Difference between revisions of "User:WillNess"
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− | I am a newbie, interested in Haskell. |
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<haskell> |
<haskell> |
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− | + | -- infinite folding idea due to Richard Bird |
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+ | -- double staged production idea due to Melissa O'Neill |
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− | where |
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+ | -- tree folding idea Dave Bayer / improved tree structure |
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− | g xs = 3 : (gaps 5 $ foldi (\x:xs -> (x:) . union xs) |
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+ | -- Heinrich Apfelmus / simplified formulation Will Ness |
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+ | primes = 2 : _Y ((3:) . gaps 5 |
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− | gaps k s@(x:xs) -- | k<=x = minus [k,k+2..] xs |
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− | + | . foldi (\(x:xs) -> (x:) . union xs) [] |
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− | + | . map (\p-> [p*p, p*p+2*p..])) |
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− | then k : gaps (k+2) s |
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− | else gaps (k+2) xs |
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− | + | _Y g = g (_Y g) -- multistage production |
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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 |
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</haskell> |
</haskell> |
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− | <code>foldi</code> is on [[Fold#Tree-like_folds|Tree-like folds]]. |
+ | <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]]. |
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+ | The constructive definition of primes is the Sieve of Eratosthenes: |
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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>   . . . or,  <math>\textstyle\mathbb{N}_{k} = \{k\} \bigcup \mathbb{N}_{k+1}</math>   :) :) . |
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+ | Trial division sieve is: |
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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> |
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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:
using standard definition
- . . . or, :) :) .
Trial division sieve is:
If you're put off by self-referentiality, just replace or on the right-hand side of equations with , but even ancient Greeks knew better.