sidef-scripts/Math/BBP-type_formulae.sf

#!/usr/bin/ruby

# The generalized Bailey–Borwein–Plouffe formula (BBP formula)

# See also:
#   https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula

func BBP(s, b, A, m=A.len, r=200) {
    sum(0..r, {|k|
        sum(1..m, {|j|
            A[j-1] / (m*k + j)**s
        }) / b**k
    })
}

say BBP(1,  2, [1])                          # 1.38629436111989061883446424291635313615100026872 = log(4)
say BBP(1, 16, [4, 0])                       # 4.08660499012792546564411277042929547902488637157 = log(390625/6561)
say BBP(1, 16, [4, 0, 0, -2, -1, -1, 0, 0])  # 3.14159265358979323846264338327950288419716939938 = Pi

# The BBP() function can also be used to efficiently compute atan(1/x)

func plouffe_atan(x) {
    BBP(1, x**4, [1, 0, -(x**-2), 0]) / x
}

say plouffe_atan(2)                          # 0.463647609000806116214256231461214402028537054286 = atan(1/2)
say plouffe_atan(3+4i)                       # 0.122489331563432077086041240605637905457072049191-0.158997191679999174364761036007018781573305474235i