-- | -- Module: Data.Chimera.ContinuousMapping -- Copyright: (c) 2017 Andrew Lelechenko -- Licence: MIT -- Maintainer: Andrew Lelechenko -- -- Helpers for continuous mappings, useful to memoize -- predicates on 'Int' (instead of 'Word' only), and -- predicates over two, three and more arguments. -- -- __ Example__ -- -- An infinite plain board of live and dead cells (common for cellular automatons, -- e. g., ) -- can be represented as a predicate @board@ :: 'Int' -> 'Int' -> 'Bool'. Assume that -- we want to convert it to memoized form. We cannot do it directly, because 'Data.Chimera.Bool.tabulate' -- accepts predicates from 'Word' to 'Bool' only. -- -- The first step is to define: -- -- > board'' :: Int -> Int -> Bool -- > board'' x y = board' (intToWord x) (intToWord y) -- > -- > board' :: Word -> Word -> Bool -- > board' x y = board (wordToInt x) (wordToInt y) -- -- This is better, but @board'@ is a predicate over two arguments, and we need it to be a predicate over one. -- Conversion to Z-curve and back does the trick: -- -- > board'' :: Int -> Int -> Bool -- > board'' x y = board1 \$ toZCurve (intToWord x) (intToWord y) -- > -- > board' :: Word -> Bool -- > board' z = let (x, y) = fromZCurve z in -- > board (wordToInt x) (wordToInt y) -- -- Now we are ready to insert memoizing layer: -- -- > board'' :: Int -> Int -> Bool -- > board'' x y = index board' \$ toZCurve (intToWord x) (intToWord y) -- > -- > board' :: Chimera -- > board' = tabulate \$ -- > \z -> let (x, y) = fromZCurve z in -- > board (wordToInt x) (wordToInt y) {-# OPTIONS_GHC -fno-warn-unused-imports #-} module Data.Chimera.ContinuousMapping ( intToWord , wordToInt , toZCurve , fromZCurve , toZCurve3 , fromZCurve3 ) where import Data.Bits import Data.Word import Unsafe.Coerce word2int :: Word -> Int word2int = unsafeCoerce int2word :: Int -> Word int2word = unsafeCoerce -- | Total map, which satisfies inequality -- abs ('intToWord' x - 'intToWord' y) ≤ 2 abs(x - y). -- -- Note that this is not the case for 'fromIntegral' :: 'Int' -> 'Word', -- because it has a discontinuity between −1 and 0. -- -- > > map intToWord [-5..5] -- > [9,7,5,3,1,0,2,4,6,8,10] intToWord :: Int -> Word intToWord i | i >= 0 = int2word i `shiftL` 1 | otherwise = int2word (-1 - i) `shiftL` 1 + 1 -- | Inverse for 'intToWord'. -- -- > > map wordToInt [0..10] -- > [0,-1,1,-2,2,-3,3,-4,4,-5,5] wordToInt :: Word -> Int wordToInt w | even w = word2int (w `shiftR` 1) | otherwise = negate (word2int (w `shiftR` 1)) - 1 -- | Total map from plain to line, continuous almost everywhere. -- See . -- -- Only lower halfs of bits of arguments are used (32 bits on 64-bit architecture). -- -- > > [ toZCurve x y | x <- [0..3], y <- [0..3] ] -- > [0,2,8,10,1,3,9,11,4,6,12,14,5,7,13,15] toZCurve :: Word -> Word -> Word toZCurve x y = part1by1 y `shiftL` 1 .|. part1by1 x -- | Inverse for 'toZCurve'. -- See . -- -- > > map fromZCurve [0..15] -- > [(0,0),(1,0),(0,1),(1,1),(2,0),(3,0),(2,1),(3,1),(0,2),(1,2),(0,3),(1,3),(2,2),(3,2),(2,3),(3,3)] fromZCurve :: Word -> (Word, Word) fromZCurve z = (compact1by1 z, compact1by1 (z `shiftR` 1)) -- | Total map from space to line, continuous almost everywhere. -- See . -- -- Only lower thirds of bits of arguments are used (21 bits on 64-bit architecture). -- -- > > [ toZCurve3 x y z | x <- [0..3], y <- [0..3], z <- [0..3] ] -- > [0,4,32,36,2,6,34,38,16,20,48,52,18,22,50,54,1,5,33,37,3,7,35,39,17,21,49,53,19,23,51,55, -- > 8,12,40,44,10,14,42,46,24,28,56,60,26,30,58,62,9,13,41,45,11,15,43,47,25,29,57,61,27,31,59,63] toZCurve3 :: Word -> Word -> Word -> Word toZCurve3 x y z = part1by2 z `shiftL` 2 .|. part1by2 y `shiftL` 1 .|. part1by2 x -- | Inverse for 'toZCurve3'. -- See . -- -- > > map fromZCurve3 [0..63] -- > [(0,0,0),(1,0,0),(0,1,0),(1,1,0),(0,0,1),(1,0,1),(0,1,1),(1,1,1),(2,0,0),(3,0,0),(2,1,0),(3,1,0),(2,0,1),(3,0,1),(2,1,1),(3,1,1), -- > (0,2,0),(1,2,0),(0,3,0),(1,3,0),(0,2,1),(1,2,1),(0,3,1),(1,3,1),(2,2,0),(3,2,0),(2,3,0),(3,3,0),(2,2,1),(3,2,1),(2,3,1),(3,3,1), -- > (0,0,2),(1,0,2),(0,1,2),(1,1,2),(0,0,3),(1,0,3),(0,1,3),(1,1,3),(2,0,2),(3,0,2),(2,1,2),(3,1,2),(2,0,3),(3,0,3),(2,1,3),(3,1,3), -- > (0,2,2),(1,2,2),(0,3,2),(1,3,2),(0,2,3),(1,2,3),(0,3,3),(1,3,3),(2,2,2),(3,2,2),(2,3,2),(3,3,2),(2,2,3),(3,2,3),(2,3,3),(3,3,3)] fromZCurve3 :: Word -> (Word, Word, Word) fromZCurve3 z = (compact1by2 z, compact1by2 (z `shiftR` 1), compact1by2 (z `shiftR` 2)) -- Inspired by https://fgiesen.wordpress.com/2009/12/13/decoding-morton-codes/ part1by1 :: Word -> Word part1by1 x = x5 where x0 = x .&. 0x00000000ffffffff x1 = (x0 `xor` (x0 `shiftL` 16)) .&. 0x0000ffff0000ffff x2 = (x1 `xor` (x1 `shiftL` 8)) .&. 0x00ff00ff00ff00ff x3 = (x2 `xor` (x2 `shiftL` 4)) .&. 0x0f0f0f0f0f0f0f0f x4 = (x3 `xor` (x3 `shiftL` 2)) .&. 0x3333333333333333 x5 = (x4 `xor` (x4 `shiftL` 1)) .&. 0x5555555555555555 -- Inspired by https://fgiesen.wordpress.com/2009/12/13/decoding-morton-codes/ part1by2 :: Word -> Word part1by2 x = x5 where x0 = x .&. 0x00000000ffffffff x1 = (x0 `xor` (x0 `shiftL` 32)) .&. 0xffff00000000ffff x2 = (x1 `xor` (x1 `shiftL` 16)) .&. 0x00ff0000ff0000ff x3 = (x2 `xor` (x2 `shiftL` 8)) .&. 0xf00f00f00f00f00f x4 = (x3 `xor` (x3 `shiftL` 4)) .&. 0x30c30c30c30c30c3 x5 = (x4 `xor` (x4 `shiftL` 2)) .&. 0x1249249249249249 -- Inspired by https://fgiesen.wordpress.com/2009/12/13/decoding-morton-codes/ compact1by1 :: Word -> Word compact1by1 x = x5 where x0 = x .&. 0x5555555555555555 x1 = (x0 `xor` (x0 `shiftR` 1)) .&. 0x3333333333333333 x2 = (x1 `xor` (x1 `shiftR` 2)) .&. 0x0f0f0f0f0f0f0f0f x3 = (x2 `xor` (x2 `shiftR` 4)) .&. 0x00ff00ff00ff00ff x4 = (x3 `xor` (x3 `shiftR` 8)) .&. 0x0000ffff0000ffff x5 = (x4 `xor` (x4 `shiftR` 16)) .&. 0x00000000ffffffff -- Inspired by https://fgiesen.wordpress.com/2009/12/13/decoding-morton-codes/ compact1by2 :: Word -> Word compact1by2 x = x5 where x0 = x .&. 0x1249249249249249 x1 = (x0 `xor` (x0 `shiftR` 2)) .&. 0x30c30c30c30c30c3 x2 = (x1 `xor` (x1 `shiftR` 4)) .&. 0xf00f00f00f00f00f x3 = (x2 `xor` (x2 `shiftR` 8)) .&. 0x00ff0000ff0000ff x4 = (x3 `xor` (x3 `shiftR` 16)) .&. 0xffff00000000ffff x5 = (x4 `xor` (x4 `shiftR` 32)) .&. 0x00000000ffffffff