DoubleDouble

Double-Double Arithmetic and Special Function Implements

Requirement

.NET 8.0

Install

Download DLL
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MultiPrecision

Type

type	mantissa bits	significant digits
ddouble	106	31

limit	bin	dec
MaxValue	2^1024	1.79769e308
Normal MinValue	2^-968	4.00833e-292
Subnormal MinValue	2^-1074	4.94066e-324

Functions

function	domain	mantissa error bits	note
ddouble.Sqrt(x)	[0,+inf)	2
ddouble.Cbrt(x)	(-inf,+inf)	2
ddouble.RootN(x, n)	(-inf,+inf)	3
ddouble.Log2(x)	(0,+inf)	2
ddouble.Log(x)	(0,+inf)	3
ddouble.Log(x, b)	(0,+inf)	3
ddouble.Log10(x)	(0,+inf)	3
ddouble.Log1p(x)	(-1,+inf)	3	log(1+x)
ddouble.Pow2(x)	(-inf,+inf)	1
ddouble.Pow2m1(x)	(-inf,+inf)	2	pow2(x)-1
ddouble.Pow(x, y)	(-inf,+inf)	2
ddouble.Pow1p(x, y)	(-inf,+inf)	2	pow(1+x, y)
ddouble.Pow10(x)	(-inf,+inf)	2
ddouble.Exp(x)	(-inf,+inf)	2
ddouble.Expm1(x)	(-inf,+inf)	2	exp(x)-1
ddouble.Sin(x)	(-inf,+inf)	2
ddouble.Cos(x)	(-inf,+inf)	2
ddouble.Tan(x)	(-inf,+inf)	3
ddouble.SinPi(x)	(-inf,+inf)	1	sin(πx)
ddouble.CosPi(x)	(-inf,+inf)	1	cos(πx)
ddouble.TanPi(x)	(-inf,+inf)	2	tan(πx)
ddouble.Sinh(x)	(-inf,+inf)	2
ddouble.Cosh(x)	(-inf,+inf)	2
ddouble.Tanh(x)	(-inf,+inf)	2
ddouble.Asin(x)	[-1,1]	2	Accuracy deteriorates near x=-1,1.
ddouble.Acos(x)	[-1,1]	2	Accuracy deteriorates near x=-1,1.
ddouble.Atan(x)	(-inf,+inf)	2
ddouble.Atan2(y, x)	(-inf,+inf)	2
ddouble.AsinPi(x)	[-1,1]	3	Accuracy deteriorates near x=-1,1.
ddouble.AcosPi(x)	[-1,1]	3	Accuracy deteriorates near x=-1,1.
ddouble.AtanPi(x)	(-inf,+inf)	3
ddouble.Atan2Pi(y, x)	(-inf,+inf)	3
ddouble.Asinh(x)	(-inf,+inf)	2
ddouble.Acosh(x)	[1,+inf)	2
ddouble.Atanh(x)	(-1,1)	4	Accuracy deteriorates near x=-1,1.
ddouble.Sinc(x, normalized)	(-inf,+inf)	2	normalized: x -> πx
ddouble.Sinhc(x)	(-inf,+inf)	3
ddouble.Gamma(x)	(-inf,+inf)	2	Accuracy deteriorates near non-positive intergers. If x is Natual number lass than 35, an exact integer value is returned.
ddouble.LogGamma(x)	(0,+inf)	4
ddouble.Digamma(x)	(-inf,+inf)	4	Near the positive root, polynomial interpolation is used.
ddouble.Polygamma(n, x)	(-inf,+inf)	4	Accuracy deteriorates near non-positive intergers. n ≤ 16
ddouble.InverseGamma(x)	[sqrt(π)/2,+inf)	2	gamma^-1(x)
ddouble.InverseDigamma(x)	(-inf,+inf)	2	digamma^-1(x)
ddouble.RcpGamma(x)	(-inf,+inf)	3	1/gamma(x)
ddouble.LowerIncompleteGamma(nu, x)	[0,+inf)	4	nu ≤ 171.625
ddouble.UpperIncompleteGamma(nu, x)	[0,+inf)	4	nu ≤ 171.625
ddouble.LowerIncompleteGammaRegularized(nu, x)	[0,+inf)	4	nu ≤ 8192
ddouble.UpperIncompleteGammaRegularized(nu, x)	[0,+inf)	4	nu ≤ 8192
ddouble.InverseLowerIncompleteGamma(nu, x)	[0,1]	8	nu ≤ 8192
ddouble.InverseUpperIncompleteGamma(nu, x)	[0,1]	8	nu ≤ 8192
ddouble.Beta(a, b)	[0,+inf)	4
ddouble.LogBeta(a, b)	[0,+inf)	4
ddouble.IncompleteBeta(x, a, b)	[0,1]	4	Accuracy decreases when the radio of a,b is too large. a+b-max(a,b) ≤ 512
ddouble.IncompleteBetaRegularized(x, a, b)	[0,1]	4	Accuracy decreases when the radio of a,b is too large. a+b-max(a,b) ≤ 8192
ddouble.InverseIncompleteBeta(x, a, b)	[0,1]	8	Accuracy decreases when the radio of a,b is too large. a+b-max(a,b) ≤ 8192
ddouble.Erf(x)	(-inf,+inf)	3
ddouble.Erfc(x)	(-inf,+inf)	3
ddouble.InverseErf(x)	(-1,1)	3
ddouble.InverseErfc(x)	(0,2)	3
ddouble.Erfcx(x)	(-inf,+inf)	3
ddouble.Erfi(x)	(-inf,+inf)	4
ddouble.DawsonF(x)	(-inf,+inf)	4
ddouble.BesselJ(nu, x)	[0,+inf)	6	Accuracy deteriorates near root. abs(nu) ≤ 256
ddouble.BesselY(nu, x)	[0,+inf)	6	Accuracy deteriorates near root. abs(nu) ≤ 256
ddouble.BesselI(nu, x)	[0,+inf)	6	Accuracy deteriorates near root. abs(nu) ≤ 256
ddouble.BesselK(nu, x)	[0,+inf)	6	abs(nu) ≤ 256
ddouble.StruveH(n, x)	(-inf,+inf)	4	0 ≤ n ≤ 8
ddouble.StruveK(n, x)	[0,+inf)	4	0 ≤ n ≤ 8
ddouble.StruveL(n, x)	(-inf,+inf)	4	0 ≤ n ≤ 8
ddouble.StruveM(n, x)	[0,+inf)	4	0 ≤ n ≤ 8
ddouble.AngerJ(n, x)	(-inf,+inf)	6
ddouble.WeberE(n, x)	(-inf,+inf)	6	0 ≤ n ≤ 8
ddouble.Jinc(x)	(-inf,+inf)	3
ddouble.EllipticK(m)	[0,1]	4	k: elliptic modulus, m=k^2
ddouble.EllipticE(m)	[0,1]	4	k: elliptic modulus, m=k^2
ddouble.EllipticPi(n, m)	[0,1]	4	k: elliptic modulus, m=k^2
ddouble.EllipticK(x, m)	[0,2π]	4	k: elliptic modulus, m=k^2
ddouble.EllipticE(x, m)	[0,2π]	4	k: elliptic modulus, m=k^2, incomplete elliptic integral
ddouble.EllipticPi(n, x, m)	[0,2π]	4	k: elliptic modulus, m=k^2 Argument order follows wolfram. incomplete elliptic integral
ddouble.EllipticTheta(a, x, q)	(-inf,+inf)	4	incomplete elliptic integral, a=1...4, q ≤ 0.995
ddouble.KeplerE(m, e, centered)	(-inf,+inf)	6	inverse kepler's equation, e(eccentricity) ≤ 256
ddouble.Agm(a, b)	[0,+inf)	2
ddouble.FresnelC(x)	(-inf,+inf)	4
ddouble.FresnelS(x)	(-inf,+inf)	4
ddouble.FresnelF(x)	(-inf,+inf)	4
ddouble.FresnelG(x)	(-inf,+inf)	4
ddouble.Ei(x)	(-inf,+inf)	4	exponential integral
ddouble.Ein(x)	(-inf,+inf)	4	complementary exponential integral
ddouble.En(n, x)	[0,+inf)	4	exponential integral, n ≤ 256
ddouble.Li(x)	[0,+inf)	5	logarithmic integral li(x)=ei(log(x))
ddouble.Si(x, limit_zero)	(-inf,+inf)	4	sin integral, limit_zero=true: si(x)
ddouble.Ci(x)	[0,+inf)	4	cos integral
ddouble.Ti(x)	(-inf,+inf)	4	arctan integral
ddouble.Shi(x)	(-inf,+inf)	5	hyperbolic sin integral
ddouble.Chi(x)	[0,+inf)	5	hyperbolic cos integral
ddouble.Clausen(x, normalized)	(-inf,+inf)	3	Clausen function of order 2, Cl_2(x), normalized: x -> πx
ddouble.BarnesG(x)	(-inf,+inf)	3
ddouble.LogBarnesG(x)	(0,+inf)	3
ddouble.LambertW(x)	[-1/e,+inf)	4
ddouble.AiryAi(x)	(-inf,+inf)	5	Accuracy deteriorates near root.
ddouble.AiryBi(x)	(-inf,+inf)	5	Accuracy deteriorates near root.
ddouble.ScorerGi(x)	(-inf,+inf)	5	Accuracy deteriorates near root.
ddouble.ScorerHi(x)	(-inf,+inf)	4
ddouble.JacobiSn(x, m)	(-inf,+inf)	4	k: elliptic modulus, m=k^2
ddouble.JacobiCn(x, m)	(-inf,+inf)	4	k: elliptic modulus, m=k^2
ddouble.JacobiDn(x, m)	(-inf,+inf)	4	k: elliptic modulus, m=k^2
ddouble.JacobiAm(x, m)	(-inf,+inf)	4	k: elliptic modulus, m=k^2
ddouble.JacobiArcSn(x, m)	[-1,+1]	4	k: elliptic modulus, m=k^2
ddouble.JacobiArcCn(x, m)	[-1,+1]	4	k: elliptic modulus, m=k^2
ddouble.JacobiArcDn(x, m)	[0,1]	4	k: elliptic modulus, m=k^2
ddouble.CarlsonRD(x, y, z)	[0,+inf)	4
ddouble.CarlsonRC(x, y)	[0,+inf)	4
ddouble.CarlsonRF(x, y, z)	[0,+inf)	4
ddouble.CarlsonRJ(x, y, z, rho)	[0,+inf)	4
ddouble.CarlsonRG(x, y, z)	[0,+inf)	4
ddouble.RiemannZeta(x)	(-inf,+inf)	3
ddouble.HurwitzZeta(x, a)	(1,+inf)	3	a ≥ 0
ddouble.DirichletEta(x)	(-inf,+inf)	3
ddouble.Polylog(n, x)	(-inf,1]	3	n ∈ [-4,8]
ddouble.OwenT(h, a)	(-inf,+inf)	5
ddouble.Bump(x)	(-inf,+inf)	4	C-infinity smoothness basis function, bump(x)=1/(exp(1/x-1/(1-x))+1)
ddouble.HermiteH(n, x)	(-inf,+inf)	3	n ≤ 64
ddouble.LaguerreL(n, x)	(-inf,+inf)	3	n ≤ 64
ddouble.LaguerreL(n, alpha, x)	(-inf,+inf)	3	n ≤ 64, associated
ddouble.LegendreP(n, x)	(-inf,+inf)	3	n ≤ 64
ddouble.LegendreP(n, m, x)	[-1,1]	3	n ≤ 64, associated
ddouble.ChebyshevT(n, x)	(-inf,+inf)	3	n ≤ 64
ddouble.ChebyshevU(n, x)	(-inf,+inf)	3	n ≤ 64
ddouble.ZernikeR(n, m, x)	[0,1]	3	n ≤ 64
ddouble.GegenbauerC(n, alpha, x)	(-inf,+inf)	3	n ≤ 64
ddouble.JacobiP(n, alpha, beta, x)	[-1,1]	3	n ≤ 64, alpha,beta > -1
ddouble.Bernoulli(n, x, centered)	[0,1]	4	n ≤ 64, centered: x->x-1/2
ddouble.Cyclotomic(n, x)	(-inf,+inf)	1	n ≤ 32
ddouble.MathieuA(n, q)	(-inf,+inf)	4	n ≤ 16
ddouble.MathieuB(n, q)	(-inf,+inf)	4	n ≤ 16
ddouble.MathieuC(n, q, x)	(-inf,+inf)	4	n ≤ 16, Accuracy deteriorates when q is very large.
ddouble.MathieuS(n, q, x)	(-inf,+inf)	4	n ≤ 16, Accuracy deteriorates when q is very large.
ddouble.EulerQ(q)	(-1,1)	4
ddouble.LogEulerQ(q)	(-1,1)	4
ddouble.Ldexp(x, y)	(-inf,+inf)	N/A
ddouble.Binomial(n, k)	N/A	1	n ≤ 1000
ddouble.Hypot(x, y)	N/A	2
ddouble.GeometricMean(x, y)	N/A	2
ddouble.Logit(x)	(0,1)	2
ddouble.Expit(x)	(-inf,+inf)	2
ddouble.Min(x, y)	N/A	N/A
ddouble.Max(x, y)	N/A	N/A
ddouble.Clamp(v, min, max)	N/A	N/A
ddouble.CopySign(value, sign)	N/A	N/A
ddouble.Floor(x)	N/A	N/A
ddouble.Ceiling(x)	N/A	N/A
ddouble.Round(x)	N/A	N/A
ddouble.Truncate(x)	N/A	N/A
IEnumerable<ddouble>.Sum()	N/A	N/A
IEnumerable<ddouble>.Average()	N/A	N/A
IEnumerable<ddouble>.Min()	N/A	N/A
IEnumerable<ddouble>.Max()	N/A	N/A

Constants

constant	value	note
ddouble.Pi	3.141592653589793238462...	Pi
ddouble.E	2.718281828459045235360...	Napier's E
ddouble.Sqrt2	1.414213562373095048801...	Sqrt(2)
ddouble.GoldenRatio	1.618033988749894848204...	Golden ratio
ddouble.Lg2	0.301029995663981195213...	log10(2)
ddouble.Lb10	3.321928094887362347870...	log2(10)
ddouble.Ln2	0.693147180559945309417...	log(2)
ddouble.LbE	1.442695040888963407359...	log2(e)
ddouble.EulerGamma	0.577215664901532860606...	Euler's Gamma
ddouble.Zeta3	1.202056903159594285399...	ζ(3), Apery const.
ddouble.Zeta5	1.036927755143369926331...	ζ(5)
ddouble.Zeta7	1.008349277381922826839...	ζ(7)
ddouble.Zeta9	1.002008392826082214418...	ζ(9)
ddouble.DigammaZero	1.461632144968362341263...	Positive root of digamma
ddouble.ErdosBorwein	1.606695152415291763783...	Erdös Borwein constant
ddouble.FeigenbaumDelta	4.669201609102990671853...	Feigenbaum constant
ddouble.FeigenbaumAlpha	2.502907875095892822283...	Feigenbaum constant
ddouble.LemniscatePi	2.622057554292119810465...	Lemniscate constant
ddouble.GlaisherA	1.282427129100622636875...	Glaisher–Kinkelin constant
ddouble.CatalanG	0.915965594177219015055...	Catalan's constant
ddouble.FransenRobinson	2.807770242028519365222...	Fransén–Robinson constant
ddouble.KhinchinK	2.685452001065306445310...	Khinchin's constant
ddouble.MeisselMertens	0.261497212847642783755...	Meissel–Mertens constant
ddouble.LambertOmega	0.567143290409783873000...	LambertW(1)
ddouble.LandauRamanujan	0.764223653589220662991...	Landau–Ramanujan constant
ddouble.MillsTheta	1.306377883863080690469...	Mills constant
ddouble.SoldnerMu	1.451369234883381050284...	Ramanujan–Soldner constant
ddouble.SierpinskiK	0.822825249678847032995...	Sierpiński's constant, Define follows wolfram.
ddouble.RcpFibonacci	3.359885666243177553172...	Reciprocal Fibonacci constant
ddouble.Niven	1.705211140105367764289...	Niven's constant
ddouble.GolombDickman	0.624329988543550870992...	Golomb–Dickman constant

Sequence

sequence	note
ddouble.TaylorSequence	Taylor,1/n!
ddouble.Factorial	Factorial,n!
ddouble.BernoulliSequence	Bernoulli,B(2k)
ddouble.HarmonicNumber	HarmonicNumber, H_n
ddouble.StieltjesGamma	StieltjesGamma, γ_n

Casts

• long (accurately)

ddouble v0 = 123;
long n0 = (long)v0;

• double (accurately)

ddouble v1 = 0.5;
double n1 = (double)v1;

• decimal (approximately)

ddouble v1 = 0.1m;
decimal n1 = (decimal)v1;

• string (approximately)

ddouble v2 = "3.14e0";
string s0 = v2.ToString();
string s1 = v2.ToString("E8");
string s2 = $"{v2:E8}";

I/O

BinaryWriter, BinaryReader

Licence

MIT

Author

T.Yoshimura