private static Series.PQ ArctanOfInverse(mpz digits, mpz x, bool isHyperbolic)
{
if (digits == null)
{
throw new ArgumentNullException("digits");
}
if (digits <= 0)
{
throw new ArgumentOutOfRangeException("digits");
}
if (x == null)
{
throw new ArgumentNullException("x");
}
if (x <= 1)
{
throw new ArgumentOutOfRangeException("x");
}
// http://numbers.computation.free.fr/Constants/Algorithms/splitting.html
Func <mpz, mpz, Series.PQT> directlyCompute = (a, b) =>
{
var t = 2 * a + 3;
var q = t * x.Square();
var p = (!isHyperbolic && a.IsEven()) ? mpz.NegativeOne : mpz.One;
return(new Series.PQT {
P = p, Q = q, T = t
});
};
Func <Series.PQT, Series.PQT, Series.PQT> recursivelyCombine = (am, mb) => new Series.PQT {
P = mb.Q * am.P + am.T * mb.P, Q = am.Q * mb.Q, T = am.T * mb.T
};
// How many terms to compute
/*
* x^(2k+1) > 10^digits
* => (2k+1) log(x) > digits * log(10)
* => 2k + 1 > digits * log(10) / log(x)
* => 2k > digits * log(10) / log(x)
* => k > [digits * log(10) / log(x)] / 2
*/
var n = (digits.ToMpf() * Math.Log(10, (double)x) / 2).ToMpz() + _ARCTAN_OF_INVERSE_NUMBER_OF_TERMS_SURCHARGE;
// Calclate P(0,N), Q(0,N) and T(0,N)
var pqt = Series.BinarySplitting(0, n, directlyCompute, recursivelyCombine);
return(new Series.PQ {
P = pqt.P + pqt.Q, Q = x * pqt.Q
});
}
/*
* Compute int(pi * 10^digits)
*
* This is done using Chudnovsky's series with binary splitting
*/
public static mpz PiZ(mpz digits)
{
if (digits == null)
{
throw new ArgumentNullException("digits");
}
if (digits <= 0)
{
throw new ArgumentOutOfRangeException("digits");
}
const long C = 640320;
const long C3_OVER_24 = C * C * C / 24;
/*
* Source: http://www.craig-wood.com/nick/articles/pi-chudnovsky/
*
* Computes the terms for binary splitting the Chudnovsky infinite series
*
* a(a) = +/- (13591409 + 545140134*a)
* p(a) = (6*a-5)*(2*a-1)*(6*a-1)
* b(a) = 1
* q(a) = a*a*a*C3_OVER_24
*/
Func <mpz, mpz, Series.PQT> directlyCompute = (a, b) =>
{
mpz Pab, Qab;
if (a == 0)
{
Pab = Qab = mpz.One;
}
else
{
var a2 = a << 1;
var a6 = a2 * 3;
Pab = (a6 - 5) * (a2 - 1) * (a6 - 1);
Qab = a.Power(3U) * C3_OVER_24;
}
// t(a) = p(a) * a(a)
var Tab = Pab * (13591409 + 545140134 * a);
if (a.IsOdd())
{
Tab = -Tab;
}
return(new Series.PQT {
P = Pab, Q = Qab, T = Tab
});
};
Func <Series.PQT, Series.PQT, Series.PQT> recursivelyCombine = (am, mb) => new Series.PQT
{
P = am.P * mb.P,
Q = am.Q * mb.Q,
T = mb.Q * am.T + am.P * mb.T
};
// How many terms to compute
var digitsPerTerm = Math.Log10(C3_OVER_24 / 6 / 2 / 6);
var n = (digits.ToMpf() / digitsPerTerm).ToMpz() + _PI_NUMBER_OF_TERMS_SURCHARGE;
// Calculate P(0, N), Q(0, N) and T(0, N)
var pqt = Series.BinarySplitting(0, n, directlyCompute, recursivelyCombine);
var oneSquared = mpz.Ten.Power(digits << 1);
var sqrtC = (10005 * oneSquared).Sqrt();
return((pqt.Q * 426880 * sqrtC) / pqt.T);
}