Difference between revisions of "User:WillNess"

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A perpetual Haskell newbie. I like ''[http://ideone.com/qpnqe this semi-one-liner]'':
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I like ''[http://ideone.com/qpnqe this one-liner]'':
   
 
<haskell>
 
<haskell>
-- inifinte folding idea due to Richard Bird
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-- infinite folding due to Richard Bird
-- double staged production idea due to Melissa O'Neill
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-- double staged primes production due to Melissa O'Neill
-- tree folding idea Dave Bayer / simplified formulation Will Ness
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-- tree folding idea Heinrich Apfelmus / Dave Bayer
primes = 2 : g (fix g)
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primes = 2 : _Y ((3:) . gaps 5
where
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. foldi (\(x:xs) -> (x:) . union xs) []
g xs = 3 : gaps 5 (foldi (\(c:cs) -> (c:) . union cs) []
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. map (\p-> [p*p, p*p+2*p..]))
[[x*x, x*x+2*x..] | x <- xs])
 
   
fix g = xs where xs = g xs -- global defn to avoid space leak
 
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_Y g = g (_Y g) -- multistage production via Y combinator
   
gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c,
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gaps k s@(c:t) -- == minus [k,k+2..] (c:t), k<=c,
| k < c = k : gaps (k+2) s -- fused to avoid a space leak
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| k < c = k : gaps (k+2) s -- fused for better performance
| True = gaps (k+2) t
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| otherwise = gaps (k+2) t -- k==c
 
</haskell>
 
</haskell>
   
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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>
 
::::<math>\textstyle\mathbb{S} = \mathbb{N}_{2} \setminus \bigcup_{p\in \mathbb{S}} \{p\,q:q \in \mathbb{N}_{p}\}</math>
 
using standard definition
 
using standard definition
::::<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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::::<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> .
   
 
Trial division sieve is:
 
Trial division sieve is:

Latest revision as of 11:45, 8 April 2015

I like this one-liner:

--   infinite folding due to Richard Bird
--   double staged primes production due to Melissa O'Neill
--   tree folding idea Heinrich Apfelmus / Dave Bayer 
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 via Y combinator

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.