99 questions/Solutions/39

(*) 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 = filter isPrime [a..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 by trial division using (up to${\sqrt {b}}$) a memoized primes list produced by sieve of Eratosthenes to which it refers internally. So it'll be slower, but immediate.

Solution 2:

primes :: Integral a => [a]
primes = let sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ] 
         in sieve [2..]

primesR :: Integral a => a -> a -> [a]
primesR a b = takeWhile (<= b) $ dropWhile (< a) primes

Another way to compute the claimed list is done by using the Sieve of Eratosthenes. The primes function generates a list of all (!) prime numbers using this algorithm and primesR filter 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 executable specification, with all the implied pitfalls of inefficiency when (ab)used as if it were an actual code.

Solution 3:

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 haskellwiki/prime_numbers, 
--  adjusted to produce primes in a given range (inclusive)
primesR a b 
  | b<a || b<2 = []
  | otherwise  = 
     (if a <= 2 then [2] else []) ++
     gaps a' (join [[x,x+step..b] | p <- takeWhile (<= z) primes'
                    , let q = p*p ; step = 2*p
                          x = snapUp (max a' q) q step ])
  where
    primes' = tail primesTME        -- external unbounded list of primes
    a'      = snapUp (max 3 a) 1 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
    gaps k []       = [k,k+2..b]
    snapUp v origin step = let r = rem (v-origin) step -- rem OK if 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
    union  a      b     = a ++ b

(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.3
[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193]
(5,497 reductions, 11,382 cells)

> takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME   -- TME of Q.31
[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.2
     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 [10100..10200]                                  -- Sol.1
[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193]
(15,750 reductions, 29,292 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 up all 49 primes in one go after just under 1 sec.

Solution 4.

For very wide ranges, specifically when $a<{\sqrt {b}}$, 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