# A Better Fibonacci (Revisited)

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 const*fib to run faster and more efficiently than loop*fib?