approximations.rb

#!/usr/bin/env ruby

def s n
  Math.sqrt n
end

approxes = [
  "3",
  "s(7 + s(6 + s(5)))",
  "(63 * (17 + 15 * s(5))) / (25 * (7 + 15 * s(5)))",
  "339 / 108.0",
  "s(15) - s(3) + 1",
  "(7.0 ** 7) / (4.0 ** 9)",
  "355 / 113.0",
  "3 + 8/60.0 + 29/(60.0 ** 2) + 44/(60.0 ** 3)",
  "3927 / 1250.0",
  "377.0 / 120",
  "25.0 / 8",
  "22.0 / 7",
  "s(2) + s(3)",
  "(2143.0 / 22) ** (0.25)",
  "31 ** (1 / 3.0)",
  "9.0/5 + s(9.0/5)"
]

vals = approxes.map { |a| [a, eval(a)] }.sort_by { |k,v| (Math::PI - v).abs }
puts vals