
// Value: 0.72181773758940517124663837013655263470277650157849...
//
{
NameShort : "ArcCoth(GoldenRatio)"
NameLong : "ArcCoth(GoldenRatio)"
AlgorithmShort : "Log-Formula"
AlgorithmLong : "Log Expansion of ArcCoth(x)"
Formula : {
Shift : [
{Log : {Scope : {
Locals : [
{arg : {GoldenRatio : {}}}
]
Formula : {
Divide : [
{LinearCombination : [[1 "arg"][1 1]]}
{LinearCombination : [[1 "arg"][1 -1]]}
]
}
}}}
-1
]
}
}
Custom Formulas/ArcSinh(e).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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 30, 2018
//
// Value: 1.72538255885231509394509797040488875627455727467293...
//
{
NameShort : "ArcSinh(e)"
NameLong : "ArcSinh(e)"
AlgorithmShort : "Log-Formula"
AlgorithmLong : "Log Expansion of ArcSinh(x)"
Formula : {
Log : {Scope : {
Locals : [
{arg : {E : {}}}
]
Formula : {
LinearCombination : [
[1 "arg"]
[1 {Sqrt : {
LinearCombination : [
[1 {Power : ["arg" 2]}]
[1 1]
]
}}]
]
}
}}
}
}
Custom Formulas/ArcTanh(Pi^-1).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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 30, 2018
//
// Value: 0.32976531495669910761786341755521860427013739114069...
//
{
NameShort : "ArcTanh(Pi^-1)"
NameLong : "ArcTanh(Pi^-1)"
AlgorithmShort : "Log-Formula"
AlgorithmLong : "Log Expansion of ArcTanh(x)"
Formula : {
Shift : [
{Scope : {
Locals : [
{pi : {Pi : {}}}
{log2 : {Log : 2}}
]
Formula : {
Log-AGM : {
Pi : "pi"
Log2 : "log2"
Argument : {
Divide : [
{LinearCombination : [[1 "pi"][1 1]]}
{LinearCombination : [[1 "pi"][1 -1]]}
]
}
}
}
}}
-1
]
}
}
Custom Formulas/ArcTanh(e^-1).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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 30, 2018
//
// Value: 0.38596841645265236253531957001759267189612899618127...
//
{
NameShort : "ArcTanh(e^-1)"
NameLong : "ArcTanh(e^-1)"
AlgorithmShort : "Log-Formula"
AlgorithmLong : "Log Expansion of ArcTanh(x)"
Formula : {
Shift : [
{Log : {Scope : {
Locals : [
{arg : {E : {Power : -1}}}
]
Formula : {
Divide : [
{LinearCombination : [[1 "arg"][1 1]]}
{LinearCombination : [[-1 "arg"][1 1]]}
]
}
}}}
-1
]
}
}
Custom Formulas/BesselI(0,1) - Series.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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 21, 2018
//
// Value: 1.26606587775200833559824462521471753760767031135496...
//
// Formula: Taylor Series of BesselI(0,x)
//
{
NameShort : "BesselI(0,1)"
NameLong : "BesselI(0,1)"
AlgorithmShort : "Series"
AlgorithmLong : "Taylor Series of BesselI(0,x)"
Formula : {
SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 0 4]
}
}
}
Custom Formulas/BesselJ(0,1) - Series.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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 21, 2018
//
// Value: 0.76519768655796655144971752610266322090927428975532...
//
// Formula: Taylor Series of BesselJ(0,x)
//
{
NameShort : "BesselJ(0,1)"
NameLong : "BesselJ(0,1)"
AlgorithmShort : "Series"
AlgorithmLong : "Taylor Series of BesselJ(0,x)"
Formula : {
SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 0 -4]
}
}
}
Custom Formulas/Catalan - Guillera (2008).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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// Original formula by Guillera (2008). Optimized as follows:
//
// 1 inf ( (-8)^k (3k+2) k!^6 )
// Catalan = --- SUM (--------------------)
// 2 k=0 ( (2k+1)!^3 )
//
// https://arxiv.org/pdf/1104.0396.pdf
// http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#catalan_guillera
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Guillera (2008)"
AlgorithmLong : "Guillera (2008)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 2
CoefficientD : 2
PolynomialP : [0 0 0 2 3]
PolynomialQ : [-1 -6 -12 -8]
PolynomialR : [0 0 0 1]
}
}
}
Custom Formulas/Catalan - Guillera (2019).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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: May 8, 2019
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// https://drive.google.com/file/d/1KUALCj_8JMnKUcABXSkRhoZ2YyrF92Mh/view
// http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#catalan_guillera2019
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Guillera (2019)"
AlgorithmLong : "Guillera (2019)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2
PolynomialP : [165 -3160 21240 -57184 45136]
PolynomialQ : [3375 -72900 597780 -2309472 4304016 -3779136 1259712]
PolynomialR : [0 0 0 512 -3072 6144 -4096]
}
}
}
Custom Formulas/Catalan - Huvent (combined).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.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 2, 2018
//
// Value = 0.91596559417721901505460351493238411077414937428167...
//
// Formula: G. Huvent (2006)
//
// 3 inf ( 1 ( 2 2 1 ))
// Catalan = --- SUM (-------- (---------- - ---------- + ----------))
// 4 k=0 ( (-4)^k ( (4k+1)^2 (4k+2)^2 (4k+3)^2 ))
//
// 1 inf ( 1 ( 8 4 1 ))
// - ---- SUM (--------- (---------- + ---------- + ----------))
// 32 k=0 ( (-64)^k ( (4k+1)^2 (4k+2)^2 (4k+3)^2 ))
//
//
// Optimized by Alex Yee (2013) to:
//
// 1 inf ( 1 ( 576 144 72 18
// Catalan = ----- SUM (--------- (----------- - ----------- - ----------- - ------------
// 384 k=0 ( (-64)^k ( (12k+1)^2 (12k+5)^2 (12k+7)^2 (12k+11)^2
//
// 144 9 64 8 8 ))
// - ---------- - ---------- - ---------- - ---------- - ---------- ))
// (6k+1)^2 (6k+5)^2 (4k+1)^2 (4k+3)^2 (2k+1)^2 ))
//
// This implementation combines all the terms into a single massive one.
// This mirrors y-cruncher's native implementation.
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Huvent (combined)"
AlgorithmLong : "Huvent (2006) (combined)"
Formula : {
SeriesBinaryBBP : {
CoefficientP : -1334025
CoefficientQ : 469137217
CoefficientD : 512265600
Alternating : "true"
PowerCoef : -6
PowerShift : 0
PolynomialP : [
-11728430425 -360406957080 -4419704161120 -24153149527360
5572409156528 1029713994232704 8098743309464448 36670081182465024
113872382317283328 256653109156773888 428868733241327616 532937589964406784
486920573914447872 318025249742389248 140614334152704000 37721048963088384
4636372233093120
]
PolynomialQ : [
33350625 2116999500 60083208700 1015899910560
11506397536624 93025165344960 558162678313024 2549178063839232
9008762620336128 24883108561207296 53971174016925696 91902035745570816
122177336673042432 125309104724901888 97123825191223296 54970731296980992
21422417343676416 5135673858195456 570630428688384
]
}
}
}
Custom Formulas/Catalan - Huvent (optimized).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.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 2, 2018
//
// Value = 0.91596559417721901505460351493238411077414937428167...
//
// Formula: G. Huvent (2006)
//
// 3 inf ( 1 ( 2 2 1 ))
// Catalan = --- SUM (-------- (---------- - ---------- + ----------))
// 4 k=0 ( (-4)^k ( (4k+1)^2 (4k+2)^2 (4k+3)^2 ))
//
// 1 inf ( 1 ( 8 4 1 ))
// - ---- SUM (--------- (---------- + ---------- + ----------))
// 32 k=0 ( (-64)^k ( (4k+1)^2 (4k+2)^2 (4k+3)^2 ))
//
//
// Optimized by Alex Yee (2013) to:
//
// 1 inf ( 1 ( 576 144 72 18
// Catalan = ----- SUM (--------- (----------- - ----------- - ----------- - ------------
// 384 k=0 ( (-64)^k ( (12k+1)^2 (12k+5)^2 (12k+7)^2 (12k+11)^2
//
// 144 9 64 8 8 ))
// - ---------- - ---------- - ---------- - ---------- - ---------- ))
// (6k+1)^2 (6k+5)^2 (4k+1)^2 (4k+3)^2 (2k+1)^2 ))
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Huvent (optimized)"
AlgorithmLong : "Huvent (2006) (optimized)"
Formula : {
LinearCombination : [
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 5
PolynomialP : [3]
PolynomialQ : [121 -264 144]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 3
PolynomialP : [3]
PolynomialQ : [49 -168 144]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 2
PolynomialP : [3]
PolynomialQ : [25 -120 144]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 0
PolynomialP : [3]
PolynomialQ : [1 -24 144]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 3
PolynomialP : [3]
PolynomialQ : [25 -60 36]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2
Alternating : "true"
PowerCoef : -6
PowerShift : 0
PolynomialP : [3]
PolynomialQ : [1 -12 36]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 3
Alternating : "true"
PowerCoef : -6
PowerShift : 5
PolynomialP : [1]
PolynomialQ : [9 -24 16]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 3
Alternating : "true"
PowerCoef : -6
PowerShift : 2
PolynomialP : [1]
PolynomialQ : [1 -8 16]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 3
Alternating : "true"
PowerCoef : -6
PowerShift : 2
PolynomialP : [1]
PolynomialQ : [1 -4 4]
}}]
]
}
}
Custom Formulas/Catalan - Huvent (original).cfg
+95
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@@ -0,0 +1,95 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 2, 2018
//
// Value = 0.91596559417721901505460351493238411077414937428167...
//
// Formula: G. Huvent (2006)
//
// 3 inf ( 1 ( 2 2 1 ))
// Catalan = --- SUM (-------- (---------- - ---------- + ----------))
// 4 k=0 ( (-4)^k ( (4k+1)^2 (4k+2)^2 (4k+3)^2 ))
//
// 1 inf ( 1 ( 8 4 1 ))
// - ---- SUM (--------- (---------- + ---------- + ----------))
// 32 k=0 ( (-64)^k ( (4k+1)^2 (4k+2)^2 (4k+3)^2 ))
//
// http://functions.wolfram.com/Constants/Catalan/06/01/0007/
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Huvent (original)"
AlgorithmLong : "Huvent (2006) (original)"
Formula : {
LinearCombination : [
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -2
PowerShift : 1
PolynomialP : [3]
PolynomialQ : [9 -24 16]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -2
PowerShift : 0
PolynomialP : [3]
PolynomialQ : [2 -8 8]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -2
PowerShift : 0
PolynomialP : [3]
PolynomialQ : [1 -8 16]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 4
PolynomialP : [1]
PolynomialQ : [9 -24 16]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 1
PolynomialP : [1]
PolynomialQ : [1 -4 4]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -6
PowerShift : 1
PolynomialP : [1]
PolynomialQ : [1 -8 16]
}}]
]
}
}
Custom Formulas/Catalan - LogPi.cfg
+49
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@@ -0,0 +1,49 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 4, 2018
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// Unknown formula: http://functions.wolfram.com/PDF/Catalan.pdf
//
// inf ( (2k)! ) Pi Log(2)
// Catalan = sqrt(2) SUM (-------------------) - -----------
// k=0 ( 8^k k!^2 (2k+1)^2 ) 4
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "LogPi"
AlgorithmLong : "Unknown Log-Pi Series"
Formula : {
LinearCombination : [
[2 {Multiply : [
{SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [-1 2]
PolynomialQ : [0 4 16 16]
PolynomialR : [-1 -2 4 8]
}}
{Invsqrt : 2}
]
}]
[-1 {Shift : [
{Multiply : [
{Pi : {}}
{Log : 2}
]}
-2
]
}]
]
}
}
Custom Formulas/Catalan - Lupas.cfg
+38
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@@ -0,0 +1,38 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// Formula by A. Lupas (2000):
//
// 1 inf ( (-1)^(k-1) 256^k (40k^2 - 24k + 3) (2k)!^3 k!^2 )
// Catalan = ---- SUM (-------------------------------------------------)
// 64 k=0 ( k^3 (2k-1) (4k)!^2 )
//
// http://www.lacim.uqam.ca/~plouffe/articles/alupas1.pdf
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Lupas"
AlgorithmLong : "Lupas (2000)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 19
CoefficientD : 18
PolynomialP : [0 0 0 -608 -576 2304 2560]
PolynomialQ : [-9 -96 -352 -512 -256]
PolynomialR : [0 0 0 -32 64]
}
}
}
Custom Formulas/Catalan - Pilehrood (long).cfg
+39
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@@ -0,0 +1,39 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// Formula by Pilehrood (2010):
//
// 1 inf ( (-256)^k (419840k^6 - 915456k^5 + 782848k^4 - 332800k^3 + 73256k^2 - 7800k + 315) k!^2 (4k)!^4 )
// Catalan = ---- SUM (------------------------------------------------------------------------------------------------)
// 64 k=0 ( k^3 (2k-1) (4k-1)^2 (4k-3)^2 (8k)!^2 (2k)! )
//
// https://hal.inria.fr/hal-00990465/document
// http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#catalan_pilehrood_long
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Pilehrood (long)"
AlgorithmLong : "Pilehrood (2010) (long)"
Formula : {
SeriesHypergeometric : {
CoefficientP : -1
CoefficientQ : 0
CoefficientD : 2
PolynomialP : [315 -7800 73256 -332800 782848 -915456 419840]
PolynomialQ : [-11025 295680 -3138304 17219584 -54222848 101711872 -112197632 67108864 -16777216]
PolynomialR : [0 0 0 -288 3648 -17408 38912 -40960 16384]
}
}
}
Custom Formulas/Catalan - Pilehrood (short).cfg
+39
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@@ -0,0 +1,39 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// Formula by Pilehrood (2010):
//
// 1 inf ( 256^k (580k^2 - 184k + 15) (2k)!^3 (3k)!^2 )
// Catalan = ---- SUM (--------------------------------------------)
// 64 k=0 ( k^3 (2k-1) (6k)!^2 )
//
// https://hal.inria.fr/hal-00990465/document
// http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#catalan_pilehrood_short
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Pilehrood (short)"
AlgorithmLong : "Pilehrood (2010) (short)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2
PolynomialP : [15 -184 580]
PolynomialQ : [225 -3240 14904 -23328 11664]
PolynomialR : [0 0 0 -32 64]
}
}
}
Custom Formulas/Catalan - Ramanujan (unoptimized).cfg
+48
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@@ -0,0 +1,48 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 1, 2018
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// Pi 3 inf ( k!^2 )
// Catalan = ---- Log(2 + sqrt(3)) + --- SUM (----------------)
// 8 8 k=0 ( (2k+1)^2 (2k)! )
//
// This is not efficient because it computes Pi twice.
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Ramanujan"
AlgorithmLong : "Ramanujan's Formula"
Formula : {
LinearCombination : [
[1 {SeriesHypergeometric : {
Power : 1
CoefficientP : 3
CoefficientQ : 3
CoefficientD : 8
PolynomialP : [0 1]
PolynomialQ : [-2 -4 8 16]
PolynomialR : [0 1 4 4]
}}]
[1 {Multiply : [
{Log : {
LinearCombination : [
[1 2]
[1 {Sqrt : 3}]
]
}}
{Shift : [{Pi : {}} -3]}
]}]
]
}
}
Custom Formulas/Catalan - Ramanujan.cfg
+58
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@@ -0,0 +1,58 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 1, 2018
//
// Value: 0.91596559417721901505460351493238411077414937428167...
//
// Pi 3 inf ( k!^2 )
// Catalan = ---- Log(2 + sqrt(3)) + --- SUM (----------------)
// 8 8 k=0 ( (2k+1)^2 (2k)! )
//
{
NameShort : "Catalan"
NameLong : "Catalan's Constant"
AlgorithmShort : "Ramanujan"
AlgorithmLong : "Ramanujan's Formula"
Formula : {
LinearCombination : [
[1 {SeriesHypergeometric : {
Power : 1
CoefficientP : 3
CoefficientQ : 3
CoefficientD : 8
PolynomialP : [0 1]
PolynomialQ : [-2 -4 8 16]
PolynomialR : [0 1 4 4]
}}]
[1 {Scope : {
Locals : [
{pi : {Pi : {}}}
{log2 : {Log : 2}}
]
Formula : {
Multiply : [
{Log-AGM : {
Pi : "pi"
Log2 : "log2"
Argument : {
LinearCombination : [
[1 2]
[1 {Sqrt : 3}]
]
}
}}
{Shift : ["pi" -3]}
]
}
}}]
]
}
}
Custom Formulas/Cbrt(2) - Native.cfg
+28
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@@ -0,0 +1,28 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Cube root of 2.
//
// Value: 1.25992104989487316476721060727822835057025146470150...
//
{
NameShort : "Cbrt(2)"
NameLong : "Cube Root of 2"
AlgorithmShort : "Native"
AlgorithmLong : "Native InvNthRoot"
Formula : {
Shift : [
{InvNthRoot : [3 4]}
1
]
}
}
Custom Formulas/Cbrt(2) - Series.cfg
+32
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@@ -0,0 +1,32 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 11, 2018
//
// Cube root of 2.
//
// Value: 1.25992104989487316476721060727822835057025146470150...
//
{
NameShort : "Cbrt(2)"
NameLong : "Cube Root of 2"
AlgorithmShort : "Series"
AlgorithmLong : "Hypergeometric Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [2 -3]
PolynomialQ : [0 -6]
PolynomialR : [2 -3]
}
}
}
Custom Formulas/Cbrt(2) - Series2.cfg
+32
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@@ -0,0 +1,32 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 13, 2018
//
// Cube root of 2.
//
// Value: 1.25992104989487316476721060727822835057025146470150...
//
{
NameShort : "Cbrt(2)"
NameLong : "Cube Root of 2"
AlgorithmShort : "Series 2"
AlgorithmLong : "Hypergeometric Series 2"
Formula : {
SeriesHypergeometric : {
CoefficientP : 5
CoefficientQ : 5
CoefficientD : 4
PolynomialP : [2 -3]
PolynomialQ : [0 -128]
PolynomialR : [2 -3]
}
}
}
Custom Formulas/Cbrt(3) - Native.cfg
+28
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@@ -0,0 +1,28 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Cube root of 3.
//
// Value: 1.44224957030740838232163831078010958839186925349935...
//
{
NameShort : "Cbrt(3)"
NameLong : "Cube Root of 3"
AlgorithmShort : "Native"
AlgorithmLong : "Native InvNthRoot"
Formula : {
LinearCombination : [[
3
{InvNthRoot : [3 9]}
]]
}
}
Custom Formulas/Cbrt(3) - Series.cfg
+32
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@@ -0,0 +1,32 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 13, 2018
//
// Cube root of 3.
//
// Value: 1.44224957030740838232163831078010958839186925349935...
//
{
NameShort : "Cbrt(3)"
NameLong : "Cube Root of 3"
AlgorithmShort : "Series"
AlgorithmLong : "Hypergeometric Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 3
CoefficientQ : 3
CoefficientD : 2
PolynomialP : [4 -3]
PolynomialQ : [0 -27]
PolynomialR : [4 -3]
}
}
}
Custom Formulas/Cos(1) - Half Angle Formula.cfg
+37
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@@ -0,0 +1,37 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 6, 2018
//
// Value: 0.54030230586813971740093660744297660373231042061792...
//
// Formula: 2 Cos(1/2)^2 - 1
//
{
NameShort : "Cos(1)"
NameLong : "Cosine of 1"
AlgorithmShort : "Half Angle Formula"
AlgorithmLong : "2 Cos(1/2)^2 - 1"
Formula : {
LinearCombination : [
[2 {Power : [
{SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 8 -16]
}}
2
]}]
[-1 1]
]
}
}
Custom Formulas/Cos(1) - Series.cfg
+31
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@@ -0,0 +1,31 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 6, 2018
//
// Value: 0.54030230586813971740093660744297660373231042061792...
//
// Formula: Taylor Series of Cos(x)
//
{
NameShort : "Cos(1)"
NameLong : "Cosine of 1"
AlgorithmShort : "Series"
AlgorithmLong : "Taylor Series of cos(1)"
Formula : {
SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 2 -4]
}
}
}
Custom Formulas/Erf(1) - Series.cfg
+35
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@@ -0,0 +1,35 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 24, 2018
//
// Value: 0.84270079294971486934122063508260925929606699796630...
//
{
NameShort : "Erf(1)"
NameLong : "Erf(1)"
AlgorithmShort : "Series"
AlgorithmLong : "Taylor Series"
Formula : {
Multiply : [
{SeriesHypergeometric : {
CoefficientP : -6
CoefficientQ : 4
CoefficientD : 3
PolynomialP : [1]
PolynomialQ : [-3 -5 -2]
PolynomialR : [3 2]
}}
{Invsqrt : {
Pi : {}
}}
]
}
}
Custom Formulas/Gamma(1d3) - AGM-Pi.cfg
+47
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@@ -0,0 +1,47 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 2.67893853470774763365569294097467764412868937795730...
//
// Source: http://arblib.org/gamma.html
//
{
NameShort : "Gamma(⅓)"
NameLong : "Gamma(1/3)"
AlgorithmShort : "AGM-Pi"
AlgorithmLong : "AGM-Pi"
Formula : {
Divide : [
{Multiply : [
{InvNthRoot : [
3
{AGM : {Invsqrt :
{LinearCombination : [
[1 8]
[-1 {Sqrt : 48}]
]}
}}
]}
{InvNthRoot : [12 3]}
]}
{Multiply : [
{InvNthRoot : [3
{Power : [
{Pi : {Algorithm : "ramanujan"}}
2
]}
]}
{InvNthRoot : [9 16]}
]}
]
}
}
Custom Formulas/Gamma(1d3) - Series-Pi.cfg
+45
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@@ -0,0 +1,45 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 2.67893853470774763365569294097467764412868937795730...
//
// Sources:
// http://iamned.com/math/infiniteseries.pdf
// http://arblib.org/gamma.html
//
{
NameShort : "Gamma(⅓)"
NameLong : "Gamma(1/3)"
AlgorithmShort : "Series-Pi (Brown)"
AlgorithmLong : "Series-Pi (Brown 2009)"
Formula : {
InvNthRoot : [
6
{Multiply : [
{SeriesHypergeometric : {
Power : -1
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 5
PolynomialP : [-5 46 -108 72]
PolynomialQ : [0 0 0 -512000]
PolynomialR : [-5 46 -108 72]
}}
{Power : [
{Pi : {Power : -1}}
4
]}
{Invsqrt : 360}
]}
]
}
}
Custom Formulas/Gamma(1d4) - AGM-Pi.cfg
+44
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@@ -0,0 +1,44 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 24, 2018
//
// Value: 3.62560990822190831193068515586767200299516768288006...
//
// ( 1 ) 1
// Gamma(---) = ----------------------------
// ( 4 ) ( AGM(1, sqrt(2)) )
// sqrt( ------------------ )
// ( sqrt( (2 Pi)^3 ) )
//
// Source: http://mathworld.wolfram.com/GausssConstant.html
//
{
NameShort : "Gamma(¼)"
NameLong : "Gamma(1/4)"
AlgorithmShort : "AGM-Pi"
AlgorithmLong : "AGM-Pi"
Formula : {
Invsqrt : {
Multiply : [
{AGM : {Sqrt : 2}}
{Invsqrt : {
Power : [
{Shift : [
{Pi : {}}
1
]}
3
]
}}
]
}
}
}
Custom Formulas/Gamma(1d4) - Lemniscate.cfg
+39
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@@ -0,0 +1,39 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 24, 2018
//
// Value: 3.62560990822190831193068515586767200299516768288006...
//
// ( 1 ) ( Lemniscate )
// Gamma(---) = sqrt(---------------)
// ( 4 ) ( ( 1 ) )
// ( sqrt(------) )
// ( ( 2 Pi ) )
//
{
NameShort : "Gamma(¼)"
NameLong : "Gamma(1/4)"
AlgorithmShort : "Lemniscate"
AlgorithmLong : "Lemniscate Formula"
Formula : {
Sqrt : {
Multiply : [
{Lemniscate : {Algorithm : "gauss"}}
{Invsqrt : {
Shift : [
{Pi : {Power : -1}}
-1
]
}}
]
}
}
}
Custom Formulas/Gamma(1d4) - Series-Pi.cfg
+49
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@@ -0,0 +1,49 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 26, 2019
//
// Value: 3.62560990822190831193068515586767200299516768288006...
//
// Source: http://iamned.com/math/infiniteseries.pdf
//
// This file uses Ramanujan's formula for Pi to make it computationally
// independent from "Gamma(1d4) - AGM-Pi.cfg".
//
{
NameShort : "Gamma(¼)"
NameLong : "Gamma(1/4)"
AlgorithmShort : "Series-Pi (Brown)"
AlgorithmLong : "Series-Pi (Brown 2009)"
Formula : {
Invsqrt : {
Multiply : [
{SeriesHypergeometric : {
Power : -1
CoefficientP : 1
CoefficientQ : 147
CoefficientD : 108
PolynomialP : [15435 -143136 20224 1241088 -856064 -2752512 2621440]
PolynomialQ : [0 0 82944 331776 331776]
PolynomialR : [105 -1408 5504 -8192 4096]
}}
{Invsqrt : {
LinearCombination : [[
3
{Power : [
{Pi : {Algorithm : "ramanujan"}}
3
]}
]]
}}
]
}
}
}
Custom Formulas/Gamma(1d6) - AGM-Pi.cfg
+51
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@@ -0,0 +1,51 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 5.56631600178023520425009689520772611139879911487285...
//
// Source: http://arblib.org/gamma.html
//
{
NameShort : "Gamma(⅙)"
NameLong : "Gamma(1/6)"
AlgorithmShort : "AGM-Pi"
AlgorithmLong : "AGM-Pi"
Formula : {
Scope : {
Locals : [
{pi : {Pi : {Algorithm : "ramanujan"}}}
]
Formula : {Multiply : [
{InvNthRoot : [
6
{Multiply : [
{Power : [
{AGM : {Invsqrt :
{LinearCombination : [
[1 8]
[-1 {Sqrt : 48}]
]}
}}
4
]}
"pi"
]}
]}
"pi"
{LinearCombination : [[
6
{InvNthRoot : [9 11664]}
]]}
]}
}
}
}
Custom Formulas/Gamma(1d6) - Series-Pi.cfg
+44
View File
@@ -0,0 +1,44 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 5.56631600178023520425009689520772611139879911487285...
//
// Sources:
// http://arblib.org/gamma.html
// http://iamned.com/math/infiniteseries.pdf
//
{
NameShort : "Gamma(⅙)"
NameLong : "Gamma(1/6)"
AlgorithmShort : "Series-Pi (Brown)"
AlgorithmLong : "Series-Pi (Brown 2009)"
Formula : {
InvNthRoot : [
3
{Multiply : [
{SeriesHypergeometric : {
Power : -1
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 5
PolynomialP : [-5 46 -108 72]
PolynomialQ : [0 0 0 -512000]
PolynomialR : [-5 46 -108 72]
}}
{Invsqrt : {LinearCombination : [[
2430
{Power : [{Pi : {}} 5]}
]]}}
]}
]
}
}
Custom Formulas/Gamma(2d3) - AGM-Pi.cfg
+40
View File
@@ -0,0 +1,40 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 1.35411793942640041694528802815451378551932726605679...
//
// Source: http://arblib.org/gamma.html
//
{
NameShort : "Gamma(⅔)"
NameLong : "Gamma(2/3)"
AlgorithmShort : "AGM-Pi"
AlgorithmLong : "AGM-Pi"
Formula : {
Divide : [
{InvNthRoot : [12 243]}
{Multiply : [
{InvNthRoot : [
3
{AGM : {Invsqrt :
{LinearCombination : [
[1 8]
[-1 {Sqrt : 48}]
]}
}}
]}
{InvNthRoot : [3 {Pi : {Algorithm : "ramanujan"}}]}
{InvNthRoot : [9 32]}
]}
]
}
}
Custom Formulas/Gamma(2d3) - Series-Pi.cfg
+44
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@@ -0,0 +1,44 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 1.35411793942640041694528802815451378551932726605679...
//
// Sources:
// http://arblib.org/gamma.html
// http://iamned.com/math/infiniteseries.pdf
//
{
NameShort : "Gamma(⅔)"
NameLong : "Gamma(2/3)"
AlgorithmShort : "Series-Pi (Brown)"
AlgorithmLong : "Series-Pi (Brown 2009)"
Formula : {
InvNthRoot : [
6
{Multiply : [
{SeriesHypergeometric : {
CoefficientP : 81
CoefficientQ : 81
CoefficientD : 1
PolynomialP : [-5 46 -108 72]
PolynomialQ : [0 0 0 -512000]
PolynomialR : [-5 46 -108 72]
}}
{Power : [
{Pi : {Power : -1}}
2
]}
{Invsqrt : 2560}
]}
]
}
}
Custom Formulas/Gamma(3d4) - AGM-Pi.cfg
+39
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@@ -0,0 +1,39 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 24, 2018
//
// Value: 1.22541670246517764512909830336289052685123924810807...
//
// ( 3 ) ( ( Pi ))
// Gamma(---) = sqrt( AGM(1, sqrt(2)) * sqrt(----))
// ( 4 ) ( ( 2 ))
//
// Source: http://arblib.org/gamma.html
//
{
NameShort : "Gamma(¾)"
NameLong : "Gamma(3/4)"
AlgorithmShort : "AGM-Pi"
AlgorithmLong : "AGM-Pi"
Formula : {
Sqrt : {
Multiply : [
{AGM : {Sqrt : 2}}
{Sqrt : {
Shift : [
{Pi : {}}
-1
]
}}
]
}
}
}
Custom Formulas/Gamma(3d4) - Lemniscate.cfg
+42
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@@ -0,0 +1,42 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 24, 2018
//
// Value: 1.22541670246517764512909830336289052685123924810807...
//
// ( 3 ) 1
// Gamma(---) = ----------------------
// ( 4 ) ( Lemniscate )
// sqrt(--------------)
// ( sqrt(2 Pi^3) )
//
{
NameShort : "Gamma(¾)"
NameLong : "Gamma(3/4)"
AlgorithmShort : "Lemniscate"
AlgorithmLong : "Lemniscate Formula"
Formula : {
Invsqrt : {
Multiply : [
{Lemniscate : {Algorithm : "gauss"}}
{Invsqrt : {
Shift : [
{Power : [
{Pi : {}}
3
]}
1
]
}}
]
}
}
}
Custom Formulas/Gamma(3d4) - Series-Pi.cfg
+45
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@@ -0,0 +1,45 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 26, 2019
//
// Value: 1.22541670246517764512909830336289052685123924810807...
//
// Source: http://iamned.com/math/infiniteseries.pdf
//
// This file uses Ramanujan's formula for Pi to make it computationally
// independent from "Gamma(1d4) - AGM-Pi.cfg".
//
{
NameShort : "Gamma(¾)"
NameLong : "Gamma(3/4)"
AlgorithmShort : "Series-Pi (Brown)"
AlgorithmLong : "Series-Pi (Brown 2009)"
Formula : {
Invsqrt : {
Multiply : [
{SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 2
PolynomialP : [9 -24 16]
PolynomialQ : [0 0 -128]
PolynomialR : [9 -24 16]
}}
{Invsqrt : {
Multiply : [
{Pi : {Algorithm : "ramanujan"}}
{Invsqrt : 32}
]
}}
]
}
}
}
Custom Formulas/Gamma(5d6) - AGM-Pi.cfg
+51
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@@ -0,0 +1,51 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 1.12878702990812596126090109025884201332678744166475...
//
// Source: http://arblib.org/gamma.html
//
{
NameShort : "Gamma(⅚)"
NameLong : "Gamma(5/6)"
AlgorithmShort : "AGM-Pi"
AlgorithmLong : "AGM-Pi"
Formula : {
Multiply : [
{InvNthRoot : [
6
{LinearCombination : [[
9
{Power : [
{Multiply : [
{Power : [
{AGM : {Invsqrt :
{LinearCombination : [
[1 8]
[-1 {Sqrt : 48}]
]}
}}
4
]}
{Pi : {Algorithm : "ramanujan"}}
]}
-1
]}
]]}
]}
{Shift : [
{InvNthRoot : [9 32]}
1
]}
]
}
}
Custom Formulas/Gamma(5d6) - Series-Pi.cfg
+43
View File
@@ -0,0 +1,43 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 22, 2019
//
// Value: 1.12878702990812596126090109025884201332678744166475...
//
// Sources:
// http://arblib.org/gamma.html
// http://iamned.com/math/infiniteseries.pdf
//
{
NameShort : "Gamma(⅚)"
NameLong : "Gamma(5/6)"
AlgorithmShort : "Series-Pi (Brown)"
AlgorithmLong : "Series-Pi (Brown 2009)"
Formula : {
InvNthRoot : [
3
{Multiply : [
{SeriesHypergeometric : {
CoefficientP : 27
CoefficientQ : 27
CoefficientD : 1
PolynomialP : [-5 46 -108 72]
PolynomialQ : [0 0 0 -512000]
PolynomialR : [-5 46 -108 72]
}}
{Invsqrt : {LinearCombination : [[
480
{Pi : {}}
]]}}
]}
]
}
}
Custom Formulas/Gauss - AGM.cfg
+30
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@@ -0,0 +1,30 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 26, 2018
//
// Value: 0.83462684167407318628142973279904680899399301349034...
//
// 1
// Gauss's Constant = -----------------
// AGM(1, sqrt(2))
//
{
NameShort : "Gauss"
NameLong : "Gauss's Constant"
AlgorithmShort : "AGM"
AlgorithmLong : "1 / AGM(1, Sqrt(2))"
Formula : {
Power : [
{AGM : {Sqrt : 2}}
-1
]
}
}
Custom Formulas/Gauss - Lemniscate.cfg
+33
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@@ -0,0 +1,33 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 6, 2018
//
// Value: 0.83462684167407318628142973279904680899399301349034...
//
// 1
// Gauss's Constant = -----------------
// AGM(1, sqrt(2))
//
{
NameShort : "Gauss"
NameLong : "Gauss's Constant"
AlgorithmShort : "Lemniscate"
AlgorithmLong : "Lemniscate / (2 Pi)"
Formula : {
Shift : [
{Multiply : [
{Lemniscate : {Algorithm : "gauss"}}
{Pi : {Power : -1}}
]}
-1
]
}
}
Custom Formulas/Gauss - Series.cfg
+43
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@@ -0,0 +1,43 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 26, 2018
//
// Value: 0.83462684167407318628142973279904680899399301349034...
//
// 1
// Gauss's Constant = -----------------
// AGM(1, sqrt(2))
//
//
// The formula used here is derived from Brown's 2009 formula
// for Gamma(1/4).
//
// Source: http://iamned.com/math/infiniteseries.pdf
//
{
NameShort : "Gauss"
NameLong : "Gauss's Constant"
AlgorithmShort : "Series"
AlgorithmLong : "Hypergeometric Series"
Formula : {
Multiply : [
{SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 147
CoefficientD : 1
PolynomialP : [15435 -143136 20224 1241088 -856064 -2752512 2621440]
PolynomialQ : [0 0 82944 331776 331776]
PolynomialR : [105 -1408 5504 -8192 4096]
}}
{Invsqrt : 31104}
]
}
}
Custom Formulas/GoldenRatio - Native Invsqrt.cfg
+29
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@@ -0,0 +1,29 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 6, 2018
//
// Value: 1.61803398874989484820458683436563811772030917980576...
//
{
NameShort : "Golden Ratio"
NameLong : "Golden Ratio"
AlgorithmShort : "Invsqrt"
AlgorithmLong : "Native Invsqrt"
Formula : {
Shift : [
{LinearCombination : [
[5 {Invsqrt : 5}]
[1 1]
]}
-1
]
}
}
Custom Formulas/GoldenRatio - Series.cfg
+30
View File
@@ -0,0 +1,30 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 11, 2018
//
// Value: 1.61803398874989484820458683436563811772030917980576...
//
{
NameShort : "Golden Ratio"
NameLong : "Golden Ratio"
AlgorithmShort : "Series"
AlgorithmLong : "Hypergeometric Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 3
CoefficientD : 2
PolynomialP : [6 -4]
PolynomialQ : [0 8]
PolynomialR : [3 -2]
}
}
}
Custom Formulas/Khinchin-Levy Constant.cfg
+35
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@@ -0,0 +1,35 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 24, 2018
//
// Value: 1.18656911041562545282172297594723712056835653647205...
//
// Pi^2
// Beta = -------------
// 12 * log(2)
//
{
NameShort : "Khinchin-Levy"
NameLong : "Khinchin-Levy Constant"
AlgorithmShort : "Definition"
AlgorithmLong : "Direct Definition"
Formula : {
Divide : [
{Power : [
{Pi : {}}
2
]}
{LinearCombination : [
[12 {Log : 2}]
]}
]
}
}
Custom Formulas/Lemniscate - AGM-Pi.cfg
+30
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@@ -0,0 +1,30 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 26, 2018
//
// Value: 5.24411510858423962092967917978223882736550990286324...
//
// 2 Pi
// Lemniscate = -----------------
// AGM(1, sqrt(2))
//
{
NameShort : "Lemniscate"
NameLong : "Lemniscate Constant"
AlgorithmShort : "AGM-Pi"
AlgorithmLong : "AGM-Pi"
Formula : {
Divide : [
{LinearCombination : [[2 {Pi : {}}]]}
{AGM : {Sqrt : 2}}
]
}
}
Custom Formulas/Lemniscate - Gauss.cfg
+32
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@@ -0,0 +1,32 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 5.24411510858423962092967917978223882736550990286324...
//
// Gauss Formula:
//
// Lemniscate = 8 ArcSinlemn(1/2) + 4 ArcSinlemn(7/23)
//
// http://www.numberworld.org/y-cruncher/internals/formulas.html#lemniscate
//
{
NameShort : "Lemniscate"
NameLong : "Lemniscate Constant"
AlgorithmShort : "Gauss"
AlgorithmLong : "Gauss Formula"
Formula : {
LinearCombination : [
[1 {ArcSinlemn : {Coefficient:8 x:1 y:2}}]
[1 {ArcSinlemn : {Coefficient:4 x:7 y:23}}]
]
}
}
Custom Formulas/Lemniscate - Sebah.cfg
+32
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@@ -0,0 +1,32 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 5.24411510858423962092967917978223882736550990286324...
//
// Sebah's Formula:
//
// Lemniscate = 8 ArcSinlemn(2/3) - 4 ArcSinlemn(7/137)
//
// http://www.numberworld.org/y-cruncher/internals/formulas.html#lemniscate
//
{
NameShort : "Lemniscate"
NameLong : "Lemniscate Constant"
AlgorithmShort : "Sebah"
AlgorithmLong : "Sebah's Formula"
Formula : {
LinearCombination : [
[1 {ArcSinlemn : {Coefficient:8 x:2 y:3}}]
[1 {ArcSinlemn : {Coefficient:-4 x:7 y:137}}]
]
}
}
Custom Formulas/Lemniscate - Series-Pi.cfg
+40
View File
@@ -0,0 +1,40 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: March 2, 2019
//
// Value: 5.24411510858423962092967917978223882736550990286324...
//
//
// The formula used here is derived from Brown's 2009 formula
// for Gamma(1/4).
//
// Source: http://iamned.com/math/infiniteseries.pdf
//
{
NameShort : "Lemniscate"
NameLong : "Lemniscate Constant"
AlgorithmShort : "Series-Pi"
AlgorithmLong : "Series-Pi"
Formula : {
Multiply : [
{SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 147
CoefficientD : 1
PolynomialP : [15435 -143136 20224 1241088 -856064 -2752512 2621440]
PolynomialQ : [0 0 82944 331776 331776]
PolynomialR : [105 -1408 5504 -8192 4096]
}}
{Pi : {Algorithm : "ramanujan"}}
{Invsqrt : 7776}
]
}
}
Custom Formulas/Log(2) - BBP.cfg
+61
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@@ -0,0 +1,61 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 5, 2018
//
// Value: 0.69314718055994530941723212145817656807550013436025...
//
// Formula: Unknown BBP Formula
//
// 1 inf ( 1 ( 8 4 1 ))
// Log(2) = --- SUM (-------- (------ - ------ - ------))
// 8 k=0 ( (-8)^k ( 3k+1 3k+2 3k+3 ))
//
// http://mathworld.wolfram.com/NaturalLogarithmof2.html
//
{
NameShort : "Log(2)"
NameLong : "Log(2)"
AlgorithmShort : "BBP (3 terms)"
AlgorithmLong : "Unknown BBP Formula (3 terms)"
Formula : {
LinearCombination : [
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -3
PowerShift : 3
PolynomialP : [1]
PolynomialQ : [-2 3]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -3
PowerShift : 2
PolynomialP : [1]
PolynomialQ : [-1 3]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -3
PowerShift : 0
PolynomialP : [1]
PolynomialQ : [0 3]
}}]
]
}
}
Custom Formulas/Log(2) - Lupas.cfg
+35
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@@ -0,0 +1,35 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 5, 2018
//
// Value: 0.69314718055994530941723212145817656807550013436025...
//
// Formula: A. Lupas (2000)
//
// 3 1 inf ( (5k + 1) (2k)! )
// Log(2) = --- + --- SUM (-------------------------)
// 4 4 k=1 ( (-16)^k k (2k + 1) k!^2 )
//
{
NameShort : "Log(2)"
NameLong : "Log(2)"
AlgorithmShort : "Lupas"
AlgorithmLong : "Lupas (2000)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 3
CoefficientD : 4
PolynomialP : [-1 -3 10]
PolynomialQ : [0 0 -8 -16]
PolynomialR : [0 -1 0 4]
}
}
}
Custom Formulas/Log(2) - Machin (3 terms).cfg
+29
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@@ -0,0 +1,29 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 0.69314718055994530941723212145817656807550013436025...
//
// Log(2) = 18 ArcCoth(26) - 2 ArcCoth(4801) + 8 ArcCoth(8749)
//
{
NameShort : "Log(2)"
NameLong : "Log(2)"
AlgorithmShort : "Machin (3 terms)"
AlgorithmLong : "Machin-like Formula (3 terms)"
Formula : {
LinearCombination : [
[1 {ArcCoth : {Coefficient:18 x:26}}]
[1 {ArcCoth : {Coefficient:-2 x:4801}}]
[1 {ArcCoth : {Coefficient:8 x:8749}}]
]
}
}
Custom Formulas/Log(2) - Machin (4 terms).cfg
+30
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@@ -0,0 +1,30 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 0.69314718055994530941723212145817656807550013436025...
//
// Log(2) = 144 ArcCoth(251) + 54 ArcCoth(449) - 38 ArcCoth(4801) + 62 ArcCoth(8749)
//
{
NameShort : "Log(2)"
NameLong : "Log(2)"
AlgorithmShort : "Machin (4 terms)"
AlgorithmLong : "Machin-like Formula (4 terms)"
Formula : {
LinearCombination : [
[1 {ArcCoth : {Coefficient:144 x:251}}]
[1 {ArcCoth : {Coefficient:54 x:449}}]
[1 {ArcCoth : {Coefficient:-38 x:4801}}]
[1 {ArcCoth : {Coefficient:62 x:8749}}]
]
}
}
Custom Formulas/Log(Pi) (unoptimized).cfg
+27
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@@ -0,0 +1,27 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 2, 2018
//
// Value: 1.14472988584940017414342735135305871164729481291531...
//
// This is not efficient because it computes Pi twice.
//
{
NameShort : "Log(Pi)"
NameLong : "Log(Pi)"
AlgorithmShort : "Definition"
AlgorithmLong : "Direct Definition"
Formula : {
Log: {
Pi : {}
}
}
}
Custom Formulas/Log(Pi).cfg
+37
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@@ -0,0 +1,37 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 30, 2018
//
// Value: 1.14472988584940017414342735135305871164729481291531...
//
// Same as "Log(Pi).cfg", but avoids computing Pi twice.
//
{
NameShort : "Log(Pi)"
NameLong : "Log(Pi)"
AlgorithmShort : "Definition"
AlgorithmLong : "Direct Definition"
Formula : {
Scope : {
Locals : [
{pi : {Pi : {}}}
{log2 : {Log : 2}}
]
Formula : {
Log-AGM : {
Pi : "pi"
Log2 : "log2"
Argument : "pi"
}
}
}
}
}
Custom Formulas/Log10(2).cfg
+36
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@@ -0,0 +1,36 @@
// 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:"constant - algorithm"
//
//
// Author: Originally from 鲍东方
// Date: February 8, 2019
//
// Value: 0.30102999566398119521373889472449302676818988146210...
//
{
NameShort : "Log10(2)"
NameLong : "Log base 10 of 2"
AlgorithmShort : "Machin (4 terms)"
AlgorithmLong : "Machin-like Formula (4 terms)"
Formula : {
Scope : {
Locals : [
{t0 : {ArcCoth : {x : 251}}}
{t1 : {ArcCoth : {x : 449}}}
{t2 : {ArcCoth : {x : 4801}}}
{t3 : {ArcCoth : {x : 8749}}}
]
Formula : {
Divide : [
{LinearCombination : [[144 "t0"][ 54 "t1"][ -38 "t2"][ 62 "t3"]]}
{LinearCombination : [[478 "t0"][180 "t1"][-126 "t2"][206 "t3"]]}
]
}
}
}
}
Custom Formulas/Pi - BBP (Original).cfg
+71
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@@ -0,0 +1,71 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 1, 2018
//
// Value = 3.14159265358979323846264338327950288419716939937510...
//
// Formula: Original BBP Formula
//
// inf ( 1 ( 4 2 1 1 ))
// Pi = SUM (------ (------ - ------ - ------ - ------))
// k=0 ( 16^k ( 8k+1 8k+4 8k+5 8k+6 ))
//
// https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "BBP (Original)"
AlgorithmLong : "Original BBP Formula"
Formula : {
LinearCombination : [
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -4
PowerShift : 6
PolynomialP : [1]
PolynomialQ : [-7 8]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -4
PowerShift : 3
PolynomialP : [1]
PolynomialQ : [-1 2]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -4
PowerShift : 4
PolynomialP : [1]
PolynomialQ : [-3 8]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -4
PowerShift : 3
PolynomialP : [1]
PolynomialQ : [-1 4]
}}]
]
}
}
Custom Formulas/Pi - Bellard.cfg
+100
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@@ -0,0 +1,100 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 1, 2018
//
// Value = 3.14159265358979323846264338327950288419716939937510...
// Formula: Fabrice Bellard's BBP Formula
//
// 1 inf ( (-1)^k ( 32 1 256 64 4 4 1 ))
// Pi = ---- SUM (-------- (- ------ - ------ + ------- - ------- - ------- - ------- + -------))
// 64 k=0 ( 1024^k ( 4k+1 4k+3 10k+1 10k+3 10k+5 10k+7 10k+9 ))
//
// https://en.wikipedia.org/wiki/Bellard%27s_formula
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Bellard"
AlgorithmLong : "Fabrice Bellard's BBP Formula"
Formula : {
LinearCombination : [
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -10
PowerShift : 9
PolynomialP : [1]
PolynomialQ : [-3 4]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -10
PowerShift : 4
PolynomialP : [1]
PolynomialQ : [-1 4]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -10
PowerShift : 12
PolynomialP : [1]
PolynomialQ : [-9 10]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -10
PowerShift : 10
PolynomialP : [1]
PolynomialQ : [-7 10]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 5
Alternating : "true"
PowerCoef : -10
PowerShift : 6
PolynomialP : [1]
PolynomialQ : [-1 2]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -10
PowerShift : 6
PolynomialP : [1]
PolynomialQ : [-3 10]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "true"
PowerCoef : -10
PowerShift : 4
PolynomialP : [1]
PolynomialQ : [-1 10]
}}]
]
}
}
Custom Formulas/Pi - Brown.cfg
+37
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@@ -0,0 +1,37 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: February 26, 2019
//
// Value: 3.14159265358979323846264338327950288419716939937510...
//
// Source: http://iamned.com/math/infiniteseries.pdf
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Brown"
AlgorithmLong : "Brown (2009)"
Formula : {
Multiply : [
{
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 109
CoefficientD : 1
PolynomialP : [0 109 -88 -260]
PolynomialQ : [90 648 648]
PolynomialR : [0 1 -2]
}
}
{Invsqrt : 1200}
]
}
}
Custom Formulas/Pi - Chudnovsky.cfg
+38
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@@ -0,0 +1,38 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 3.14159265358979323846264338327950288419716939937510...
//
// https://en.wikipedia.org/wiki/Chudnovsky_algorithm
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Chudnovsky"
AlgorithmLong : "Chudnovsky (1988)"
Formula : {
Multiply : [
{
SeriesHypergeometric : {
Power : -1
CoefficientP : 1
CoefficientQ : 13591409
CoefficientD : 4270934400
PolynomialP : [-67957045 -2100495856 23608573992 -57896553024 39250089648]
PolynomialQ : [0 0 0 -10939058860032000]
PolynomialR : [-5 46 -108 72]
}
}
{Invsqrt : 10005}
]
}
}
Custom Formulas/Pi - Hwang (1997).cfg
+79
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@@ -0,0 +1,79 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 3.14159265358979323846264338327950288419716939937510...
//
// Formula: Chien-Lih Hwang's 6-term formula (1997)
//
// Pi = 732 ArcTan(1/239) + 128 ArcTan(1/1023) - 272 ArcTan(1/5832)
// + 48 ArcTan(1/110443) - 48 ArcTan(1/4841182) - 400 ArcTan(1/6826318)
//
// This uses Euler's ArcTan formula which is slightly faster
// than the usual Taylor series.
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Hwang"
AlgorithmLong : "Hwang's 6-term Formula (1997)"
Formula : {
LinearCombination : [
[87474 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 28561
PolynomialP : [0 1]
PolynomialQ : [28561 57122]
PolynomialR : [0 1]
}}]
[65472 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 523265
PolynomialP : [0 1]
PolynomialQ : [523265 1046530]
PolynomialR : [0 1]
}}]
[-1586304 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 34012225
PolynomialP : [0 2]
PolynomialQ : [34012225 68024450]
PolynomialR : [0 2]
}}]
[2650632 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 6098828125
PolynomialP : [0 1]
PolynomialQ : [6098828125 12197656250]
PolynomialR : [0 1]
}}]
[-232376736 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 23437043157125
PolynomialP : [0 2]
PolynomialQ : [23437043157125 46874086314250]
PolynomialR : [0 2]
}}]
[-109221088 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1863944697485
PolynomialP : [0 2]
PolynomialQ : [46598617437125 93197234874250]
PolynomialR : [0 2]
}}]
]
}
}
Custom Formulas/Pi - Machin.cfg
+46
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@@ -0,0 +1,46 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 3.14159265358979323846264338327950288419716939937510...
//
// Formula: Machin's original 2-term formula
//
// Pi = 16 ArcTan(1/5) - 4 ArcTan(1/239)
//
// This uses Euler's ArcTan formula which is slightly faster
// than the usual Taylor series.
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Machin"
AlgorithmLong : "Machin's Formula"
Formula : {
LinearCombination : [
[80 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 26
PolynomialP : [0 1]
PolynomialQ : [13 26]
PolynomialR : [0 1]
}}]
[-956 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 57122
PolynomialP : [0 1]
PolynomialQ : [28561 57122]
PolynomialR : [0 1]
}}]
]
}
}
Custom Formulas/Pi - Ramanujan.cfg
+38
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@@ -0,0 +1,38 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 3.14159265358979323846264338327950288419716939937510...
//
// http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#pi_ramanujan
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Ramanujan"
AlgorithmLong : "Ramanujan (1910)"
Formula : {
Multiply : [
{
SeriesHypergeometric : {
Power : -1
CoefficientP : 1
CoefficientQ : 1103
CoefficientD : 9801
PolynomialP : [-3309 -54904 527636 -1231424 844480]
PolynomialQ : [0 0 0 3073907232]
PolynomialR : [-3 22 -48 32]
}
}
{Invsqrt : 8}
]
}
}
Custom Formulas/Pi - Sharp.cfg
+38
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@@ -0,0 +1,38 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value = 3.14159265358979323846264338327950288419716939937510...
//
// Formula: Abraham Sharp (1717)
//
// Which corresponds to ArcTan(1/sqrt(3))
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Sharp"
AlgorithmLong : "Abraham Sharp (1717)"
Formula : {
Multiply : [
{
SeriesHypergeometric : {
CoefficientP : 6
CoefficientQ : 6
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [-3 -6]
PolynomialR : [1 2]
}
}
{Invsqrt : 3}
]
}
}
Custom Formulas/Pi - Stormer.cfg
+62
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@@ -0,0 +1,62 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 3.14159265358979323846264338327950288419716939937510...
//
// Formula: Carl Stormer's 4-term formula
//
// Pi = 176 ArcTan(1/57) + 28 ArcTan(1/239) - 48 ArcTan(1/682) + 96 ArcTan(1/12943)
//
// This uses Euler's ArcTan formula which is slightly faster
// than the usual Taylor series.
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Stormer"
AlgorithmLong : "Stormer's 4-term formula"
Formula : {
LinearCombination : [
[5016 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1625
PolynomialP : [0 1]
PolynomialQ : [1625 3250]
PolynomialR : [0 1]
}}]
[3346 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 28561
PolynomialP : [0 1]
PolynomialQ : [28561 57122]
PolynomialR : [0 1]
}}]
[-32736 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 465125
PolynomialP : [0 2]
PolynomialQ : [465125 930250]
PolynomialR : [0 2]
}}]
[621264 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 83760625
PolynomialP : [0 1]
PolynomialQ : [83760625 167521250]
PolynomialR : [0 1]
}}]
]
}
}
Custom Formulas/Pi - Takano.cfg
+62
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@@ -0,0 +1,62 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 3.14159265358979323846264338327950288419716939937510...
//
// Formula: Kikuo Takano's 4-term formula
//
// Pi = 48 ArcTan(1/49) + 128 ArcTan(1/57) - 20 ArcTan(1/239) + 48 ArcTan(1/110443)
//
// This uses Euler's ArcTan formula which is slightly faster
// than the usual Taylor series.
//
{
NameShort : "Pi"
NameLong : "Pi"
AlgorithmShort : "Takano"
AlgorithmLong : "Takano's 4-term Formula"
Formula : {
LinearCombination : [
[1176 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1201
PolynomialP : [0 1]
PolynomialQ : [1201 2402]
PolynomialR : [0 1]
}}]
[3648 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1625
PolynomialP : [0 1]
PolynomialQ : [1625 3250]
PolynomialR : [0 1]
}}]
[-2390 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 28561
PolynomialP : [0 1]
PolynomialQ : [28561 57122]
PolynomialR : [0 1]
}}]
[2650632 {SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 6098828125
PolynomialP : [0 1]
PolynomialQ : [6098828125 12197656250]
PolynomialR : [0 1]
}}]
]
}
}
Custom Formulas/Pi^-1 - Native.cfg
+25
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@@ -0,0 +1,25 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 2, 2018
//
// Value: 0.31830988618379067153776752674502872406891929148091...
//
{
NameShort : "Pi^-1"
NameLong : "1 / Pi"
AlgorithmShort : "Native"
AlgorithmLong : "Native Implementation"
Formula : {
Pi : {
Power : -1
}
}
}
Custom Formulas/Sin(1) - Half Angle Formula.cfg
+40
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@@ -0,0 +1,40 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 10, 2018
//
// Value: 0.84147098480789650665250232163029899962256306079837...
//
// Formula: 2 Sin(1/2) Cos(1/2)
//
{
NameShort : "Sin(1)"
NameLong : "Sine of 1"
AlgorithmShort : "Half Angle Formula"
AlgorithmLong : "2 Sin(1/2) Cos(1/2)"
Formula : {
Multiply : [
{SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 -8 -16]
}}
{SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 8 -16]
}}
]
}
}
Custom Formulas/Sin(1) - Series.cfg
+31
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@@ -0,0 +1,31 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 0.84147098480789650665250232163029899962256306079837...
//
// Formula: Taylor Series of Sin(x)
//
{
NameShort : "Sin(1)"
NameLong : "Sine of 1"
AlgorithmShort : "Series"
AlgorithmLong : "Taylor Series of sin(1)"
Formula : {
SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 -2 -4]
}
}
}
Custom Formulas/Sqrt(2) - Native Invsqrt.cfg
+26
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@@ -0,0 +1,26 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 6, 2018
//
// Value: 1.41421356237309504880168872420969807856967187537694...
//
{
NameShort : "Sqrt(2)"
NameLong : "Sqrt(2)"
AlgorithmShort : "Invsqrt"
AlgorithmLong : "Native Invsqrt"
Formula : {
Shift : [
{Invsqrt : 2}
1
]
}
}
Custom Formulas/Sqrt(2) - Series.cfg
+30
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@@ -0,0 +1,30 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 11, 2018
//
// Value: 1.41421356237309504880168872420969807856967187537694...
//
{
NameShort : "Sqrt(2)"
NameLong : "Sqrt(2)"
AlgorithmShort : "Series"
AlgorithmLong : "Hypergeometric Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1 -2]
PolynomialQ : [0 -4]
PolynomialR : [1 -2]
}
}
}
Custom Formulas/Sqrt(Pi).cfg
+25
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@@ -0,0 +1,25 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 24, 2018
//
// Value: 1.77245385090551602729816748334114518279754945612238...
//
{
NameShort : "Sqrt(Pi)"
NameLong : "Sqrt(Pi)"
AlgorithmShort : "Simple"
AlgorithmLong : "Simple"
Formula : {
Invsqrt : {
Pi : { Power : -1 }
}
}
}
Custom Formulas/Universal Parabolic Constant (unoptimized).cfg
+35
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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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 4, 2018
//
// Value: 2.29558714939263807403429804918949038759783220363858...
//
// P = Log(1 + sqrt(2)) + sqrt(2)
//
// https://en.wikipedia.org/wiki/Universal_parabolic_constant
//
{
NameShort : "Universal Parabolic Constant"
NameLong : "Universal Parabolic Constant"
AlgorithmShort : "Definition"
AlgorithmLong : "Direct Definition"
Formula : {
LinearCombination : [
[1 {Log : {
LinearCombination : [
[1 1]
[1 {Sqrt : 2}]
]
}}]
[1 {Sqrt : 2}]
]
}
}
Custom Formulas/Universal Parabolic Constant.cfg
+45
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@@ -0,0 +1,45 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 4, 2018
//
// Value: 2.29558714939263807403429804918949038759783220363858...
//
// P = Log(1 + sqrt(2)) + sqrt(2)
//
// https://en.wikipedia.org/wiki/Universal_parabolic_constant
//
// Same as "Universal Parabolic Constant (unoptimized).cfg",
// but avoids computing sqrt(2) twice.
//
{
NameShort : "Universal Parabolic Constant"
NameLong : "Universal Parabolic Constant"
AlgorithmShort : "Definition"
AlgorithmLong : "Direct Definition"
Formula : {
Scope : {
Locals : [
{sqrt2 : {Sqrt : 2}}
]
Formula : {
LinearCombination : [
[1 {Log : {
LinearCombination : [
[1 1]
[1 "sqrt2"]
]
}}]
[1 "sqrt2"]
]
}
}
}
}
Custom Formulas/Unnormalized Fresnel C(1).cfg
+34
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@@ -0,0 +1,34 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 24, 2018
//
// Value: 0.90452423790027208147478836683255714580799182059550...
//
// 1 ( )
// C(1) = Integrate( Cos(x^2) ) dx
// 0 ( )
//
{
NameShort : "Unnormalized Fresnel C(1)"
NameLong : "Unnormalized Fresnel C(1)"
AlgorithmShort : "Series"
AlgorithmLong : "Taylor Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 2 4 -16]
PolynomialR : [1 4]
}
}
}
Custom Formulas/Unnormalized Fresnel S(1).cfg
+34
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@@ -0,0 +1,34 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 24, 2018
//
// Value: 0.31026830172338110180815242316539650757450938883244...
//
// 1 ( )
// S(1) = Integrate( Sin(x^2) ) dx
// 0 ( )
//
{
NameShort : "Unnormalized Fresnel S(1)"
NameLong : "Unnormalized Fresnel S(1)"
AlgorithmShort : "Series"
AlgorithmLong : "Taylor Series"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 3
PolynomialP : [3]
PolynomialQ : [0 -6 -20 -16]
PolynomialR : [3 4]
}
}
}
Custom Formulas/Zeta(2) - Chudnovsky.cfg
+43
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@@ -0,0 +1,43 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 6, 2018
//
// Value: 1.64493406684822643647241516664602518921894990120679...
//
// Zeta(2) = Pi^2 / 6
//
// https://en.wikipedia.org/wiki/Chudnovsky_algorithm
//
{
NameShort : "Zeta(2)"
NameLong : "Zeta(2)"
AlgorithmShort : "Chudnovsky"
AlgorithmLong : "Chudnovsky (1988)"
Formula : {
LinearCombination : [
[
303862746112000
{Power : [
{SeriesHypergeometric : {
Power : -1
CoefficientP : 1
CoefficientQ : 13591409
CoefficientD : 1
PolynomialP : [-67957045 -2100495856 23608573992 -57896553024 39250089648]
PolynomialQ : [0 0 0 -10939058860032000]
PolynomialR : [-5 46 -108 72]
}}
2
]}
]
]
}
}
Custom Formulas/Zeta(2) - Direct.cfg
+34
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@@ -0,0 +1,34 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 10, 2018
//
// Value: 1.64493406684822643647241516664602518921894990120679...
//
// Zeta(2) = Pi^2 / 6
//
// This uses Ramanujan's formula for Pi so that it's independent
// from the other implementation.
//
{
NameShort : "Zeta(2)"
NameLong : "Zeta(2)"
AlgorithmShort : "Direct"
AlgorithmLong : "Direct Formula"
Formula : {
Divide : [
{Power : [
{Pi : {Algorithm : "ramanujan"}}
2
]}
6
]
}
}
Custom Formulas/Zeta(3) - Amdeberhan-Zeilberger.cfg
+38
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@@ -0,0 +1,38 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 1.20205690315959428539973816151144999076498629234049...
//
// Formula: Amdeberhan-Zeilberger (1997)
//
// 1 inf ( (-1)^k (205k^2 + 250k + 77) k!^10 )
// Zeta(3) = ---- SUM (-----------------------------------)
// 64 k=0 ( (2k + 1)!^5 )
//
// https://arxiv.org/pdf/math/9804121.pdf
//
{
NameShort : "Zeta(3)"
NameLong : "Zeta(3) - Apery's Constant"
AlgorithmShort : "AZ"
AlgorithmLong : "Amdeberhan-Zeilberger (1997)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 77
CoefficientD : 64
PolynomialP : [0 0 0 0 0 77 250 205]
PolynomialQ : [-32 -320 -1280 -2560 -2560 -1024]
PolynomialR : [0 0 0 0 0 1]
}
}
}
Custom Formulas/Zeta(3) - Amdeberhan.cfg
+36
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@@ -0,0 +1,36 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 1.20205690315959428539973816151144999076498629234049...
//
// Formula: Amdeberhan (1996)
//
// 1 inf ( (-1)^k (56k^2 - 32k + 5) (k - 1)!^3 )
// Zeta(3) = - --- SUM (-------------------------------------)
// 4 k=1 ( (2k - 1)^2 (3k)! )
//
{
NameShort : "Zeta(3)"
NameLong : "Zeta(3) - Apery's Constant"
AlgorithmShort : "Amdeberhan"
AlgorithmLong : "Amdeberhan (1996)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 29
CoefficientD : 24
PolynomialP : [0 0 0 29 80 56]
PolynomialQ : [-6 -57 -210 -375 -324 -108]
PolynomialR : [0 0 0 1 4 4]
}
}
}
Custom Formulas/Zeta(3) - Hjortnaes.cfg
+36
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@@ -0,0 +1,36 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 1.20205690315959428539973816151144999076498629234049...
//
// Formula: Hjortnaes (1953), rediscovered by Apery (1979)
//
// 5 inf ( (-1)^k k!^2 )
// Zeta(3) = - --- SUM (--------------)
// 2 k=0 ( (2k)! k^3 )
//
{
NameShort : "Zeta(3)"
NameLong : "Zeta(3) - Apery's Constant"
AlgorithmShort : "Hjortnaes"
AlgorithmLong : "Hjortnaes (1953)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2
PolynomialP : [-5]
PolynomialQ : [0 0 2 -4]
PolynomialR : [0 0 0 1]
}
}
}
Custom Formulas/Zeta(3) - Mohammed.cfg
+38
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@@ -0,0 +1,38 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 30, 2018
//
// Value: 1.20205690315959428539973816151144999076498629234049...
//
// Formula: Mohamud Mohammed (2005)
//
// 1 inf ( (-1)^k (2k)!^3 (k + 1)!^6 P(k) )
// Zeta(3) = --- SUM (--------------------------------)
// 2 k=0 ( (k + 1)^2 (3k + 3)!^4 )
//
// P(k) = 40885k^5 + 124346k^4 + 150160k^3 + 89888k^2 + 26629k + 3116
//
{
NameShort : "Zeta(3)"
NameLong : "Zeta(3) - Apery's Constant"
AlgorithmShort : "Mohammed"
AlgorithmLong : "Mohammed (2005)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 3116
CoefficientD : 2592
PolynomialP : [0 0 0 -24928 -63464 259952 756384 -712080 -3021000 -364496 4033184 2616640]
PolynomialQ : [-1296 -23328 -180792 -787320 -2106081 -3542940 -3661038 -2125764 -531441]
PolynomialR : [0 0 0 -8 32 -8 -80 32 64]
}
}
}
Custom Formulas/Zeta(3) - Wedeniwski.cfg
+40
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@@ -0,0 +1,40 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 1.20205690315959428539973816151144999076498629234049...
//
// Formula: Wedeniwski (1998)
//
// 1 inf ( (-1)^k ((2k + 1)! (2k)! k!)^3 P(k) )
// Zeta(3) = ---- SUM (------------------------------------)
// 24 k=0 ( (3k + 2)! (4k + 3)!^3 )
//
// P(k) = 126392k^5 + 412708k^4 + 531578k^3 + 336367k^2 + 104000k + 12463
//
// http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/Zeta3.txt
//
{
NameShort : "Zeta(3)"
NameLong : "Zeta(3) - Apery's Constant"
AlgorithmShort : "Wedeniwski"
AlgorithmLong : "Wedeniwski (1998)"
Formula : {
SeriesHypergeometric : {
CoefficientP : 1
CoefficientQ : 12463
CoefficientD : 10368
PolynomialP : [0 0 0 0 0 -12463 -29222 138077 338328 -427644 -1338144 58480 1784960 1011136]
PolynomialQ : [-1296 -26568 -230472 -1102080 -3171456 -5621760 -6002688 -3538944 -884736]
PolynomialR : [0 0 0 0 0 -1 6 -12 8]
}
}
}
Custom Formulas/Zeta(4) - Chudnovsky.cfg
+43
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@@ -0,0 +1,43 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 6, 2018
//
// Value: 1.08232323371113819151600369654116790277475095191872...
//
// Zeta(2) = Pi^4 / 90
//
// https://en.wikipedia.org/wiki/Chudnovsky_algorithm
//
{
NameShort : "Zeta(4)"
NameLong : "Zeta(4)"
AlgorithmShort : "Chudnovsky"
AlgorithmLong : "Chudnovsky (1988)"
Formula : {
LinearCombination : [
[
17795560
{Power : [
{SeriesHypergeometric : {
Power : -1
CoefficientP : 1
CoefficientQ : 13591409
CoefficientD : 213440
PolynomialP : [-67957045 -2100495856 23608573992 -57896553024 39250089648]
PolynomialQ : [0 0 0 -10939058860032000]
PolynomialR : [-5 46 -108 72]
}}
4
]}
]
]
}
}
Custom Formulas/Zeta(4) - Direct.cfg
+34
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@@ -0,0 +1,34 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: November 10, 2018
//
// Value: 1.08232323371113819151600369654116790277475095191872...
//
// Zeta(4) = Pi^4 / 90
//
// This uses Ramanujan's formula for Pi so that it's independent
// from the other implementation.
//
{
NameShort : "Zeta(4)"
NameLong : "Zeta(4)"
AlgorithmShort : "Direct"
AlgorithmLong : "Direct Formula"
Formula : {
Divide : [
{Power : [
{Pi : {Algorithm : "ramanujan"}}
4
]}
90
]
}
}
Custom Formulas/Zeta(5) - BBP-Kruse.cfg
+118
View File
@@ -0,0 +1,118 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Oliver Kruse // DSG
// Date: October 30, 2019
//
// Value = 1.03692775514336992633136548645703416805708091950191...
//
// Formula: Bailey, Borwein and Plouffe (1998), [Kruse (2019)]
// https://www.davidhbailey.com/dhbpapers/digits.pdf
//
// Formula optimized by Alex Yee (2019).
//
{
NameShort : "Zeta(5)"
NameLong : "Zeta(5)"
AlgorithmShort : "BBP-Kruse"
AlgorithmLong : "BBP (1998), Kruse (2019)"
Formula : {
Divide : [
{LinearCombination : [
[-1 {SeriesBinaryBBP : {
Power : 1
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -6
PowerShift : 0
PolynomialP : [5]
PolynomialQ : [0 0 0 0 0 1]
}}]
[-1 {SeriesBinaryBBP : {
Power : 1
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -6
PowerShift : 8
PolynomialP : [23]
PolynomialQ : [-1 10 -40 80 -80 32]
}}]
[-1 {SeriesBinaryBBP : {
Power : 1
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -6
PowerShift : 7
PolynomialP : [243]
PolynomialQ : [-1 15 -90 270 -405 243]
}}]
[-1 {SeriesBinaryBBP : {
Power : 1
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -6
PowerShift : 9
PolynomialP : [243]
PolynomialQ : [-32 240 -720 1080 -810 243]
}}]
[1 {SeriesBinaryBBP : {
Power : 1
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -6
PowerShift : 8
PolynomialP : [243]
PolynomialQ : [-1 30 -360 2160 -6480 7776]
}}]
[1 {SeriesBinaryBBP : {
Power : 1
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1
Alternating : "false"
PowerCoef : -6
PowerShift : 12
PolynomialP : [243]
PolynomialQ : [-3125 18750 -45000 54000 -32400 7776]
}}]
[1 {Scope : {
Locals : [
{ log1 : {Log : 2}}
{ log2 : {Power : ["log1" 2]}}
]
Formula : {
Multiply : [
{LinearCombination : [
[1 {Power : [
{LinearCombination : [
[12 {Power : [{Pi : {}} 2]}]
[-9 "log2"]
]}
2
]}]
[243 {Power : ["log2" 2]}]
]}
"log1"
]
}
}}]
]}
21762
]
}
}
Custom Formulas/Zeta(5) - Broadhurst (Huvent 2006).cfg
+305
View File
@@ -0,0 +1,305 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: February 19, 2019
//
// Value = 1.03692775514336992633136548645703416805708091950191...
//
// Formula: Broadhurst (1998) rearranged as Huvent (2006)
// http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/06/05/0002/
//
// Huvent's 2006 formula is mathematically the same as Broadhurst's 1998 formula.
//
// ***Therefore, Broadhurst's and Huvent's formulas cannot be used as a compute/verify pair.***
//
// The use of the optimized 74-term rearrangement by Alex Yee (2018) to pair with either
// Broadhurst (1998) or Huvent (2006) is also questionable. But the reformulation is
// significant enough that it can be considered *computationally* independent, though
// still *mathematically* the same.
//
{
NameShort : "Zeta(5)"
NameLong : "Zeta(5)"
AlgorithmShort : "Broadhurst (Huvent 2006)"
AlgorithmLong : "Broadhurst (1998), Huvent (2006)"
Formula : {
LinearCombination : [
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "true"
PowerCoef : -10
PowerShift : 12
PolynomialP : [369]
PolynomialQ : [-1 20 -160 640 -1280 1024]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "true"
PowerCoef : -10
PowerShift : 0
PolynomialP : [369]
PolynomialQ : [0 0 0 0 0 1]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "true"
PowerCoef : -10
PowerShift : 17
PolynomialP : [369]
PolynomialQ : [-243 1620 -4320 5760 -3840 1024]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 18
PolynomialP : [7263]
PolynomialQ : [-161051 878460 -1916640 2090880 -1140480 248832]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "false"
PowerCoef : -12
PowerShift : 20
PolynomialP : [32635]
PolynomialQ : [-16807 96040 -219520 250880 -143360 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 13
PolynomialP : [13977]
PolynomialQ : [-3125 18750 -45000 54000 -32400 7776]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 20
PolynomialP : [9]
PolynomialQ : [-2476099 15638520 -39507840 49904640 -31518720 7962624]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 563859
Alternating : "false"
PowerCoef : -12
PowerShift : 18
PolynomialP : [1051]
PolynomialQ : [-243 1620 -4320 5760 -3840 1024]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 19
PolynomialP : [9]
PolynomialQ : [-1419857 10022520 -28298880 39951360 -28200960 7962624]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 7
PolynomialP : [83871]
PolynomialQ : [-32 240 -720 1080 -810 243]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "false"
PowerCoef : -12
PowerShift : 17
PolynomialP : [32635]
PolynomialQ : [-3125 25000 -80000 128000 -102400 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 14
PolynomialP : [7263]
PolynomialQ : [-16807 144060 -493920 846720 -725760 248832]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 17
PolynomialP : [9]
PolynomialQ : [-371293 3427320 -12654720 23362560 -21565440 7962624]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 54567
Alternating : "false"
PowerCoef : -12
PowerShift : 6
PolynomialP : [3187]
PolynomialQ : [-1 10 -40 80 -80 32]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 16
PolynomialP : [9]
PolynomialQ : [-161051 1756920 -7666560 16727040 -18247680 7962624]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 12
PolynomialP : [7263]
PolynomialQ : [-3125 37500 -180000 432000 -518400 248832]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "false"
PowerCoef : -12
PowerShift : 14
PolynomialP : [32635]
PolynomialQ : [-243 3240 -17280 46080 -61440 32768]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 3
PolynomialP : [83871]
PolynomialQ : [-1 15 -90 270 -405 243]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 14
PolynomialP : [9]
PolynomialQ : [-16807 288120 -1975680 6773760 -11612160 7962624]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 563859
Alternating : "false"
PowerCoef : -12
PowerShift : 12
PolynomialP : [1051]
PolynomialQ : [-1 20 -160 640 -1280 1024]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 13
PolynomialP : [9]
PolynomialQ : [-3125 75000 -720000 3456000 -8294400 7962624]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 5
PolynomialP : [13977]
PolynomialQ : [-1 30 -360 2160 -6480 7776]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "false"
PowerCoef : -12
PowerShift : 11
PolynomialP : [32635]
PolynomialQ : [-1 40 -640 5120 -20480 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 8
PolynomialP : [7263]
PolynomialQ : [-1 60 -1440 17280 -103680 248832]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 11
PolynomialP : [9]
PolynomialQ : [-1 120 -5760 138240 -1658880 7962624]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "false"
PowerCoef : -12
PowerShift : -5
PolynomialP : [128125]
PolynomialQ : [0 0 0 0 0 1]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -12
PowerShift : 22
PolynomialP : [9]
PolynomialQ : [-6436343 33580920 -70081920 73128960 -38154240 7962624]
}}]
]
}
}
Custom Formulas/Zeta(5) - Broadhurst (optimized).cfg
+655
View File
@@ -0,0 +1,655 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 4, 2018
//
// Value = 1.03692775514336992633136548645703416805708091950191...
//
// Original Formula: Broadhurst (1998)
// https://arxiv.org/pdf/math/9803067.pdf
//
// 1152 inf 1 ( 248 12912 124 24848 62 3228 31 74552 )
// Zeta(5) = ------- SUM ------ (---------- - ---------- - ---------- - ---------- - ---------- - ---------- + ---------- + ----------)
// 62651 k=0 16^k ( (8k+1)^5 (8k+2)^5 (8k+3)^5 (8k+4)^5 (8k+5)^5 (8k+6)^5 (8k+7)^5 (8k+8)^5 )
//
// 7 inf 1 ( 177152 145408 22144 29248 2768 2272 346 111 )
// + -------- SUM -------- (---------- + ---------- - ---------- - ---------- - ---------- + ---------- + ---------- - ----------)
// 250604 k=0 4096^k ( (8k+1)^5 (8k+2)^5 (8k+3)^5 (8k+4)^5 (8k+5)^5 (8k+6)^5 (8k+7)^5 (8k+8)^5 )
//
// 369 inf 1 ( 131072 4096 1024 128 4 1 )
// + ---------- SUM --------- (- ---------- + ---------- + ---------- + ---------- - ---------- - ----------)
// 64154624 k=0 2^(20k) ( (8k+1)^5 (8k+3)^5 (8k+4)^5 (8k+5)^5 (8k+7)^5 (8k+8)^5 )
//
//
// Optimized into a massive 74-term BBP formula by Alex Yee (2018).
// The following implementation groups the 74 terms into sets of 2 or 3 terms each.
//
{
NameShort : "Zeta(5)"
NameLong : "Zeta(5)"
AlgorithmShort : "Broadhurst (optimized)"
AlgorithmLong : "Broadhurst (1998) (optimized)"
Formula : {
LinearCombination : [
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1762059375
Alternating : "false"
PowerCoef : -30
PowerShift : 12
PolynomialP : [
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2743554653514720 -11903795644396032 34544269695818880 -66159912056279040
80125279389331200 -55550303705640960 16795225145889792
]
PolynomialQ : [
-3125 143750 -2982500 36979000
-306192400 1793303776 -7678751040 24504739200
-58849079040 106537628160 -144464145408 144349931520
-103061237760 49716633600 -14511882240 1934917632
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 117470625
Alternating : "true"
PowerCoef : -30
PowerShift : 18
PolynomialP : [
-34509025 693618140 -5558112160 22244671360 -44497491200 35600165888
]
PolynomialQ : [
243 -6480 75600 -506880
2158080 -6086656 11509760 -14417920
11468800 -5242880 1048576
]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 563859
Alternating : "false"
PowerCoef : -30
PowerShift : 30
PolynomialP : [
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]
PolynomialQ : [
992436543 -12602368800 71888512500 -242585280000
536262300000 -811465600000 851210000000 -611200000000
287500000000 -80000000000 10000000000
]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 563859
Alternating : "false"
PowerCoef : -30
PowerShift : 12
PolynomialP : [
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]
PolynomialQ : [
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84300000 -594400000 2810000000 -8800000000
17500000000 -20000000000 10000000000
]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 195784375
Alternating : "true"
PowerCoef : -30
PowerShift : 27
PolynomialP : [
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-856076349070080 293675932965888
]
PolynomialQ : [
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2127741073920 -3501690310656 3979152138240 -3082968760320
1558683648000 -464380231680 61917364224
]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 195784375
Alternating : "true"
PowerCoef : -30
PowerShift : 12
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-121886516655360 293675932965888
]
PolynomialQ : [
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1257638400 -8429432832 36219985920 -98895790080
165542952960 -154793410560 61917364224
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "true"
PowerCoef : -30
PowerShift : 35
PolynomialP : [
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-3054636000000 731024000000
]
PolynomialQ : [
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1136939844800000 -1518525619200000 1407975040000000 -894873600000000
373120000000000 -92160000000000 10240000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "true"
PowerCoef : -30
PowerShift : 26
PolynomialP : [
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-2636908000000 939888000000
]
PolynomialQ : [
59797108943 -1003587842400 7573696130000 -33843912960000
99171300800000 -199112140800000 277402240000000 -264806400000000
165760000000000 -61440000000000 10240000000000
]
}}]
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CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "true"
PowerCoef : -30
PowerShift : 20
PolynomialP : [
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-1697020000000 939888000000
]
PolynomialQ : [
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8580196800000 -25966899200000 54477440000000 -78233600000000
73600000000000 -40960000000000 10240000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "true"
PowerCoef : -30
PowerShift : 11
PolynomialP : [
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-130540000000 731024000000
]
PolynomialQ : [
243 -32400 1890000 -63360000
1348800000 -19020800000 179840000000 -1126400000000
4480000000000 -10240000000000 10240000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -30
PowerShift : 30
PolynomialP : [
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-44152140150000 11471908500000
]
PolynomialQ : [
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51387246182700000 -71683580685600000 69338113290000000 -45921751200000000
19929037500000000 -5117580000000000 590490000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -30
PowerShift : 24
PolynomialP : [
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-2559118050000 882454500000
]
PolynomialQ : [
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5755757964300000 -11531303920800000 16037715690000000 -15289754400000000
9562657500000000 -3542940000000000 590490000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -30
PowerShift : 18
PolynomialP : [
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-1676663550000 882454500000
]
PolynomialQ : [
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502054710300000 -1512007819200000 3159784890000000 -4524465600000000
4248247500000000 -2361960000000000 590490000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -30
PowerShift : 8
PolynomialP : [
-122534073 2685494250 -26604369000 221594130000
-2088475650000 11471908500000
]
PolynomialQ : [
16807 -2881200 208372500 -8255520000
195104700000 -2823854400000 25084890000000 -136468800000000
442867500000000 -787320000000000 590490000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 11
PolynomialP : [
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-313103081469600000 3757237383542400000
]
PolynomialQ : [
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54447575198400000 -807728734080000000 3322224927360000000 -6244812288000000000
6096166272000000000 -3023308800000000000 604661760000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 14
PolynomialP : [
-1265122252467 54359524782900 -932194548108000 7990597829520000
-34245622389600000 58706827228800000
]
PolynomialQ : [
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510840122385600000 -2043590186880000000 4956939192960000000 -7292892672000000000
6358186368000000000 -3023308800000000000 604661760000000000
]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 16
PolynomialP : [
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-3363383800800000 3669170140800000
]
PolynomialQ : [
45492921326699 -1519244125938000 21813414268338000 -176133818044800000
879587115422400000 -2821924241280000000 5874793349760000000 -7857243648000000000
6499274112000000000 -3023308800000000000 604661760000000000
]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 17
PolynomialP : [
-435930939801 10094402819700 -93267080964000 430575762960000
-993708064800000 917287286400000
]
PolynomialQ : [
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1065214272657600000 -3179682449280000000 6276221573760000000 -8099108352000000000
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]
}}]
[-1 {SeriesBinaryBBP : {
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CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 19
PolynomialP : [
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-81193687200000 57323894400000
]
PolynomialQ : [
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1412458131153600000 -3808194756480000000 6956018156160000000 -8502216192000000000
6660517248000000000 -3023308800000000000 604661760000000000
]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 20
PolynomialP : [
-44596750959 712993446900 -4528972764000 14339857680000
-22669567200000 14325724800000
]
PolynomialQ : [
286871423439899 -6628607987058000 67894304401458000 -405659129788800000
1564859339294400000 -4070886698880000000 7231699128960000000 -8663459328000000000
6700828032000000000 -3023308800000000000 604661760000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 22
PolynomialP : [
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-1695362400000 888796800000
]
PolynomialQ : [
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1808198102289600000 -4477017847680000000 7649251666560000000 -8905324032000000000
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]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "true"
PowerCoef : -30
PowerShift : 25
PolynomialP : [
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-15746400000 6998400000
]
PolynomialQ : [
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1980373989470400000 -4755834103680000000 7930307410560000000 -9066567168000000000
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]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1762059375
Alternating : "false"
PowerCoef : -60
PowerShift : 15
PolynomialP : [
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107976311014313520 -57826406256512400 13874549065109280
]
PolynomialQ : [
-32 1040 -15680 145480
-928930 4324093 -15159195 40757940
-84738960 136242405 -168022107 156100770
-105763320 49338720 -14171760 1889568
]
}}]
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CoefficientP : 1
CoefficientQ : 0
CoefficientD : 1691577
Alternating : "false"
PowerCoef : -60
PowerShift : 31
PolynomialP : [
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]
PolynomialQ : [
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233137500 -404271875 485703125 -399218750
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]
}}]
[1 {SeriesBinaryBBP : {
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CoefficientQ : 0
CoefficientD : 1691577
Alternating : "false"
PowerCoef : -60
PowerShift : 7
PolynomialP : [
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-1640800781250 1640400390625
]
PolynomialQ : [
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]
}}]
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CoefficientQ : 0
CoefficientD : 195784375
Alternating : "false"
PowerCoef : -60
PowerShift : 10
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]
PolynomialQ : [
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]
}}]
[-1 {SeriesBinaryBBP : {
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CoefficientQ : 0
CoefficientD : 54567
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PowerCoef : -60
PowerShift : 42
PolynomialP : [
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PolynomialQ : [
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}}]
[-1 {SeriesBinaryBBP : {
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CoefficientQ : 0
CoefficientD : 54567
Alternating : "false"
PowerCoef : -60
PowerShift : 6
PolynomialP : [
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]
PolynomialQ : [
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]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 51
PolynomialP : [
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]
PolynomialQ : [
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]
}}]
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CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 31
PolynomialP : [
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]
PolynomialQ : [
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}}]
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CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 15
PolynomialP : [
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PolynomialQ : [
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}}]
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CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 3
PolynomialP : [
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]
PolynomialQ : [
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]
}}]
[-1 {SeriesBinaryBBP : {
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CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 49
PolynomialP : [
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PolynomialQ : [
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]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 37
PolynomialP : [
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]
PolynomialQ : [
3515706497843 -58776517301400 442037017372500 -1969339242960000
5755757964300000 -11531303920800000 16037715690000000 -15289754400000000
9562657500000000 -3542940000000000 590490000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 25
PolynomialP : [
-41205719493 551008981350 -2955254949000 7943827950000
-10698694650000 5773898700000
]
PolynomialQ : [
59797108943 -1505381763600 17040816292500 -114223206240000
502054710300000 -1512007819200000 3159784890000000 -4524465600000000
4248247500000000 -2361960000000000 590490000000000
]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -60
PowerShift : 5
PolynomialP : [
-292161231 13621285350 -558395127000 15642359550000
-232257905550000 1391509586700000
]
PolynomialQ : [
16807 -2881200 208372500 -8255520000
195104700000 -2823854400000 25084890000000 -136468800000000
442867500000000 -787320000000000 590490000000000
]
}}]
]
}
}
Custom Formulas/Zeta(5) - Broadhurst.cfg
+258
View File
@@ -0,0 +1,258 @@
// y-cruncher Custom Formula File
//
// This can be loaded directly from the Custom Compute menu or
// entered from the command line as "custom:filename.cfg".
// y-cruncher custom custom:"filename.cfg"
//
//
// Author: Alexander J. Yee
// Date: October 2, 2018
//
// Value = 1.03692775514336992633136548645703416805708091950191...
//
// Formula: Broadhurst (1998)
// https://arxiv.org/pdf/math/9803067.pdf
//
// 1152 inf 1 ( 248 12912 124 24848 62 3228 31 74552 )
// Zeta(5) = ------- SUM ------ (---------- - ---------- - ---------- - ---------- - ---------- - ---------- + ---------- + ----------)
// 62651 k=0 16^k ( (8k+1)^5 (8k+2)^5 (8k+3)^5 (8k+4)^5 (8k+5)^5 (8k+6)^5 (8k+7)^5 (8k+8)^5 )
//
// 7 inf 1 ( 177152 145408 22144 29248 2768 2272 346 111 )
// + -------- SUM -------- (---------- + ---------- - ---------- - ---------- - ---------- + ---------- + ---------- - ----------)
// 250604 k=0 4096^k ( (8k+1)^5 (8k+2)^5 (8k+3)^5 (8k+4)^5 (8k+5)^5 (8k+6)^5 (8k+7)^5 (8k+8)^5 )
//
// 369 inf 1 ( 131072 4096 1024 128 4 1 )
// + ---------- SUM --------- (- ---------- + ---------- + ---------- + ---------- - ---------- - ----------)
// 64154624 k=0 2^(20k) ( (8k+1)^5 (8k+3)^5 (8k+4)^5 (8k+5)^5 (8k+7)^5 (8k+8)^5 )
//
{
NameShort : "Zeta(5)"
NameLong : "Zeta(5)"
AlgorithmShort : "Broadhurst"
AlgorithmLong : "Broadhurst (1998)"
Formula : {
LinearCombination : [
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -4
PowerShift : 14
PolynomialP : [9]
PolynomialQ : [-16807 96040 -219520 250880 -143360 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -4
PowerShift : 13
PolynomialP : [9]
PolynomialQ : [-3125 25000 -80000 128000 -102400 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -4
PowerShift : 12
PolynomialP : [9]
PolynomialQ : [-243 3240 -17280 46080 -61440 32768]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2021
Alternating : "false"
PowerCoef : -4
PowerShift : 11
PolynomialP : [9]
PolynomialQ : [-1 40 -640 5120 -20480 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -4
PowerShift : 10
PolynomialP : [7263]
PolynomialQ : [-243 1620 -4320 5760 -3840 1024]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -4
PowerShift : 8
PolynomialP : [7263]
PolynomialQ : [-1 20 -160 640 -1280 1024]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -4
PowerShift : 5
PolynomialP : [13977]
PolynomialQ : [-1 10 -40 80 -80 32]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 125302
Alternating : "false"
PowerCoef : -4
PowerShift : 0
PolynomialP : [83871]
PolynomialQ : [0 0 0 0 0 1]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 20
PolynomialP : [1211]
PolynomialQ : [-16807 96040 -219520 250880 -143360 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 17
PolynomialP : [1211]
PolynomialQ : [-3125 25000 -80000 128000 -102400 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 14
PolynomialP : [1211]
PolynomialQ : [-243 3240 -17280 46080 -61440 32768]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 11
PolynomialP : [1211]
PolynomialQ : [-1 40 -640 5120 -20480 32768]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 16
PolynomialP : [497]
PolynomialQ : [-243 1620 -4320 5760 -3840 1024]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 10
PolynomialP : [497]
PolynomialQ : [-1 20 -160 640 -1280 1024]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -12
PowerShift : 6
PolynomialP : [3199]
PolynomialQ : [-1 10 -40 80 -80 32]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2004832
Alternating : "false"
PowerCoef : -12
PowerShift : 0
PolynomialP : [777]
PolynomialQ : [0 0 0 0 0 1]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -20
PowerShift : 27
PolynomialP : [369]
PolynomialQ : [-16807 96040 -219520 250880 -143360 32768]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -20
PowerShift : 22
PolynomialP : [369]
PolynomialQ : [-3125 25000 -80000 128000 -102400 32768]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -20
PowerShift : 17
PolynomialP : [369]
PolynomialQ : [-243 3240 -17280 46080 -61440 32768]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -20
PowerShift : 12
PolynomialP : [369]
PolynomialQ : [-1 40 -640 5120 -20480 32768]
}}]
[1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 62651
Alternating : "false"
PowerCoef : -20
PowerShift : 10
PolynomialP : [369]
PolynomialQ : [-1 10 -40 80 -80 32]
}}]
[-1 {SeriesBinaryBBP : {
CoefficientP : 1
CoefficientQ : 0
CoefficientD : 2004832
Alternating : "false"
PowerCoef : -20
PowerShift : 0
PolynomialP : [369]
PolynomialQ : [0 0 0 0 0 1]
}}]
]
}
}
Custom Formulas/e - exp(-1).cfg
+30
View File
@@ -0,0 +1,30 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 2.71828182845904523536028747135266249775724709369995...
//
{
NameShort : "e"
NameLong : "e"
AlgorithmShort : "exp(-1)"
AlgorithmLong : "Taylor Series of exp(-1)"
Formula : {
SeriesHyperdescent : {
Power : -1
CoefficientP : -1
CoefficientQ : 0
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [-1 -1]
}
}
}
Custom Formulas/e - exp(0.25)^4.cfg
+32
View File
@@ -0,0 +1,32 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 2.71828182845904523536028747135266249775724709369995...
//
{
NameShort : "e"
NameLong : "e"
AlgorithmShort : "exp(¼)^4"
AlgorithmLong : "Taylor Series of exp(1/4)"
Formula : {
Power : [
{SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 4]
}}
4
]
}
}
Custom Formulas/e - exp(0.5)^2.cfg
+32
View File
@@ -0,0 +1,32 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 2.71828182845904523536028747135266249775724709369995...
//
{
NameShort : "e"
NameLong : "e"
AlgorithmShort : "exp(½)^2"
AlgorithmLong : "Taylor Series of exp(1/2)"
Formula : {
Power : [
{SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 3
CoefficientD : 2
PolynomialP : [1]
PolynomialQ : [2 2]
}}
2
]
}
}
Custom Formulas/e - exp(1).cfg
+29
View File
@@ -0,0 +1,29 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: September 29, 2018
//
// Value: 2.71828182845904523536028747135266249775724709369995...
//
{
NameShort : "e"
NameLong : "e"
AlgorithmShort : "exp(1)"
AlgorithmLong : "Taylor Series of exp(1)"
Formula : {
SeriesHyperdescent : {
CoefficientP : 1
CoefficientQ : 1
CoefficientD : 1
PolynomialP : [1]
PolynomialQ : [0 1]
}
}
}
Custom Formulas/e^-1 - Native.cfg
+25
View File
@@ -0,0 +1,25 @@
// 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:"constant - algorithm"
//
//
// Author: Alexander J. Yee
// Date: October 2, 2018
//
// Value: 0.36787944117144232159552377016146086744581113103176...
//
{
NameShort : "e^-1"
NameLong : "1 / e"
AlgorithmShort : "Native"
AlgorithmLong : "Native Implementation"
Formula : {
E : {
Power : -1
}
}
}