function math.Round (x)
  if x >= 0 then
    return math.floor (x + 0.5)
  end  -- if positive

  return math.ceil (x - 0.5)
end -- function round

function fib(n)
    local x = 0
    local xlast = 1
    for i=1,n,1 do
        x,xlast = x + xlast,x
    end
    return x
end

function findPhi()
    local n = 1
    local res
    repeat
        n = n+1
        res = fib(n)/fib(n-1)
    until not (res > 0)
    n = n - 2
    phi = fib(n)/fib(n-1)
    return phi, n
end

local phi = findPhi()
local sqrtFive = math.sqrt(5)

function fibAlt(n)
    return math.Round(math.pow(((n >= 0) and 1 or -1)*phi, n) / sqrtFive)
end

local n = 0
repeat
    n=n+1
until(fib(n) ~= fibAlt(n))
print("Accurate up until and including the "..(n - 1).."th iteration.")
print("Actual:  ",fib(n-1))
print("Estimate:",fibAlt(n-1))