\begin{code}

    {-# OPTIONS_GHC -XNoImplicitPrelude #-}
    {-# OPTIONS_GHC -fno-warn-orphans #-}
    {-# OPTIONS_HADDOCK hide #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  GHC.Num
    -- Copyright   :  (c) The University of Glasgow 1994-2002
    -- License     :  see libraries/base/LICENSE
    --
    -- Maintainer  :  cvs-ghc@haskell.org
    -- Stability   :  internal
    -- Portability :  non-portable (GHC Extensions)
    --
    -- The 'Num' class and the 'Integer' type.
    --
    -----------------------------------------------------------------------------

    #include "MachDeps.h"
    #if SIZEOF_HSWORD == 4
    #define DIGITS       9
    #define BASE         1000000000
    #elif SIZEOF_HSWORD == 8
    #define DIGITS       18
    #define BASE         1000000000000000000
    #else
    #error Please define DIGITS and BASE
    -- DIGITS should be the largest integer such that
    --     10^DIGITS < 2^(SIZEOF_HSWORD * 8 - 1)
    -- BASE should be 10^DIGITS. Note that ^ is not available yet.
    #endif

    -- #hide
    module GHC.Num (module GHC.Num, module GHC.Integer) where

    import GHC.Base
    import GHC.Enum
    import GHC.Show
    import GHC.Integer

    infixl 7  *
    infixl 6  +, -

    default ()              -- Double isn't available yet,
                            -- and we shouldn't be using defaults anyway

\end{code} %********************************************************* %* * \subsection{Standard numeric class} %* * %********************************************************* \begin{code}

    -- | Basic numeric class.
    --
    -- Minimal complete definition: all except 'negate' or @(-)@
    class  (Eq a, Show a) => Num a  where
        (+), (-), (*)       :: a -> a -> a
        -- | Unary negation.
        negate              :: a -> a
        -- | Absolute value.
        abs                 :: a -> a
        -- | Sign of a number.
        -- The functions 'abs' and 'signum' should satisfy the law:
        --
        -- > abs x * signum x == x
        --
        -- For real numbers, the 'signum' is either @-1@ (negative), @0@ (zero)
        -- or @1@ (positive).
        signum              :: a -> a
        -- | Conversion from an 'Integer'.
        -- An integer literal represents the application of the function
        -- 'fromInteger' to the appropriate value of type 'Integer',
        -- so such literals have type @('Num' a) => a@.
        fromInteger         :: Integer -> a

        x - y               = x + negate y
        negate x            = 0 - x

    -- | the same as @'flip' ('-')@.
    --
    -- Because @-@ is treated specially in the Haskell grammar,
    -- @(-@ /e/@)@ is not a section, but an application of prefix negation.
    -- However, @('subtract'@ /exp/@)@ is equivalent to the disallowed section.
    {-# INLINE subtract #-}
    subtract :: (Num a) => a -> a -> a
    subtract x y = y - x

\end{code} %********************************************************* %* * \subsection{Instances for @Int@} %* * %********************************************************* \begin{code}

    instance  Num Int  where
        (+)    = plusInt
        (-)    = minusInt
        negate = negateInt
        (*)    = timesInt
        abs n  = if n `geInt` 0 then n else negateInt n

        signum n | n `ltInt` 0 = negateInt 1
                 | n `eqInt` 0 = 0
                 | otherwise   = 1

        fromInteger i = I# (toInt# i)

    quotRemInt :: Int -> Int -> (Int, Int)
    quotRemInt a@(I# _) b@(I# _) = (a `quotInt` b, a `remInt` b)
        -- OK, so I made it a little stricter.  Shoot me.  (WDP 94/10)

    divModInt ::  Int -> Int -> (Int, Int)
    divModInt x@(I# _) y@(I# _) = (x `divInt` y, x `modInt` y)
        -- Stricter.  Sorry if you don't like it.  (WDP 94/10)

\end{code} %********************************************************* %* * \subsection{The @Integer@ instances for @Eq@, @Ord@} %* * %********************************************************* \begin{code}

    instance  Eq Integer  where
        (==) = eqInteger
        (/=) = neqInteger

    ------------------------------------------------------------------------
    instance Ord Integer where
        (<=) = leInteger
        (>)  = gtInteger
        (<)  = ltInteger
        (>=) = geInteger
        compare = compareInteger

\end{code} %********************************************************* %* * \subsection{The @Integer@ instances for @Show@} %* * %********************************************************* \begin{code}

    instance Show Integer where
        showsPrec p n r
            | p > 6 && n < 0 = '(' : integerToString n (')' : r)
            -- Minor point: testing p first gives better code
            -- in the not-uncommon case where the p argument
            -- is a constant
            | otherwise = integerToString n r
        showList = showList__ (showsPrec 0)

    -- Divide an conquer implementation of string conversion
    integerToString :: Integer -> String -> String
    integerToString n0 cs0
        | n0 < 0    = '-' : integerToString' (- n0) cs0
        | otherwise = integerToString' n0 cs0
        where
        integerToString' :: Integer -> String -> String
        integerToString' n cs
            | n < BASE  = jhead (fromInteger n) cs
            | otherwise = jprinth (jsplitf (BASE*BASE) n) cs

        -- Split n into digits in base p. We first split n into digits
        -- in base p*p and then split each of these digits into two.
        -- Note that the first 'digit' modulo p*p may have a leading zero
        -- in base p that we need to drop - this is what jsplith takes care of.
        -- jsplitb the handles the remaining digits.
        jsplitf :: Integer -> Integer -> [Integer]
        jsplitf p n
            | p > n     = [n]
            | otherwise = jsplith p (jsplitf (p*p) n)

        jsplith :: Integer -> [Integer] -> [Integer]
        jsplith p (n:ns) =
            case n `quotRemInteger` p of
            (# q, r #) ->
                if q > 0 then fromInteger q : fromInteger r : jsplitb p ns
                         else fromInteger r : jsplitb p ns
        jsplith _ [] = error "jsplith: []"

        jsplitb :: Integer -> [Integer] -> [Integer]
        jsplitb _ []     = []
        jsplitb p (n:ns) = case n `quotRemInteger` p of
                           (# q, r #) ->
                               q : r : jsplitb p ns

        -- Convert a number that has been split into digits in base BASE^2
        -- this includes a last splitting step and then conversion of digits
        -- that all fit into a machine word.
        jprinth :: [Integer] -> String -> String
        jprinth (n:ns) cs =
            case n `quotRemInteger` BASE of
            (# q', r' #) ->
                let q = fromInteger q'
                    r = fromInteger r'
                in if q > 0 then jhead q $ jblock r $ jprintb ns cs
                            else jhead r $ jprintb ns cs
        jprinth [] _ = error "jprinth []"

        jprintb :: [Integer] -> String -> String
        jprintb []     cs = cs
        jprintb (n:ns) cs = case n `quotRemInteger` BASE of
                            (# q', r' #) ->
                                let q = fromInteger q'
                                    r = fromInteger r'
                                in jblock q $ jblock r $ jprintb ns cs

        -- Convert an integer that fits into a machine word. Again, we have two
        -- functions, one that drops leading zeros (jhead) and one that doesn't
        -- (jblock)
        jhead :: Int -> String -> String
        jhead n cs
            | n < 10    = case unsafeChr (ord '0' + n) of
                c@(C# _) -> c : cs
            | otherwise = case unsafeChr (ord '0' + r) of
                c@(C# _) -> jhead q (c : cs)
            where
            (q, r) = n `quotRemInt` 10

        jblock = jblock' {- ' -} DIGITS

        jblock' :: Int -> Int -> String -> String
        jblock' d n cs
            | d == 1    = case unsafeChr (ord '0' + n) of
                 c@(C# _) -> c : cs
            | otherwise = case unsafeChr (ord '0' + r) of
                 c@(C# _) -> jblock' (d - 1) q (c : cs)
            where
            (q, r) = n `quotRemInt` 10

\end{code} %********************************************************* %* * \subsection{The @Integer@ instances for @Num@} %* * %********************************************************* \begin{code}

    instance  Num Integer  where
        (+) = plusInteger
        (-) = minusInteger
        (*) = timesInteger
        negate         = negateInteger
        fromInteger x  =  x

        abs = absInteger
        signum = signumInteger

\end{code} %********************************************************* %* * \subsection{The @Integer@ instance for @Enum@} %* * %********************************************************* \begin{code}

    instance  Enum Integer  where
        succ x               = x + 1
        pred x               = x - 1
        toEnum (I# n)        = smallInteger n
        fromEnum n           = I# (toInt# n)

        {-# INLINE enumFrom #-}
        {-# INLINE enumFromThen #-}
        {-# INLINE enumFromTo #-}
        {-# INLINE enumFromThenTo #-}
        enumFrom x             = enumDeltaInteger  x 1
        enumFromThen x y       = enumDeltaInteger  x (y-x)
        enumFromTo x lim       = enumDeltaToInteger x 1     lim
        enumFromThenTo x y lim = enumDeltaToInteger x (y-x) lim

    {-# RULES
    "enumDeltaInteger"      [~1] forall x y.  enumDeltaInteger x y     = build (\c _ -> enumDeltaIntegerFB c x y)
    "efdtInteger"           [~1] forall x y l.enumDeltaToInteger x y l = build (\c n -> enumDeltaToIntegerFB c n x y l)
    "enumDeltaInteger"      [1] enumDeltaIntegerFB   (:)    = enumDeltaInteger
    "enumDeltaToInteger"    [1] enumDeltaToIntegerFB (:) [] = enumDeltaToInteger
     #-}

    enumDeltaIntegerFB :: (Integer -> b -> b) -> Integer -> Integer -> b
    enumDeltaIntegerFB c x d = x `seq` (x `c` enumDeltaIntegerFB c (x+d) d)

    enumDeltaInteger :: Integer -> Integer -> [Integer]
    enumDeltaInteger x d = x `seq` (x : enumDeltaInteger (x+d) d)
    -- strict accumulator, so
    --     head (drop 1000000 [1 .. ]
    -- works

    enumDeltaToIntegerFB :: (Integer -> a -> a) -> a
                         -> Integer -> Integer -> Integer -> a
    enumDeltaToIntegerFB c n x delta lim
      | delta >= 0 = up_fb c n x delta lim
      | otherwise  = dn_fb c n x delta lim

    enumDeltaToInteger :: Integer -> Integer -> Integer -> [Integer]
    enumDeltaToInteger x delta lim
      | delta >= 0 = up_list x delta lim
      | otherwise  = dn_list x delta lim

    up_fb :: (Integer -> a -> a) -> a -> Integer -> Integer -> Integer -> a
    up_fb c n x0 delta lim = go (x0 :: Integer)
                          where
                            go x | x > lim   = n
                                 | otherwise = x `c` go (x+delta)
    dn_fb :: (Integer -> a -> a) -> a -> Integer -> Integer -> Integer -> a
    dn_fb c n x0 delta lim = go (x0 :: Integer)
                          where
                            go x | x < lim   = n
                                 | otherwise = x `c` go (x+delta)

    up_list :: Integer -> Integer -> Integer -> [Integer]
    up_list x0 delta lim = go (x0 :: Integer)
                        where
                            go x | x > lim   = []
                                 | otherwise = x : go (x+delta)
    dn_list :: Integer -> Integer -> Integer -> [Integer]
    dn_list x0 delta lim = go (x0 :: Integer)
                        where
                            go x | x < lim   = []
                                 | otherwise = x : go (x+delta)

\end{code}