Haskell slower than Python in naive integer factorization?

Question

I am doing a math course where we needed to do some integer factorization as an intermediate step to the problem. I decided to write a Python program to do this for me (we were not tested for our ability to factor, so this is completely off the board). The program is as follows:

#!/usr/bin/env python3

import math
import sys

# Return a list representing the prime factorization of n. The factorization is
#   found using trial division (highly inefficient).
def factorize(n):

    def factorize_helper(n, min_poss_factor):
        if n <= 1:
            return []
        prime_factors = []
        smallest_prime_factor = -1
        for i in range(min_poss_factor, math.ceil(math.sqrt(n)) + 1):
            if n % i == 0:
                smallest_prime_factor = i
                break
        if smallest_prime_factor != -1:
            return [smallest_prime_factor] \
                   + factorize_helper(n // smallest_prime_factor,
                                      smallest_prime_factor)
        else:
            return [n]

    if n < 0:
        print("Usage: " + sys.argv[0] + " n   # where n >= 0")
        return []
    elif n == 0 or n == 1:
        return [n]
    else:
        return factorize_helper(n, 2)

if __name__ == "__main__":
    factorization = factorize(int(sys.argv[1]))
    if len(factorization) > 0:
        print(factorization)

I also taught myself Haskell, so I decided to try rewriting the program in Haskell. This program is as follows:

import System.Environment

-- Return a list containing all factors of n at least x.
factorize' :: (Integral a) => a -> a -> [a]
factorize' n x = smallestFactor
                 : (if smallestFactor == n
                    then []
                    else factorize' (n `quot` smallestFactor) smallestFactor)
    where
        smallestFactor = getSmallestFactor n x
        getSmallestFactor :: (Integral a) => a -> a -> a
        getSmallestFactor n x
            | n `rem` x == 0                          = x
            | x > (ceiling . sqrt . fromIntegral $ n) = n
            | otherwise                               = getSmallestFactor n (x+1)

-- Return a list representing the prime factorization of n.
factorize :: (Integral a) => a -> [a]
factorize n = factorize' n 2

main = do
    argv <- getArgs
    let n = read (argv !! 0) :: Int
    let factorization = factorize n
    putStrLn $ show (factorization)
    return ()

(note: this requires a 64-bit environment. In the 32-bit version, import Data.Intand use Int64as type annotation on read (argv !! 0))

, , , , , . , :

$ ghc --make -O2 factorize.hs
$ /usr/bin/time -f "%Uu %Ss %E" ./factorize 89273487253497
[3,723721,41117819]
0.18u 0.00s 0:00.23

, Python:

$ /usr/bin/time -f "%Uu %Ss %E" ./factorize.py 89273487253497
[3, 723721, 41117819]
0.09u 0.00s 0:00.09

, , , , , Python , Haskell. , Haskell , , , , .

Haskell, , , . , ? Haskell I/O ? ? Haskell, .

Answer 1

limit = ceiling . sqrt . fromIntegral $ n , , , Haskell :

limit = ceiling . sqrt . fromIntegral $ n
smallestFactor = getSmallestFactor x

getSmallestFactor x
    | n `rem` x == 0 = x
    | x > limit      = n
    | otherwise      = getSmallestFactor (x+1)

, :

$ time ./factorizePy.py 89273487253497
[3, 723721, 41117819]

real    0m0.236s
user    0m0.171s
sys     0m0.062s

$ time ./factorizeHs  89273487253497
[3,723721,41117819]

real    0m0.190s
user    0m0.000s
sys     0m0.031s

Answer 2

, , , . , , factorize :

factorize' undefined undefined = undefined : undefined

, GHC thunks. . , :

{-# LANGUAGE BangPatterns #-}

factorize' :: Integral a => a -> a -> [a]
factorize' n x
  | smallestFactor == n = [smallestFactor]
  | otherwise = smallestFactor : factorize' (n `quot` smallestFactor) smallestFactor
  where
    smallestFactor = getSmallestFactor n (ceiling . sqrt . fromIntegral $ n) x
    getSmallestFactor n !limit x
       | n `rem` x == 0 = x
       | x > limit = n
       | otherwise = getSmallestFactor n limit (x+1)

-- Return a list representing the prime factorization of n.
factorize :: Integral a => a -> [a]
factorize n = factorize' n 2

getSmallestFactor n, . getSmallestFactor . , ; .