Data/Bits.hs

    {-# LANGUAGE Trustworthy #-}
    {-# LANGUAGE CPP, NoImplicitPrelude, BangPatterns, MagicHash #-}

    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.Bits
    -- Copyright   :  (c) The University of Glasgow 2001
    -- License     :  BSD-style (see the file libraries/base/LICENSE)
    --
    -- Maintainer  :  libraries@haskell.org
    -- Stability   :  experimental
    -- Portability :  portable
    --
    -- This module defines bitwise operations for signed and unsigned
    -- integers.  Instances of the class 'Bits' for the 'Int' and
    -- 'Integer' types are available from this module, and instances for
    -- explicitly sized integral types are available from the
    -- "Data.Int" and "Data.Word" modules.
    --
    -----------------------------------------------------------------------------

    module Data.Bits (
      Bits(
        (.&.), (.|.), xor,
        complement,
        shift,
        rotate,
        zeroBits,
        bit,
        setBit,
        clearBit,
        complementBit,
        testBit,
        bitSizeMaybe,
        bitSize,
        isSigned,
        shiftL, shiftR,
        unsafeShiftL, unsafeShiftR,
        rotateL, rotateR,
        popCount
      ),
      FiniteBits(
        finiteBitSize,
        countLeadingZeros,
        countTrailingZeros
      ),

      bitDefault,
      testBitDefault,
      popCountDefault,
      toIntegralSized
     ) where

    -- Defines the @Bits@ class containing bit-based operations.
    -- See library document for details on the semantics of the
    -- individual operations.

    #include "MachDeps.h"

    #ifdef MIN_VERSION_integer_gmp
    # define HAVE_INTEGER_GMP1 MIN_VERSION_integer_gmp(1,0,0)
    #endif

    import Data.Maybe
    import GHC.Enum
    import GHC.Num
    import GHC.Base
    import GHC.Real

    #if HAVE_INTEGER_GMP1
    import GHC.Integer.GMP.Internals (bitInteger, popCountInteger)
    #endif

    infixl 8 `shift`, `rotate`, `shiftL`, `shiftR`, `rotateL`, `rotateR`
    infixl 7 .&.
    infixl 6 `xor`
    infixl 5 .|.

    {-# DEPRECATED bitSize "Use 'bitSizeMaybe' or 'finiteBitSize' instead" #-} -- deprecated in 7.8

    -- | The 'Bits' class defines bitwise operations over integral types.
    --
    -- * Bits are numbered from 0 with bit 0 being the least
    --   significant bit.
    class Eq a => Bits a where
        {-# MINIMAL (.&.), (.|.), xor, complement,
                    (shift | (shiftL, shiftR)),
                    (rotate | (rotateL, rotateR)),
                    bitSize, bitSizeMaybe, isSigned, testBit, bit, popCount #-}

        -- | Bitwise \"and\"
        (.&.) :: a -> a -> a

        -- | Bitwise \"or\"
        (.|.) :: a -> a -> a

        -- | Bitwise \"xor\"
        xor :: a -> a -> a

        {-| Reverse all the bits in the argument -}
        complement        :: a -> a

        {-| @'shift' x i@ shifts @x@ left by @i@ bits if @i@ is positive,
            or right by @-i@ bits otherwise.
            Right shifts perform sign extension on signed number types;
            i.e. they fill the top bits with 1 if the @x@ is negative
            and with 0 otherwise.

            An instance can define either this unified 'shift' or 'shiftL' and
            'shiftR', depending on which is more convenient for the type in
            question. -}
        shift             :: a -> Int -> a

        x `shift`   i | i<0       = x `shiftR` (-i)
                      | i>0       = x `shiftL` i
                      | otherwise = x

        {-| @'rotate' x i@ rotates @x@ left by @i@ bits if @i@ is positive,
            or right by @-i@ bits otherwise.

            For unbounded types like 'Integer', 'rotate' is equivalent to 'shift'.

            An instance can define either this unified 'rotate' or 'rotateL' and
            'rotateR', depending on which is more convenient for the type in
            question. -}
        rotate            :: a -> Int -> a

        x `rotate`  i | i<0       = x `rotateR` (-i)
                      | i>0       = x `rotateL` i
                      | otherwise = x

        {-
        -- Rotation can be implemented in terms of two shifts, but care is
        -- needed for negative values.  This suggested implementation assumes
        -- 2's-complement arithmetic.  It is commented out because it would
        -- require an extra context (Ord a) on the signature of 'rotate'.
        x `rotate`  i | i<0 && isSigned x && x<0
                             = let left = i+bitSize x in
                               ((x `shift` i) .&. complement ((-1) `shift` left))
                               .|. (x `shift` left)
                      | i<0  = (x `shift` i) .|. (x `shift` (i+bitSize x))
                      | i==0 = x
                      | i>0  = (x `shift` i) .|. (x `shift` (i-bitSize x))
        -}

        -- | 'zeroBits' is the value with all bits unset.
        --
        -- The following laws ought to hold (for all valid bit indices @/n/@):
        --
        --   * @'clearBit' 'zeroBits' /n/ == 'zeroBits'@
        --   * @'setBit'   'zeroBits' /n/ == 'bit' /n/@
        --   * @'testBit'  'zeroBits' /n/ == False@
        --   * @'popCount' 'zeroBits'   == 0@
        --
        -- This method uses @'clearBit' ('bit' 0) 0@ as its default
        -- implementation (which ought to be equivalent to 'zeroBits' for
        -- types which possess a 0th bit).
        --
        -- @since 4.7.0.0
        zeroBits :: a
        zeroBits = clearBit (bit 0) 0

        -- | @bit /i/@ is a value with the @/i/@th bit set and all other bits clear.
        --
        -- Can be implemented using `bitDefault' if @a@ is also an
        -- instance of 'Num'.
        --
        -- See also 'zeroBits'.
        bit               :: Int -> a

        -- | @x \`setBit\` i@ is the same as @x .|. bit i@
        setBit            :: a -> Int -> a

        -- | @x \`clearBit\` i@ is the same as @x .&. complement (bit i)@
        clearBit          :: a -> Int -> a

        -- | @x \`complementBit\` i@ is the same as @x \`xor\` bit i@
        complementBit     :: a -> Int -> a

        -- | Return 'True' if the @n@th bit of the argument is 1
        --
        -- Can be implemented using `testBitDefault' if @a@ is also an
        -- instance of 'Num'.
        testBit           :: a -> Int -> Bool

        {-| Return the number of bits in the type of the argument.  The actual
            value of the argument is ignored.  Returns Nothing
            for types that do not have a fixed bitsize, like 'Integer'.

            @since 4.7.0.0
            -}
        bitSizeMaybe      :: a -> Maybe Int

        {-| Return the number of bits in the type of the argument.  The actual
            value of the argument is ignored.  The function 'bitSize' is
            undefined for types that do not have a fixed bitsize, like 'Integer'.
            -}
        bitSize           :: a -> Int

        {-| Return 'True' if the argument is a signed type.  The actual
            value of the argument is ignored -}
        isSigned          :: a -> Bool

        {-# INLINE setBit #-}
        {-# INLINE clearBit #-}
        {-# INLINE complementBit #-}
        x `setBit` i        = x .|. bit i
        x `clearBit` i      = x .&. complement (bit i)
        x `complementBit` i = x `xor` bit i

        {-| Shift the argument left by the specified number of bits
            (which must be non-negative).

            An instance can define either this and 'shiftR' or the unified
            'shift', depending on which is more convenient for the type in
            question. -}
        shiftL            :: a -> Int -> a
        {-# INLINE shiftL #-}
        x `shiftL`  i = x `shift`  i

        {-| Shift the argument left by the specified number of bits.  The
            result is undefined for negative shift amounts and shift amounts
            greater or equal to the 'bitSize'.

            Defaults to 'shiftL' unless defined explicitly by an instance.

            @since 4.5.0.0 -}
        unsafeShiftL            :: a -> Int -> a
        {-# INLINE unsafeShiftL #-}
        x `unsafeShiftL` i = x `shiftL` i

        {-| Shift the first argument right by the specified number of bits. The
            result is undefined for negative shift amounts and shift amounts
            greater or equal to the 'bitSize'.

            Right shifts perform sign extension on signed number types;
            i.e. they fill the top bits with 1 if the @x@ is negative
            and with 0 otherwise.

            An instance can define either this and 'shiftL' or the unified
            'shift', depending on which is more convenient for the type in
            question. -}
        shiftR            :: a -> Int -> a
        {-# INLINE shiftR #-}
        x `shiftR`  i = x `shift`  (-i)

        {-| Shift the first argument right by the specified number of bits, which
            must be non-negative an smaller than the number of bits in the type.

            Right shifts perform sign extension on signed number types;
            i.e. they fill the top bits with 1 if the @x@ is negative
            and with 0 otherwise.

            Defaults to 'shiftR' unless defined explicitly by an instance.

            @since 4.5.0.0 -}
        unsafeShiftR            :: a -> Int -> a
        {-# INLINE unsafeShiftR #-}
        x `unsafeShiftR` i = x `shiftR` i

        {-| Rotate the argument left by the specified number of bits
            (which must be non-negative).

            An instance can define either this and 'rotateR' or the unified
            'rotate', depending on which is more convenient for the type in
            question. -}
        rotateL           :: a -> Int -> a
        {-# INLINE rotateL #-}
        x `rotateL` i = x `rotate` i

        {-| Rotate the argument right by the specified number of bits
            (which must be non-negative).

            An instance can define either this and 'rotateL' or the unified
            'rotate', depending on which is more convenient for the type in
            question. -}
        rotateR           :: a -> Int -> a
        {-# INLINE rotateR #-}
        x `rotateR` i = x `rotate` (-i)

        {-| Return the number of set bits in the argument.  This number is
            known as the population count or the Hamming weight.

            Can be implemented using `popCountDefault' if @a@ is also an
            instance of 'Num'.

            @since 4.5.0.0 -}
        popCount          :: a -> Int

    -- |The 'FiniteBits' class denotes types with a finite, fixed number of bits.
    --
    -- @since 4.7.0.0
    class Bits b => FiniteBits b where
        -- | Return the number of bits in the type of the argument.
        -- The actual value of the argument is ignored. Moreover, 'finiteBitSize'
        -- is total, in contrast to the deprecated 'bitSize' function it replaces.
        --
        -- @
        -- 'finiteBitSize' = 'bitSize'
        -- 'bitSizeMaybe' = 'Just' . 'finiteBitSize'
        -- @
        --
        -- @since 4.7.0.0
        finiteBitSize :: b -> Int

        -- | Count number of zero bits preceding the most significant set bit.
        --
        -- @
        -- 'countLeadingZeros' ('zeroBits' :: a) = finiteBitSize ('zeroBits' :: a)
        -- @
        --
        -- 'countLeadingZeros' can be used to compute log base 2 via
        --
        -- @
        -- logBase2 x = 'finiteBitSize' x - 1 - 'countLeadingZeros' x
        -- @
        --
        -- Note: The default implementation for this method is intentionally
        -- naive. However, the instances provided for the primitive
        -- integral types are implemented using CPU specific machine
        -- instructions.
        --
        -- @since 4.8.0.0
        countLeadingZeros :: b -> Int
        countLeadingZeros x = (w-1) - go (w-1)
          where
            go i | i < 0       = i -- no bit set
                 | testBit x i = i
                 | otherwise   = go (i-1)

            w = finiteBitSize x

        -- | Count number of zero bits following the least significant set bit.
        --
        -- @
        -- 'countTrailingZeros' ('zeroBits' :: a) = finiteBitSize ('zeroBits' :: a)
        -- 'countTrailingZeros' . 'negate' = 'countTrailingZeros'
        -- @
        --
        -- The related
        -- <http://en.wikipedia.org/wiki/Find_first_set find-first-set operation>
        -- can be expressed in terms of 'countTrailingZeros' as follows
        --
        -- @
        -- findFirstSet x = 1 + 'countTrailingZeros' x
        -- @
        --
        -- Note: The default implementation for this method is intentionally
        -- naive. However, the instances provided for the primitive
        -- integral types are implemented using CPU specific machine
        -- instructions.
        --
        -- @since 4.8.0.0
        countTrailingZeros :: b -> Int
        countTrailingZeros x = go 0
          where
            go i | i >= w      = i
                 | testBit x i = i
                 | otherwise   = go (i+1)

            w = finiteBitSize x

    -- The defaults below are written with lambdas so that e.g.
    --     bit = bitDefault
    -- is fully applied, so inlining will happen

    -- | Default implementation for 'bit'.
    --
    -- Note that: @bitDefault i = 1 `shiftL` i@
    --
    -- @since 4.6.0.0
    bitDefault :: (Bits a, Num a) => Int -> a
    bitDefault = \i -> 1 `shiftL` i
    {-# INLINE bitDefault #-}

    -- | Default implementation for 'testBit'.
    --
    -- Note that: @testBitDefault x i = (x .&. bit i) /= 0@
    --
    -- @since 4.6.0.0
    testBitDefault ::  (Bits a, Num a) => a -> Int -> Bool
    testBitDefault = \x i -> (x .&. bit i) /= 0
    {-# INLINE testBitDefault #-}

    -- | Default implementation for 'popCount'.
    --
    -- This implementation is intentionally naive. Instances are expected to provide
    -- an optimized implementation for their size.
    --
    -- @since 4.6.0.0
    popCountDefault :: (Bits a, Num a) => a -> Int
    popCountDefault = go 0
     where
       go !c 0 = c
       go c w = go (c+1) (w .&. (w - 1)) -- clear the least significant
    {-# INLINABLE popCountDefault #-}

    -- | Interpret 'Bool' as 1-bit bit-field
    --
    --  @since 4.7.0.0
    instance Bits Bool where
        (.&.) = (&&)

        (.|.) = (||)

        xor = (/=)

        complement = not

        shift x 0 = x
        shift _ _ = False

        rotate x _ = x

        bit 0 = True
        bit _ = False

        testBit x 0 = x
        testBit _ _ = False

        bitSizeMaybe _ = Just 1

        bitSize _ = 1

        isSigned _ = False

        popCount False = 0
        popCount True  = 1

    -- | @since 4.7.0.0
    instance FiniteBits Bool where
        finiteBitSize _ = 1
        countTrailingZeros x = if x then 0 else 1
        countLeadingZeros  x = if x then 0 else 1

    -- | @since 2.01
    instance Bits Int where
        {-# INLINE shift #-}
        {-# INLINE bit #-}
        {-# INLINE testBit #-}

        zeroBits = 0

        bit     = bitDefault

        testBit = testBitDefault

        (I# x#) .&.   (I# y#)          = I# (x# `andI#` y#)
        (I# x#) .|.   (I# y#)          = I# (x# `orI#`  y#)
        (I# x#) `xor` (I# y#)          = I# (x# `xorI#` y#)
        complement (I# x#)             = I# (notI# x#)
        (I# x#) `shift` (I# i#)
            | isTrue# (i# >=# 0#)      = I# (x# `iShiftL#` i#)
            | otherwise                = I# (x# `iShiftRA#` negateInt# i#)
        (I# x#) `shiftL` (I# i#)       = I# (x# `iShiftL#` i#)
        (I# x#) `unsafeShiftL` (I# i#) = I# (x# `uncheckedIShiftL#` i#)
        (I# x#) `shiftR` (I# i#)       = I# (x# `iShiftRA#` i#)
        (I# x#) `unsafeShiftR` (I# i#) = I# (x# `uncheckedIShiftRA#` i#)

        {-# INLINE rotate #-}       -- See Note [Constant folding for rotate]
        (I# x#) `rotate` (I# i#) =
            I# ((x# `uncheckedIShiftL#` i'#) `orI#` (x# `uncheckedIShiftRL#` (wsib -# i'#)))
          where
            !i'# = i# `andI#` (wsib -# 1#)
            !wsib = WORD_SIZE_IN_BITS#   {- work around preprocessor problem (??) -}
        bitSizeMaybe i         = Just (finiteBitSize i)
        bitSize i              = finiteBitSize i

        popCount (I# x#) = I# (word2Int# (popCnt# (int2Word# x#)))

        isSigned _             = True

    -- | @since 4.6.0.0
    instance FiniteBits Int where
        finiteBitSize _ = WORD_SIZE_IN_BITS
        countLeadingZeros  (I# x#) = I# (word2Int# (clz# (int2Word# x#)))
        countTrailingZeros (I# x#) = I# (word2Int# (ctz# (int2Word# x#)))

    -- | @since 2.01
    instance Bits Word where
        {-# INLINE shift #-}
        {-# INLINE bit #-}
        {-# INLINE testBit #-}

        (W# x#) .&.   (W# y#)    = W# (x# `and#` y#)
        (W# x#) .|.   (W# y#)    = W# (x# `or#`  y#)
        (W# x#) `xor` (W# y#)    = W# (x# `xor#` y#)
        complement (W# x#)       = W# (x# `xor#` mb#)
            where !(W# mb#) = maxBound
        (W# x#) `shift` (I# i#)
            | isTrue# (i# >=# 0#)      = W# (x# `shiftL#` i#)
            | otherwise                = W# (x# `shiftRL#` negateInt# i#)
        (W# x#) `shiftL` (I# i#)       = W# (x# `shiftL#` i#)
        (W# x#) `unsafeShiftL` (I# i#) = W# (x# `uncheckedShiftL#` i#)
        (W# x#) `shiftR` (I# i#)       = W# (x# `shiftRL#` i#)
        (W# x#) `unsafeShiftR` (I# i#) = W# (x# `uncheckedShiftRL#` i#)
        (W# x#) `rotate` (I# i#)
            | isTrue# (i'# ==# 0#) = W# x#
            | otherwise  = W# ((x# `uncheckedShiftL#` i'#) `or#` (x# `uncheckedShiftRL#` (wsib -# i'#)))
            where
            !i'# = i# `andI#` (wsib -# 1#)
            !wsib = WORD_SIZE_IN_BITS#  {- work around preprocessor problem (??) -}
        bitSizeMaybe i           = Just (finiteBitSize i)
        bitSize i                = finiteBitSize i
        isSigned _               = False
        popCount (W# x#)         = I# (word2Int# (popCnt# x#))
        bit                      = bitDefault
        testBit                  = testBitDefault

    -- | @since 4.6.0.0
    instance FiniteBits Word where
        finiteBitSize _ = WORD_SIZE_IN_BITS
        countLeadingZeros  (W# x#) = I# (word2Int# (clz# x#))
        countTrailingZeros (W# x#) = I# (word2Int# (ctz# x#))

    -- | @since 2.01
    instance Bits Integer where
       (.&.) = andInteger
       (.|.) = orInteger
       xor = xorInteger
       complement = complementInteger
       shift x i@(I# i#) | i >= 0    = shiftLInteger x i#
                         | otherwise = shiftRInteger x (negateInt# i#)
       testBit x (I# i) = testBitInteger x i
       zeroBits   = 0

    #if HAVE_INTEGER_GMP1
       bit (I# i#) = bitInteger i#
       popCount x  = I# (popCountInteger x)
    #else
       bit        = bitDefault
       popCount   = popCountDefault
    #endif

       rotate x i = shift x i   -- since an Integer never wraps around

       bitSizeMaybe _ = Nothing
       bitSize _  = errorWithoutStackTrace "Data.Bits.bitSize(Integer)"
       isSigned _ = True

    -----------------------------------------------------------------------------

    -- | Attempt to convert an 'Integral' type @a@ to an 'Integral' type @b@ using
    -- the size of the types as measured by 'Bits' methods.
    --
    -- A simpler version of this function is:
    --
    -- > toIntegral :: (Integral a, Integral b) => a -> Maybe b
    -- > toIntegral x
    -- >   | toInteger x == y = Just (fromInteger y)
    -- >   | otherwise        = Nothing
    -- >   where
    -- >     y = toInteger x
    --
    -- This version requires going through 'Integer', which can be inefficient.
    -- However, @toIntegralSized@ is optimized to allow GHC to statically determine
    -- the relative type sizes (as measured by 'bitSizeMaybe' and 'isSigned') and
    -- avoid going through 'Integer' for many types. (The implementation uses
    -- 'fromIntegral', which is itself optimized with rules for @base@ types but may
    -- go through 'Integer' for some type pairs.)
    --
    -- @since 4.8.0.0

    toIntegralSized :: (Integral a, Integral b, Bits a, Bits b) => a -> Maybe b
    toIntegralSized x                 -- See Note [toIntegralSized optimization]
      | maybe True (<= x) yMinBound
      , maybe True (x <=) yMaxBound = Just y
      | otherwise                   = Nothing
      where
        y = fromIntegral x

        xWidth = bitSizeMaybe x
        yWidth = bitSizeMaybe y

        yMinBound
          | isBitSubType x y = Nothing
          | isSigned x, not (isSigned y) = Just 0
          | isSigned x, isSigned y
          , Just yW <- yWidth = Just (negate $ bit (yW-1)) -- Assumes sub-type
          | otherwise = Nothing

        yMaxBound
          | isBitSubType x y = Nothing
          | isSigned x, not (isSigned y)
          , Just xW <- xWidth, Just yW <- yWidth
          , xW <= yW+1 = Nothing -- Max bound beyond a's domain
          | Just yW <- yWidth = if isSigned y
                                then Just (bit (yW-1)-1)
                                else Just (bit yW-1)
          | otherwise = Nothing
    {-# INLINABLE toIntegralSized #-}

    -- | 'True' if the size of @a@ is @<=@ the size of @b@, where size is measured
    -- by 'bitSizeMaybe' and 'isSigned'.
    isBitSubType :: (Bits a, Bits b) => a -> b -> Bool
    isBitSubType x y
      -- Reflexive
      | xWidth == yWidth, xSigned == ySigned = True

      -- Every integer is a subset of 'Integer'
      | ySigned, Nothing == yWidth                  = True
      | not xSigned, not ySigned, Nothing == yWidth = True

      -- Sub-type relations between fixed-with types
      | xSigned == ySigned,   Just xW <- xWidth, Just yW <- yWidth = xW <= yW
      | not xSigned, ySigned, Just xW <- xWidth, Just yW <- yWidth = xW <  yW

      | otherwise = False
      where
        xWidth  = bitSizeMaybe x
        xSigned = isSigned     x

        yWidth  = bitSizeMaybe y
        ySigned = isSigned     y
    {-# INLINE isBitSubType #-}

    {-      Note [Constant folding for rotate]
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The INLINE on the Int instance of rotate enables it to be constant
    folded.  For example:
         sumU . mapU (`rotate` 3) . replicateU 10000000 $ (7 :: Int)
    goes to:
       Main.$wfold =
         \ (ww_sO7 :: Int#) (ww1_sOb :: Int#) ->
           case ww1_sOb of wild_XM {
             __DEFAULT -> Main.$wfold (+# ww_sO7 56) (+# wild_XM 1);
             10000000 -> ww_sO7
    whereas before it was left as a call to $wrotate.

    All other Bits instances seem to inline well enough on their
    own to enable constant folding; for example 'shift':
         sumU . mapU (`shift` 3) . replicateU 10000000 $ (7 :: Int)
     goes to:
         Main.$wfold =
           \ (ww_sOb :: Int#) (ww1_sOf :: Int#) ->
             case ww1_sOf of wild_XM {
               __DEFAULT -> Main.$wfold (+# ww_sOb 56) (+# wild_XM 1);
               10000000 -> ww_sOb
             }
    -}

    -- Note [toIntegralSized optimization]
    -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    -- The code in 'toIntegralSized' relies on GHC optimizing away statically
    -- decidable branches.
    --
    -- If both integral types are statically known, GHC will be able optimize the
    -- code significantly (for @-O1@ and better).
    --
    -- For instance (as of GHC 7.8.1) the following definitions:
    --
    -- > w16_to_i32 = toIntegralSized :: Word16 -> Maybe Int32
    -- >
    -- > i16_to_w16 = toIntegralSized :: Int16 -> Maybe Word16
    --
    -- are translated into the following (simplified) /GHC Core/ language:
    --
    -- > w16_to_i32 = \x -> Just (case x of _ { W16# x# -> I32# (word2Int# x#) })
    -- >
    -- > i16_to_w16 = \x -> case eta of _
    -- >   { I16# b1 -> case tagToEnum# (<=# 0 b1) of _
    -- >       { False -> Nothing
    -- >       ; True -> Just (W16# (narrow16Word# (int2Word# b1)))
    -- >       }
    -- >   }