Implementing an algorithm to compute pi in R - r

Question

I am trying to implement a variation of the Brent-Salamin algorithm in R. It works well for the first 25 iterations, but then, it behaves unexpectedly, returning negative results.
The algorithm I want to implement is:
initial values:
x_0 = 1; y_0 = 1/sqrt(2); z_0 = 1/2
x_n = (x_n-1 + y_n-1)/2
y_n = sqrt(x_n-1 * y_n-1)
z_n = z_n-1 - 2^n * (x_n^2-y_n^2)
p_n = (2 * x_n^2) / z_n
where n is the current iteration.
A more beautifully formatted formula is here.
The code I figured out is:
mypi <- function(n){
x = 1
y = 1/sqrt(2)
z = 1/2
iteration = 0
while(iteration < n){
iteration = iteration + 1
newx = (x + y) / 2
y = sqrt(x * y)
x = newx
z = z-(2^iteration * (x^2 - y^2))
p = (2 * x^2) / z
}
return(p)
}
Output:
> mypi(10)
[1] 3.141593
> mypi(20)
[1] 3.141593
> mypi(50)
[1] -33.34323
So as I am new to R, is there a bug in my code or is it the algorithm?

Answer 1

Your code simply messes up because it does not agree with the algorithm as written in the wiki page. A correct version looks like this:
mypi <- function(n){
x = 1
y = 1/sqrt(2)
z = 1/4
p <- 1
iteration = 0
while(iteration < n){
iteration = iteration + 1
newx = (x + y) / 2
y = sqrt(x * y)
# x = newx
# z = z-(2^iteration * (x^2 - y^2))
z = z- p* (x-newx)^2
p = 2*p
x = newx
}
(newx + y)^2/(4*z)
}
Gives
> mypi(10)
[1] 3.141593
> mypi(20)
[1] 3.141593
> mypi(50)
[1] 3.141593