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I'm interested in Haskell.
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A perpetual Haskell newbie. I like ''[http://ideone.com/qpnqe this one-liner]'':
 
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I like ''[http://ideone.com/qpnqe this]'':
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<haskell>
 
<haskell>
--  inifinte folding idea due to Richard Bird
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--  infinite folding idea due to Richard Bird
 
--  double staged production idea due to Melissa O'Neill
 
--  double staged production idea due to Melissa O'Neill
--  tree folding idea Dave Bayer / simplified formulation Will Ness
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--  tree folding idea Dave Bayer / improved tree structure
primes = 2 : g (fix g)
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--    Heinrich Apfelmus / simplified formulation Will Ness
  where               
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primes = 2 : _Y ((3:) . gaps 5
    g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs)  
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                      . foldi (\(x:xs) -> (x:) . union xs) []
                            [[x*x, x*x+2*x..] | x <- xs])
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                      . map (\p-> [p*p, p*p+2*p..]))  
    gaps k s@(c:t)                                       
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      | k < c = k : gaps (k+2) s    -- minus [k,k+2..] (c:t), k<=c
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      | True  =    gaps (k+2) t    --  fused to avoid a space leak
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fix g = xs where xs = g xs            -- global defn to avoid space leak
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_Y g = g (_Y g)  -- multistage production
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gaps k s@(c:t)                        -- == minus [k,k+2..] (c:t), k<=c,
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  | k < c    = k : gaps (k+2) s    --    fused for better performance
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  | otherwise =    gaps (k+2) t    -- k==c
 
</haskell>
 
</haskell>
  
 
<code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. <code>union</code> and more at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]].
 
<code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]] page. <code>union</code> and more at [[Prime numbers#Sieve_of_Eratosthenes|Prime numbers]].
  
The math formula for Sieve of Eratosthenes,
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The constructive definition of primes is the Sieve of Eratosthenes:
  
::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{n p:n \in \mathbb{N}_{p}\}</math>  
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::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}</math>  
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using standard definition
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::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math> &emsp; . . . or, &ensp;<math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> &emsp; :)&emsp;:) .
  
where
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Trial division sieve is:
 
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::::<math>\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}</math> &emsp; . . . or, &ensp;<math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math> &emsp; :)&emsp;:) .
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Trial division sieve:
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::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}</math>
  
::::<math>\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\not\exists p \in \mathbb{T}) (p\leq \sqrt{n} \and p\mid n)\}</math>
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If you're put off by self-referentiality, just replace <math>\mathbb{S}</math> or <math>\mathbb{T}</math> on the right-hand side of equations with <math>\mathbb{N}_{2}</math>, but even ancient Greeks knew better.

Revision as of 09:30, 6 August 2013

A perpetual Haskell newbie. I like this one-liner:

--   infinite folding idea due to Richard Bird
--   double staged production idea due to Melissa O'Neill
--   tree folding idea Dave Bayer / improved tree structure 
--     Heinrich Apfelmus / simplified formulation Will Ness
primes = 2 : _Y ((3:) . gaps 5  
                      . foldi (\(x:xs) -> (x:) . union xs) []
                      . map (\p-> [p*p, p*p+2*p..])) 
 
_Y g = g (_Y g)  -- multistage production
 
gaps k s@(c:t)                        -- == minus [k,k+2..] (c:t), k<=c,
   | k < c     = k : gaps (k+2) s     --     fused for better performance
   | otherwise =     gaps (k+2) t     -- k==c

foldi is on Tree-like folds page. union and more at Prime numbers.

The constructive definition of primes is the Sieve of Eratosthenes:

\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}

using standard definition

\textstyle\mathbb{N}_{k} = \{ n \in \mathbb{N} : n \geq k \}   . . . or,  \textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}   :) :) .

Trial division sieve is:

\textstyle\mathbb{T} = \{n \in \mathbb{N}_{2}: (\forall p \in \mathbb{T})(2\leq p\leq \sqrt{n}\, \Rightarrow \neg{(p \mid n)})\}

If you're put off by self-referentiality, just replace \mathbb{S} or \mathbb{T} on the right-hand side of equations with \mathbb{N}_{2}, but even ancient Greeks knew better.