# 99 questions/Solutions/39

### From HaskellWiki

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<haskell> | <haskell> | ||

primesR :: Integral a => a -> a -> [a] | primesR :: Integral a => a -> a -> [a] | ||

− | primesR a b = filter isPrime [a..b] | + | primesR a b | even a = filter isPrime [a+1,a+3..b] |

+ | | True = filter isPrime [a,a+2..b] | ||

</haskell> | </haskell> | ||

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'''Solution 3.''' | '''Solution 3.''' | ||

+ | |||

+ | Another way to compute the claimed list is done by using the ''Sieve of Eratosthenes''. | ||

<haskell> | <haskell> | ||

primesR :: Integral a => a -> a -> [a] | primesR :: Integral a => a -> a -> [a] | ||

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</haskell> | </haskell> | ||

− | + | The <code>sieve [2..]</code> function call generates a list of all (!) prime numbers using this algorithm and <code>primesR</code> filters the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nicely the ''Sieve of Eratosthenes'' can be implemented in Haskell :)] | |

''this is of course a famous case of (mislabeled) executable specification, with all the implied pitfalls of inefficiency when (ab)used as if it were an actual code''. | ''this is of course a famous case of (mislabeled) executable specification, with all the implied pitfalls of inefficiency when (ab)used as if it were an actual code''. | ||

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| b<a || b<2 = [] | | b<a || b<2 = [] | ||

| otherwise = | | otherwise = | ||

− | (if a <= 2 then [2] else []) ++ | + | (if a <= 2 then [2] else []) ++ |

− | gaps a' | + | (takeWhile (<= b) $ gaps a' $ |

− | + | join [[x,x+step..] | p <- takeWhile (<= z) primes' -- p^2 <= b | |

− | + | , let q = p*p ; step = 2*p | |

+ | x = snap (max a' q) q step ]) | ||

where | where | ||

primes' = tail primesTME -- external unbounded list of primes | primes' = tail primesTME -- external unbounded list of primes | ||

− | a' = | + | a' = snap (max 3 a) 3 2 |

z = floor $ sqrt $ fromIntegral b + 1 | z = floor $ sqrt $ fromIntegral b + 1 | ||

join (xs:t) = union xs (join (pairs t)) | join (xs:t) = union xs (join (pairs t)) | ||

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gaps k xs@(x:t) | k==x = gaps (k+2) t | gaps k xs@(x:t) | k==x = gaps (k+2) t | ||

| True = k : gaps (k+2) xs | | True = k : gaps (k+2) xs | ||

− | + | snap v origin step = let r = rem (v-origin) step -- v >= origin | |

− | + | in if r==0 then v else v+(step-r) | |

− | + | ||

-- duplicates-removing union of two ordered increasing lists | -- duplicates-removing union of two ordered increasing lists | ||

union (x:xs) (y:ys) = case (compare x y) of | union (x:xs) (y:ys) = case (compare x y) of | ||

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EQ -> x : union xs ys | EQ -> x : union xs ys | ||

GT -> y : union (x:xs) ys | GT -> y : union (x:xs) ys | ||

− | |||

</haskell> | </haskell> | ||

''(This turned out to be quite a project, with some quite subtle points.)'' It should be much better then taking a slice of a full sequential list of primes, as it won't try to generate any primes between the ''square root of b'' and ''a''. To wit, | ''(This turned out to be quite a project, with some quite subtle points.)'' It should be much better then taking a slice of a full sequential list of primes, as it won't try to generate any primes between the ''square root of b'' and ''a''. To wit, | ||

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> primesR 10100 10200 -- Sol.4 | > primesR 10100 10200 -- Sol.4 | ||

[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] | [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] | ||

− | ( | + | (4,656 reductions, 10,859 cells) |

> takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME -- Sol.2 | > takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME -- Sol.2 | ||

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(54,893,566 reductions, 79,935,263 cells, 6 garbage collections) | (54,893,566 reductions, 79,935,263 cells, 6 garbage collections) | ||

− | > filter isPrime [ | + | > filter isPrime [10101,10103..10200] -- Sol.1 |

[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] | [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] | ||

− | ( | + | (12,927 reductions, 24,703 cells) -- isPrime: Q.31 |

</haskell> | </haskell> | ||

## Revision as of 09:50, 5 June 2011

(*) A list of prime numbers.

Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range.

**Solution 1.**

primesR :: Integral a => a -> a -> [a] primesR a b | even a = filter isPrime [a+1,a+3..b] | True = filter isPrime [a,a+2..b]

If we are challenged to give all primes in the range between a and b we simply take all numbers from a up to b and filter all the primes through.

This is good for *very narrow ranges* as Q.31's `isPrime`

tests numbers by *trial division* using (up to) a memoized primes list produced by sieve of Eratosthenes to which it refers internally. So it'll be slower, but immediate, testing the numbers one by one.

**Solution 2.**

For *very wide* ranges, specifically when , we're better off just using the primes sequence itself, without any post-processing:

primes :: Integral a => [a] primes = primesTME -- of Q.31 primesR :: Integral a => a -> a -> [a] primesR a b = takeWhile (<= b) $ dropWhile (< a) primes

**Solution 3.**

Another way to compute the claimed list is done by using the *Sieve of Eratosthenes*.

primesR :: Integral a => a -> a -> [a] primesR a b = takeWhile (<= b) $ dropWhile (< a) $ sieve [2..] where sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ]

The `sieve [2..]`

function call generates a list of all (!) prime numbers using this algorithm and `primesR`

filters the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nicely the *Sieve of Eratosthenes* can be implemented in Haskell :)]

*this is of course a famous case of (mislabeled) executable specification, with all the implied pitfalls of inefficiency when (ab)used as if it were an actual code*.

**Solution 4.**

Use the *proper* Sieve of Eratosthenes from e.g. 31st question's solution (instead of the above sieve of Turner), adjusted to start its multiples production from the given start point:

{-# OPTIONS_GHC -O2 -fno-cse #-} -- tree-merging Eratosthenes sieve, primesTME of Q.31, -- adjusted to produce primes in a given range (inclusive) primesR a b | b<a || b<2 = [] | otherwise = (if a <= 2 then [2] else []) ++ (takeWhile (<= b) $ gaps a' $ join [[x,x+step..] | p <- takeWhile (<= z) primes' -- p^2 <= b , let q = p*p ; step = 2*p x = snap (max a' q) q step ]) where primes' = tail primesTME -- external unbounded list of primes a' = snap (max 3 a) 3 2 z = floor $ sqrt $ fromIntegral b + 1 join (xs:t) = union xs (join (pairs t)) join [] = [] pairs (xs:ys:t) = (union xs ys) : pairs t pairs t = t gaps k xs@(x:t) | k==x = gaps (k+2) t | True = k : gaps (k+2) xs snap v origin step = let r = rem (v-origin) step -- v >= origin in if r==0 then v else v+(step-r) -- duplicates-removing union of two ordered increasing lists union (x:xs) (y:ys) = case (compare x y) of LT -> x : union xs (y:ys) EQ -> x : union xs ys GT -> y : union (x:xs) ys

*(This turned out to be quite a project, with some quite subtle points.)* It should be much better then taking a slice of a full sequential list of primes, as it won't try to generate any primes between the *square root of b* and *a*. To wit,

> primesR 10100 10200 -- Sol.4 [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (4,656 reductions, 10,859 cells) > takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME -- Sol.2 [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (140,313 reductions, 381,058 cells) > takeWhile (<= 10200) $ dropWhile (< 10100) $ sieve [2..] -- Sol.3 where sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ] [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (54,893,566 reductions, 79,935,263 cells, 6 garbage collections) > filter isPrime [10101,10103..10200] -- Sol.1 [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (12,927 reductions, 24,703 cells) -- isPrime: Q.31

(testing with Hugs of Nov 2002).

This solution is faster but not immediate. It has a certain preprocessing stage but then goes on fast to produce the whole range. To illustrate, to produce the 49 primes in 1000-wide range above 120200300100 it takes about 18 seconds on my oldish notebook for the 1st version, with the first number produced almost immediately (~ 0.4 sec); but this version spews out all 49 primes in one go after just under 1 sec.