A Better Fibonacci (Revisited)

cross-posted

Previously, I posted a non-recursive algorithm to compute an arbitrary Fibonacci number:

PHI = 1.6180339887498948482045868
const_fib = lambda do |n|
  (
    ( PHI**n - ( 1 - PHI )**n ) /
    Math.sqrt(5)
  ).
  floor
end

Since the mathematical constant PHI is an infinite, non-repeating decimal, it can never be exactly represented in code. Indeed, the Fibonacci series computed by the above code and the exact series diverge after only the 70th value.

computed = []
(1..1474).each { |n| computed << const_fib[n].to_s }

# list of Fibonnaci number can be downloaded [here](http://oeis.org/A000045/b000045.txt)
exact = []
File.open('./exact.txt', 'r').each_line do |line|
  exact << line.split.last
end

exact.shift # the first entry on the downloaded list is zero, get rid of it

index_of_last_match = (computed & exact).size
#=> 70

computed[index_of_last_match] == exact[index_of_last_match]
#=> true

computed[index_of_last_match + 1] == exact[index_of_last_match + 1]
#=> false

When I first hacked at this hackneyed code exercise, I assumed using the constant PHI would still be more efficient than a loop or a recursive algorithm. Turns out, modern hardware make the loop-based solution quite tolerable.

loop_fib = lambda do |n|
  fib = []
  n.times do |n|
    if n < 2
      fib = [0, 1, 1]
    else
      fib << ( fib[-1] + fib[-2] ).to_s
    end
  end
  return fib.last
end

loop_fib[70] == exact[70]
#=> true

loop_fib[71] == exact[71]
#=> true

My computer hardly notices. How about the recursive method?

recur_fib = lambda do |n|
  case n
  when 0
    1
  when 1
    1
  else
    recur_fib[n - 1] + recur_fib[n - 2]
  end
end

recur_fib[70]

Beautiful math; horrifying code! Running it quickly ate up my memory and swap space.

I'm going to pose a challenge for readers here. Read section 1.2.1 in Structure and Interpretation of Computer Programs for a much more reasonable recursive template, and try combining the approach in that book with the first algorithm above. Specifically, pack your recursion into a separate function that computes an arbitrary precision value for PHI. Can we get constfib to run faster and more efficiently than loopfib?