module Tins::Attempt

def compute_duration_base(sleep, attempts)
  x1, x2  = 1, sleep
  attempts <= sleep or raise ArgumentError,
    "need less or equal number of attempts than sleep duration #{sleep}"
  x1 >= x2 and raise ArgumentError, "invalid sleep argument: #{sleep.inspect}"
  function = -> x { (0...attempts).inject { |s, i| s + x ** i } - sleep }
  f, fmid = function[x1], function[x2]
  f * fmid >= 0 and raise ArgumentError, "invalid sleep argument: #{sleep.inspect}"
  n       = 1 << 16
  epsilon = 1E-16
  root = if f < 0
           dx = x2 - x1
           x1
         else
           dx = x1 - x2
           x2
         end
  n.times do
    fmid = function[xmid = root + (dx *= 0.5)]
    fmid < 0 and root = xmid
    dx.abs < epsilon or fmid == 0 and return root
  end
  raise ArgumentError, "too many iterations (#{n})"
  result
end