rational_approximations.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 16 January 2019
# https://github.com/trizen

# Simple and efficient algorithm for finding the first continued-fraction convergents to a given real constant.

# Continued-fraction convergents for PI:
#   https://oeis.org/A002485
#   https://oeis.org/A002486

# See also:
#   https://en.wikipedia.org/wiki/Continued_fraction

func rational_approximations (x, callback, first = 10) {

    x = x.float || return nil

    var (n1, n2) = (0, 1)
    var (d1, d2) = (1, 0)

    var f = x

    first.times {
        var z = f.int

        n1 += (n2 * z)
        d1 += (d2 * z)

        (n1, n2) = (n2, n1)
        (d1, d2) = (d2, d1)

        callback(n2 / d2);

        f -= z
        f.is_zero && break
        f = 1/f
    }
}

var x = Num.pi
var f = func (q) { say "PI =~ #{q.as_rat}" }

rational_approximations(x, f, 20)

__END__
PI =~ 3
PI =~ 22/7
PI =~ 333/106
PI =~ 355/113
PI =~ 103993/33102
PI =~ 104348/33215
PI =~ 208341/66317
PI =~ 312689/99532
PI =~ 833719/265381
PI =~ 1146408/364913
PI =~ 4272943/1360120
PI =~ 5419351/1725033
PI =~ 80143857/25510582
PI =~ 165707065/52746197
PI =~ 245850922/78256779
PI =~ 411557987/131002976
PI =~ 1068966896/340262731
PI =~ 2549491779/811528438
PI =~ 6167950454/1963319607
PI =~ 14885392687/4738167652