Gauss–Legendre algorithm

pi.rb

require 'pry'

# implements https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm
def calculate_pi(iterations)
  a = 1.0
  b = 1 / Math::sqrt(2)
  t = 1.0 / 4
  p = 1.0

  while iterations > 0
    a_next = ( a + b ) / 2.0
    b_next = Math::sqrt( a * b )
    t_next = t - p * ( a - a_next ) ** 2
    p_next = 2 * p

    a = a_next
    b = b_next
    t = t_next
    p = p_next

    iterations -= 1
  end

  pi = ( (a_next + b_next) ** 2 ) / ( 4 * t_next )

  pi
end

binding.pry