The algorithm used as a basis to calculate
Pi to 206,158,430,000 decimal digits on September 18 to 20 1999, along with [Borweins Quartically Convergent Algorithm]
?. Based on the individual work of [Johann Gauss]
? (1777-1855) and [Adrien-Marie Legendre]
? (1752-1833) combined with modern algorithms for multiplication and square roots
1. Initial value setting;
- a = 1 b = 1 / SqRt?(2) t = 1/4 x = 1
2. Repeat the following statements until the difference of a and b is within the desired accuracy;
- y = a
- a = (a+b) / 2
- b = SqRt?(b*y)
- t = t - x * (y-a)^2
- x = 2 * x
3. Pi is approximated with a, b and t as;
- Pi = ((a+b)^2) / (4*t)
The algorithm has second order convergent nature.