Calculating Pi (Gauss–Legendre’s Arithmetic-Geometric Mean Method)

We will examine the AGM method (Gauss-Legendre algorithm, arithmetic-geometric mean method), the algorithm used by Mr. Takahashi and his colleagues at the University of Tsukuba to calculate pi to 2,576,980,370,000 digits.
Using the initial values a0=1, b0=1/√2, t0=1/4, and p0=1,
an+1 = (an + bn) / 2
bn+1 = √(anbn)
tn+1 = tn - pn * (an+1 - an)^2
pn+1 = 2*pn
Thus,
π ≈ (an + bn)^2 / (4*tn)
is used for the calculation.