The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi" (/paɪ/).

Viète's Formula

In mathematics, Viète's formula is the following infinite product of nested radicals representing the mathematical constant π:

It is named after François Viète (1540–1603), who published it in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII. Using his formula, Viète calculated π to an accuracy of nine decimal digits.

    function vietesSeries() result(pi)
        real :: pi

        integer, parameter :: ITERATIONS = 15
        integer index
        integer jndex

        real factor

        pi = 1
        do index = ITERATIONS, 2, -1
            factor = 2

            do jndex = 1, index - 1
                factor = 2 + sqrt(factor)
            end do

            factor = sqrt(factor)
            pi = pi * factor / 2
        end do

        pi = pi * sqrt(2.0) / 2
        pi = 2 / pi
    end function

Wallis Formula

In mathematics, Wallis' product for π, written down in 1655 by John Wallis, states that

    function wallisSeries() result(pi)
        real(8) :: pi

        integer, parameter :: ITERATIONS = 900000000
        real(8) index

        index = 3
        pi    = 4
        do while (index <= (ITERATIONS + 2))
            pi = pi * ((index - 1) / index) * ((index + 1) / index)

            index = index + 2
        end do
    end function

Leibniz Formula

In mathematics, the Leibniz formula for π, named after Gottfried Leibniz, states that

    function leibnizSeries() result(pi)
        real(8) :: pi

        integer, parameter :: ITERATIONS = 900000000
        integer sign
        integer index

        index = 1
        sign  = 1
        pi    = 0
        do while (index <= (ITERATIONS * 2))
            pi   = pi + sign * (4.0 / index)
            sign = -sign

            index = index + 2
        end do
    end function

Nilakantha Formula

An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:

    function nilakanthaSeries() result(pi)
        real(8) :: pi

        integer, parameter :: ITERATIONS = 500
        integer sign
        real(8) index

        index = 2
        sign  = 1
        pi    = 3
        do while (index <= (ITERATIONS * 2))
            pi   = pi + sign * (4 / (index * (index + 1) * (index + 2)))
            sign = -sign

            index = index + 2
        end do
    end function

Results

Here are calculation performed with each of the algorithms for calculating π number to first eight decimal digits (3.14159265).