Re: [Haskell-cafe] The maths behind the Fastest Fibb In The West.

On Saturday, May 7, 2016 at 12:59:10 AM UTC-5, Michael Litchard wrote:

Thanks for your response Erik. It appears I have not articulated my question well enough. When p is odd, why is the return value

(f*(f+2*g), f^2 + g^2)

as opposed to the return value of

(f^2+g^2, g*(2*f-g))

What is it about the boolean value that requires two entirely seperate things to be done?

The algorithm is based on the following Fibonacci identities (which hopefully I've recited correctly): F_{2n} = F_n^2 + F_{n+1}^2 F_{2n+1) = F_{n+1}*(2*F_n + F_{n+1}) and there is a similar formula for F_{2n-1} in terms of F_n and F_{n+1}. So given a pair of consecutive Fibonacci numbers, (F_n, F_{n+1}), we can use these formulas to compute another pair of consecutive Fibonacci's, either (F_{2n-1}, F_2n) or (F_2n, F_{2n+1}). This is what fib' is doing -- f and g are a pair of consecutive Fibonacci's and it returns one of these new pairs based on whatever p is. So, starting from the pair (F_1, F_2) we have to figure out the sequence of p's which will get us to our target F_n, and that appears to be related to the binary representation of n.

On Fri, May 6, 2016 at 7:38 PM, Erik Rantapaa javascript:> wrote:

...

On Friday, May 6, 2016 at 6:46:26 PM UTC-5, Michael Litchard wrote:

...

I've been working on a project that needs a good fibonacci generator, and I'm to the point where can now improve upon this one: https://wiki.haskell.org/The_Fibonacci_sequence#Fastest_Fib_in_the_West https://www.google.com/url?q=https%3A%2F%2Fwiki.haskell.org%2FThe_Fibonacci_sequence%23Fastest_Fib_in_the_West&sa=D&sntz=1&usg=AFQjCNFzix5dbPVZ36_StXEF6yTYuaAg4w

thanks to this guy: https://groups.google.com/forum/#!topic/haskell-cafe/HUgbAUCvCp4

He suggested breaking up a guard into two diffeent functions, which I can do, but I don't know what to call them because I don't know why the operations are different. I'm referring to this section:

fib' (f, g) p | p = (f*(f+2*g), f^2 + g^2) | otherwise = (f^2+g^2, g*(2*f-g))

I'd like to know the reason why each guard does two entirely different things, so I know what to call the functions when I seperate them out.

Clearly `p` is a Bool, and it comes from the expression:

map (toEnum . fromIntegral) $ unfoldl divs n

What's going on in `toEnum . fromIntegral` is that a remainder (either 0 or 1 - it blows up for anything else) is being converted to a Bool, with 0 mapping to False and 1 mapping to True. So `isOdd` would be a more descriptive name for `p`.