I wrote a simple (and unserious) prime number generator in Haskell, with mutually-recursive definitions for generating the primes and for determining the primeness of a number:
primes :: (Integral a) => [a]
primes = 2 : filter isPrime [3, 5..]
isPrime :: (Integral a) => a -> Bool
isPrime m = all (indiv m) (takeWhile (<= (intSqrt m)) primes)
intSqrt :: (Integral a) => a -> a
intSqrt 1 = 1
intSqrt n = div (m + (div (n - 1) m)) 2
where m = intSqrt (n - 1)
indiv :: (Integral a) => a -> a -> Bool
indiv m n = rem m n /= 0
I noticed that it seems to reconstruct the sublist of already-generated primes with every reference to primes:
*Main> take 200 primes
[2,3,5,7, ..., 1223]
(2.70 secs, 446142856 bytes)
*Main> take 200 primes
[2,3,5,7, ..., 1223]
(2.49 secs, 445803544 bytes)
But when I change the type annotations to use a concrete integral type, such as
primes :: [Integer]
isPrime :: Integer -> Bool
each prime is only generated once:
*Main> :r
[1 of 1] Compiling Main ( Primes.hs, interpreted )
Ok, modules loaded: Main.
*Main> take 200 primes
[2,3,5,7, ..., 1223]
(2.15 secs, 378377848 bytes)
*Main> take 200 primes
[2,3,5,7, ..., 1223]
(0.01 secs, 1626984 bytes)
Seems curious to me. Any particular reason this is happening?
When you say
primes :: [Integer]
then primes is a constant, but when you say
primes :: (Integral a) => [a]
then it is a function with a hidden parameter: the Integral instance for whichever type a is. And like other functions, it will recompute its results when you call it with the same parameters (unless you've explicitly memoised it).