## Programming Praxis – Sieve Of Euler

In today’s Programming Praxis exercise, our goal is to implement the Sieve of Euler to generate primes. Let’s get started, shall we?

Some imports:

```import Data.List.Ordered
import Data.Time.Clock```

Euler’s sieve is pretty simple: just filter out all remaining multiples of each successive prime number. We add the minor optimization of only considering odd numbers.

```primes_euler :: [Integer]
primes_euler = 2 : euler [3,5..] where
euler ~(x:xs) = x : euler (minus xs \$ map (* x) (x:xs))```

To compare, we use the Sieve of Eratosthenes implementation found here:

```primes_eratosthenes :: [Integer]
primes_eratosthenes = 2 : 3 : sieve (tail primes_eratosthenes) 3 [] where
sieve ~(p:ps) x fs = let q=p*p in
foldr (flip minus) [x+2,x+4..q-2] [[y+s,y+2*s..q] | (s,y) <- fs]
++ sieve ps q ((2*p,q):[(s,q-rem (q-y) s) | (s,y) <- fs])```

A quick and dirty benchmarking function:

```benchPrimes :: Num a => [a] -> IO ()
benchPrimes f = do start <- getCurrentTime
print . sum \$ take 10000 f
print . flip diffUTCTime start =<< getCurrentTime```

Time to run the benchmarks.

```main :: IO ()
main = do benchPrimes primes_eratosthenes
benchPrimes primes_euler```

The sieve of Eratosthenes runs in about 0.02 seconds, while Euler’s sieve takes around 6.6 seconds, which means this implementation of Euler’s Sieve is about 300-400 times slower. Clearly you don’t want to use this version in practice.

### 3 Responses to “Programming Praxis – Sieve Of Euler”

1. cldy Says:

I’m Haskell newbie and I was wondering why did you used ‘~’ in definitions of ‘euler’ and ‘sieve’? As far as I know this is lazy pattern matching but I failed to understand where to use it and why.

2. Remco Niemeijer Says:

Correct, the ~ indicates lazy pattern binding. More importantly, it also makes the pattern irrefutable. Normally, defining a function f (x:xs) = … will produce a warning at compile time saying that the matches for f are non-exhaustive (since f [] is not defined). You could of course fix this by adding something like f [] = undefined, but that adds a useless line of code. I prefer to use ~ when I know a function will only over be applied to an infinite list. It means that it will always try to match the given pattern, producing a run-time error if it can’t. This gets rid of the compiler warning without needless extra code. For more information, see http://www.haskell.org/tutorial/patterns.html#tut-lazy-patterns

• cldy Says:

Thanks! Now it’s clear.