Random.hs

    {-# LANGUAGE CPP #-}
    #if __GLASGOW_HASKELL__ >= 701
    {-# LANGUAGE Trustworthy #-}
    #endif

    module Random (

            -- $intro

            -- * Random number generators

              RandomGen(next, split, genRange)

            -- ** Standard random number generators
            , StdGen
            , mkStdGen

            -- ** The global random number generator

            -- $globalrng

            , getStdRandom
            , getStdGen
            , setStdGen
            , newStdGen

            -- * Random values of various types
            , Random ( random,   randomR,
                       randoms,  randomRs,
                       randomIO, randomRIO )

            -- * References
            -- $references

        ) where

    import Prelude

    import Data.Int

    import System.CPUTime   ( getCPUTime )
    import Data.Time        ( getCurrentTime, UTCTime(..) )
    import Data.Ratio       ( numerator, denominator )
    import Data.Char        ( isSpace, chr, ord )
    import System.IO.Unsafe ( unsafePerformIO )
    import Data.IORef
    import Numeric          ( readDec )

    -- The standard nhc98 implementation of Time.ClockTime does not match
    -- the extended one expected in this module, so we lash-up a quick
    -- replacement here.
    getTime :: IO (Integer, Integer)
    getTime = do
      utc <- getCurrentTime
      let daytime = toRational $ utctDayTime utc
      return $ quotRem (numerator daytime) (denominator daytime)

    -- | The class 'RandomGen' provides a common interface to random number
    -- generators.
    --
    -- Minimal complete definition: 'next' and 'split'.

    class RandomGen g where

       -- |The 'next' operation returns an 'Int' that is uniformly distributed
       -- in the range returned by 'genRange' (including both end points),
       -- and a new generator.
       next     :: g -> (Int, g)

       -- |The 'split' operation allows one to obtain two distinct random number
       -- generators. This is very useful in functional programs (for example, when
       -- passing a random number generator down to recursive calls), but very
       -- little work has been done on statistically robust implementations of
       -- 'split' (["Random\#Burton", "Random\#Hellekalek"]
       -- are the only examples we know of).
       split    :: g -> (g, g)

       -- |The 'genRange' operation yields the range of values returned by
       -- the generator.
       --
       -- It is required that:
       --
       -- * If @(a,b) = 'genRange' g@, then @a < b@.
       --
       -- * 'genRange' always returns a pair of defined 'Int's.
       --
       -- The second condition ensures that 'genRange' cannot examine its
       -- argument, and hence the value it returns can be determined only by the
       -- instance of 'RandomGen'.  That in turn allows an implementation to make
       -- a single call to 'genRange' to establish a generator's range, without
       -- being concerned that the generator returned by (say) 'next' might have
       -- a different range to the generator passed to 'next'.
       --
       -- The default definition spans the full range of 'Int'.
       genRange :: g -> (Int,Int)

       -- default method
       genRange _ = (minBound, maxBound)

    {- |
    The 'StdGen' instance of 'RandomGen' has a 'genRange' of at least 30 bits.

    The result of repeatedly using 'next' should be at least as statistically
    robust as the /Minimal Standard Random Number Generator/ described by
    ["Random\#Park", "Random\#Carta"].
    Until more is known about implementations of 'split', all we require is
    that 'split' deliver generators that are (a) not identical and
    (b) independently robust in the sense just given.

    The 'Show' and 'Read' instances of 'StdGen' provide a primitive way to save the
    state of a random number generator.
    It is required that @'read' ('show' g) == g@.

    In addition, 'reads' may be used to map an arbitrary string (not necessarily one
    produced by 'show') onto a value of type 'StdGen'. In general, the 'Read'
    instance of 'StdGen' has the following properties:

    * It guarantees to succeed on any string.

    * It guarantees to consume only a finite portion of the string.

    * Different argument strings are likely to result in different results.

    -}

    data StdGen
     = StdGen Int32 Int32

    instance RandomGen StdGen where
      next  = stdNext
      split = stdSplit
      genRange _ = stdRange

    instance Show StdGen where
      showsPrec p (StdGen s1 s2) =
         showsPrec p s1 .
         showChar ' ' .
         showsPrec p s2

    instance Read StdGen where
      readsPrec _p = \ r ->
         case try_read r of
           r'@[_] -> r'
           _   -> [stdFromString r] -- because it shouldn't ever fail.
        where
          try_read r = do
             (s1, r1) <- readDec (dropWhile isSpace r)
             (s2, r2) <- readDec (dropWhile isSpace r1)
             return (StdGen s1 s2, r2)

    {-
     If we cannot unravel the StdGen from a string, create
     one based on the string given.
    -}
    stdFromString         :: String -> (StdGen, String)
    stdFromString s        = (mkStdGen num, rest)
            where (cs, rest) = splitAt 6 s
                  num        = foldl (\a x -> x + 3 * a) 1 (map ord cs)

    {- |
    The function 'mkStdGen' provides an alternative way of producing an initial
    generator, by mapping an 'Int' into a generator. Again, distinct arguments
    should be likely to produce distinct generators.
    -}
    mkStdGen :: Int -> StdGen -- why not Integer ?
    mkStdGen s = mkStdGen32 $ fromIntegral s

    mkStdGen32 :: Int32 -> StdGen
    mkStdGen32 s
     | s < 0     = mkStdGen32 (-s)
     | otherwise = StdGen (s1+1) (s2+1)
          where
            (q, s1) = s `divMod` 2147483562
            s2      = q `mod` 2147483398

    createStdGen :: Integer -> StdGen
    createStdGen s = mkStdGen32 $ fromIntegral s

    -- FIXME: 1/2/3 below should be ** (vs@30082002) XXX

    {- |
    With a source of random number supply in hand, the 'Random' class allows the
    programmer to extract random values of a variety of types.

    Minimal complete definition: 'randomR' and 'random'.

    -}

    class Random a where
      -- | Takes a range /(lo,hi)/ and a random number generator
      -- /g/, and returns a random value uniformly distributed in the closed
      -- interval /[lo,hi]/, together with a new generator. It is unspecified
      -- what happens if /lo>hi/. For continuous types there is no requirement
      -- that the values /lo/ and /hi/ are ever produced, but they may be,
      -- depending on the implementation and the interval.
      randomR :: RandomGen g => (a,a) -> g -> (a,g)

      -- | The same as 'randomR', but using a default range determined by the type:
      --
      -- * For bounded types (instances of 'Bounded', such as 'Char'),
      --   the range is normally the whole type.
      --
      -- * For fractional types, the range is normally the semi-closed interval
      -- @[0,1)@.
      --
      -- * For 'Integer', the range is (arbitrarily) the range of 'Int'.
      random  :: RandomGen g => g -> (a, g)

      -- | Plural variant of 'randomR', producing an infinite list of
      -- random values instead of returning a new generator.
      randomRs :: RandomGen g => (a,a) -> g -> [a]
      randomRs ival g = x : randomRs ival g' where (x,g') = randomR ival g

      -- | Plural variant of 'random', producing an infinite list of
      -- random values instead of returning a new generator.
      randoms  :: RandomGen g => g -> [a]
      randoms  g      = (\(x,g') -> x : randoms g') (random g)

      -- | A variant of 'randomR' that uses the global random number generator
      -- (see "Random#globalrng").
      randomRIO :: (a,a) -> IO a
      randomRIO range  = getStdRandom (randomR range)

      -- | A variant of 'random' that uses the global random number generator
      -- (see "Random#globalrng").
      randomIO  :: IO a
      randomIO         = getStdRandom random

    instance Random Int where
      randomR (a,b) g = randomIvalInteger (toInteger a, toInteger b) g
      random g        = randomR (minBound,maxBound) g

    instance Random Char where
      randomR (a,b) g =
          case (randomIvalInteger (toInteger (ord a), toInteger (ord b)) g) of
            (x,g') -> (chr x, g')
      random g        = randomR (minBound,maxBound) g

    instance Random Bool where
      randomR (a,b) g =
          case (randomIvalInteger (bool2Int a, bool2Int b) g) of
            (x, g') -> (int2Bool x, g')
           where
             bool2Int :: Bool -> Integer
             bool2Int False = 0
             bool2Int True  = 1

             int2Bool :: Int -> Bool
             int2Bool 0     = False
             int2Bool _     = True

      random g        = randomR (minBound,maxBound) g

    instance Random Integer where
      randomR ival g = randomIvalInteger ival g
      random g       = randomR (toInteger (minBound::Int), toInteger (maxBound::Int)) g

    instance Random Double where
      randomR ival g = randomIvalDouble ival id g
      random g       = randomR (0::Double,1) g

    -- hah, so you thought you were saving cycles by using Float?
    instance Random Float where
      random g        = randomIvalDouble (0::Double,1) realToFrac g
      randomR (a,b) g = randomIvalDouble (realToFrac a, realToFrac b) realToFrac g

    mkStdRNG :: Integer -> IO StdGen
    mkStdRNG o = do
        ct          <- getCPUTime
        (sec, psec) <- getTime
        return (createStdGen (sec * 12345 + psec + ct + o))

    randomIvalInteger :: (RandomGen g, Num a) => (Integer, Integer) -> g -> (a, g)
    randomIvalInteger (l,h) rng
     | l > h     = randomIvalInteger (h,l) rng
     | otherwise = case (f n 1 rng) of (v, rng') -> (fromInteger (l + v `mod` k), rng')
         where
           k = h - l + 1
           b = 2147483561
           n = iLogBase b k

           f 0 acc g = (acc, g)
           f n' acc g =
              let
               (x,g')   = next g
              in
              f (n' - 1) (fromIntegral x + acc * b) g'

    randomIvalDouble :: (RandomGen g, Fractional a) => (Double, Double) -> (Double -> a) -> g -> (a, g)
    randomIvalDouble (l,h) fromDouble rng
      | l > h     = randomIvalDouble (h,l) fromDouble rng
      | otherwise =
           case (randomIvalInteger (toInteger (minBound::Int32), toInteger (maxBound::Int32)) rng) of
             (x, rng') ->
                let
                 scaled_x =
                    fromDouble ((l+h)/2) +
                    fromDouble ((h-l) / realToFrac int32Count) *
                    fromIntegral (x::Int32)
                in
                (scaled_x, rng')

    int32Count :: Integer
    int32Count = toInteger (maxBound::Int32) - toInteger (minBound::Int32) + 1

    iLogBase :: Integer -> Integer -> Integer
    iLogBase b i = if i < b then 1 else 1 + iLogBase b (i `div` b)

    stdRange :: (Int,Int)
    stdRange = (0, 2147483562)

    stdNext :: StdGen -> (Int, StdGen)
    -- Returns values in the range stdRange
    stdNext (StdGen s1 s2) = (fromIntegral z', StdGen s1'' s2'')
            where   z'   = if z < 1 then z + 2147483562 else z
                    z    = s1'' - s2''

                    k    = s1 `quot` 53668
                    s1'  = 40014 * (s1 - k * 53668) - k * 12211
                    s1'' = if s1' < 0 then s1' + 2147483563 else s1'

                    k'   = s2 `quot` 52774
                    s2'  = 40692 * (s2 - k' * 52774) - k' * 3791
                    s2'' = if s2' < 0 then s2' + 2147483399 else s2'

    stdSplit            :: StdGen -> (StdGen, StdGen)
    stdSplit std@(StdGen s1 s2)
                         = (left, right)
                           where
                            -- no statistical foundation for this!
                            left    = StdGen new_s1 t2
                            right   = StdGen t1 new_s2

                            new_s1 | s1 == 2147483562 = 1
                                   | otherwise        = s1 + 1

                            new_s2 | s2 == 1          = 2147483398
                                   | otherwise        = s2 - 1

                            StdGen t1 t2 = snd (next std)

    -- The global random number generator

    {- $globalrng #globalrng#

    There is a single, implicit, global random number generator of type
    'StdGen', held in some global variable maintained by the 'IO' monad. It is
    initialised automatically in some system-dependent fashion, for example, by
    using the time of day, or Linux's kernel random number generator. To get
    deterministic behaviour, use 'setStdGen'.
    -}

    -- |Sets the global random number generator.
    setStdGen :: StdGen -> IO ()
    setStdGen sgen = writeIORef theStdGen sgen

    -- |Gets the global random number generator.
    getStdGen :: IO StdGen
    getStdGen  = readIORef theStdGen

    theStdGen :: IORef StdGen
    theStdGen  = unsafePerformIO $ do
       rng <- mkStdRNG 0
       newIORef rng

    -- |Applies 'split' to the current global random generator,
    -- updates it with one of the results, and returns the other.
    newStdGen :: IO StdGen
    newStdGen = atomicModifyIORef theStdGen split

    {- |Uses the supplied function to get a value from the current global
    random generator, and updates the global generator with the new generator
    returned by the function. For example, @rollDice@ gets a random integer
    between 1 and 6:

    >  rollDice :: IO Int
    >  rollDice = getStdRandom (randomR (1,6))

    -}

    getStdRandom :: (StdGen -> (a,StdGen)) -> IO a
    getStdRandom f = atomicModifyIORef theStdGen (swap . f)
      where swap (v,g) = (g,v)

    {- $references

    1. FW #Burton# Burton and RL Page, /Distributed random number generation/,
    Journal of Functional Programming, 2(2):203-212, April 1992.

    2. SK #Park# Park, and KW Miller, /Random number generators -
    good ones are hard to find/, Comm ACM 31(10), Oct 1988, pp1192-1201.

    3. DG #Carta# Carta, /Two fast implementations of the minimal standard
    random number generator/, Comm ACM, 33(1), Jan 1990, pp87-88.

    4. P #Hellekalek# Hellekalek, /Don\'t trust parallel Monte Carlo/,
    Department of Mathematics, University of Salzburg,
    <http://random.mat.sbg.ac.at/~peter/pads98.ps>, 1998.

    5. Pierre #LEcuyer# L'Ecuyer, /Efficient and portable combined random
    number generators/, Comm ACM, 31(6), Jun 1988, pp742-749.

    The Web site <http://random.mat.sbg.ac.at/> is a great source of information.

    -}