src/GHC/Integer/Type.hs

    {-# LANGUAGE NoImplicitPrelude #-}
    {-# LANGUAGE BangPatterns #-}
    {-# LANGUAGE CPP #-}
    {-# LANGUAGE DeriveDataTypeable #-}
    {-# LANGUAGE GHCForeignImportPrim #-}
    {-# LANGUAGE MagicHash #-}
    {-# LANGUAGE UnboxedTuples #-}
    {-# LANGUAGE UnliftedFFITypes #-}
    {-# LANGUAGE RebindableSyntax #-}
    {-# LANGUAGE NegativeLiterals #-}
    {-# LANGUAGE ExplicitForAll #-}

    -- |
    -- Module      :  GHC.Integer.Type
    -- Copyright   :  (c) Herbert Valerio Riedel 2014
    -- License     :  BSD3
    --
    -- Maintainer  :  ghc-devs@haskell.org
    -- Stability   :  provisional
    -- Portability :  non-portable (GHC Extensions)
    --
    -- GHC needs this module to be named "GHC.Integer.Type" and provide
    -- all the low-level 'Integer' operations.

    module GHC.Integer.Type where

    #include "MachDeps.h"

    -- Sanity check as CPP defines are implicitly 0-valued when undefined
    #if !(defined(SIZEOF_LONG) && defined(SIZEOF_HSWORD) \
               && defined(WORD_SIZE_IN_BITS))
    # error missing defines
    #endif

    import GHC.Classes
    import GHC.Magic
    import GHC.Prim
    import GHC.Types
    #if WORD_SIZE_IN_BITS < 64
    import GHC.IntWord64
    #endif

    default ()

    -- Most high-level operations need to be marked `NOINLINE` as
    -- otherwise GHC doesn't recognize them and fails to apply constant
    -- folding to `Integer`-typed expression.
    --
    -- To this end, the CPP hack below allows to write the pseudo-pragma
    --
    --   {-# CONSTANT_FOLDED plusInteger #-}
    --
    -- which is simply expaned into a
    --
    --   {-# NOINLINE plusInteger #-}
    --
    #define CONSTANT_FOLDED NOINLINE

    ----------------------------------------------------------------------------
    -- type definitions

    -- NB: all code assumes GMP_LIMB_BITS == WORD_SIZE_IN_BITS
    -- The C99 code in cbits/wrappers.c will fail to compile if this doesn't hold

    -- | Type representing a GMP Limb
    type GmpLimb = Word -- actually, 'CULong'
    type GmpLimb# = Word#

    -- | Count of 'GmpLimb's, must be positive (unless specified otherwise).
    type GmpSize = Int  -- actually, a 'CLong'
    type GmpSize# = Int#

    narrowGmpSize# :: Int# -> Int#
    #if SIZEOF_LONG == SIZEOF_HSWORD
    narrowGmpSize# x = x
    #elif (SIZEOF_LONG == 4) && (SIZEOF_HSWORD == 8)
    -- On IL32P64 (i.e. Win64), we have to be careful with CLong not being
    -- 64bit.  This is mostly an issue on values returned from C functions
    -- due to sign-extension.
    narrowGmpSize# = narrow32Int#
    #endif

    type GmpBitCnt = Word -- actually, 'CULong'
    type GmpBitCnt# = Word# -- actually, 'CULong'

    -- Pseudo FFI CType
    type CInt = Int
    type CInt# = Int#

    narrowCInt# :: Int# -> Int#
    narrowCInt# = narrow32Int#

    -- | Bits in a 'GmpLimb'. Same as @WORD_SIZE_IN_BITS@.
    gmpLimbBits :: Word -- 8 `shiftL` gmpLimbShift
    gmpLimbBits = W# WORD_SIZE_IN_BITS##

    #if WORD_SIZE_IN_BITS == 64
    # define GMP_LIMB_SHIFT   3
    # define GMP_LIMB_BYTES   8
    # define GMP_LIMB_BITS    64
    # define INT_MINBOUND     -0x8000000000000000
    # define INT_MAXBOUND      0x7fffffffffffffff
    # define ABS_INT_MINBOUND  0x8000000000000000
    # define SQRT_INT_MAXBOUND 0xb504f333
    #elif WORD_SIZE_IN_BITS == 32
    # define GMP_LIMB_SHIFT   2
    # define GMP_LIMB_BYTES   4
    # define GMP_LIMB_BITS    32
    # define INT_MINBOUND     -0x80000000
    # define INT_MAXBOUND      0x7fffffff
    # define ABS_INT_MINBOUND  0x80000000
    # define SQRT_INT_MAXBOUND 0xb504
    #else
    # error unsupported WORD_SIZE_IN_BITS config
    #endif

    -- | Type representing /raw/ arbitrary-precision Naturals
    --
    -- This is common type used by 'Natural' and 'Integer'.  As this type
    -- consists of a single constructor wrapping a 'ByteArray#' it can be
    -- unpacked.
    --
    -- Essential invariants:
    --
    --  - 'ByteArray#' size is an exact multiple of 'Word#' size
    --  - limbs are stored in least-significant-limb-first order,
    --  - the most-significant limb must be non-zero, except for
    --  - @0@ which is represented as a 1-limb.
    data BigNat = BN# ByteArray#

    instance Eq BigNat where
        (==) = eqBigNat

    instance Ord BigNat where
        compare = compareBigNat

    -- | Invariant: 'Jn#' and 'Jp#' are used iff value doesn't fit in 'S#'
    --
    -- Useful properties resulting from the invariants:
    --
    --  - @abs ('S#' _) <= abs ('Jp#' _)@
    --  - @abs ('S#' _) <  abs ('Jn#' _)@
    --
    data Integer  = S#                !Int#
                    -- ^ iff value in @[minBound::'Int', maxBound::'Int']@ range
                  | Jp# {-# UNPACK #-} !BigNat
                    -- ^ iff value in @]maxBound::'Int', +inf[@ range
                  | Jn# {-# UNPACK #-} !BigNat
                    -- ^ iff value in @]-inf, minBound::'Int'[@ range

    -- TODO: experiment with different constructor-ordering

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

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

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

    -- | Construct 'Integer' value from list of 'Int's.
    --
    -- This function is used by GHC for constructing 'Integer' literals.
    mkInteger :: Bool   -- ^ sign of integer ('True' if non-negative)
              -> [Int]  -- ^ absolute value expressed in 31 bit chunks, least
                        --   significant first (ideally these would be machine-word
                        --   'Word's rather than 31-bit truncated 'Int's)
              -> Integer
    mkInteger nonNegative is
      | nonNegative = f is
      | True        = negateInteger (f is)
      where
        f [] = S# 0#
        f (I# i : is') = smallInteger (i `andI#` 0x7fffffff#) `orInteger`
                             shiftLInteger (f is') 31#
    {-# CONSTANT_FOLDED mkInteger #-}

    -- | Test whether all internal invariants are satisfied by 'Integer' value
    --
    -- Returns @1#@ if valid, @0#@ otherwise.
    --
    -- This operation is mostly useful for test-suites and/or code which
    -- constructs 'Integer' values directly.
    isValidInteger# :: Integer -> Int#
    isValidInteger# (S#  _) = 1#
    isValidInteger# (Jp# bn)
        = isValidBigNat# bn `andI#` (bn `gtBigNatWord#` INT_MAXBOUND##)
    isValidInteger# (Jn# bn)
        = isValidBigNat# bn `andI#` (bn `gtBigNatWord#` ABS_INT_MINBOUND##)

    -- | Should rather be called @intToInteger@
    smallInteger :: Int# -> Integer
    smallInteger i# = S# i#
    {-# CONSTANT_FOLDED smallInteger #-}

    ----------------------------------------------------------------------------
    -- Int64/Word64 specific primitives

    #if WORD_SIZE_IN_BITS < 64
    int64ToInteger :: Int64# -> Integer
    int64ToInteger i
      | isTrue# (i `leInt64#` intToInt64#  0x7FFFFFFF#)
      , isTrue# (i `geInt64#` intToInt64# -0x80000000#)
        = S# (int64ToInt# i)
      | isTrue# (i `geInt64#` intToInt64# 0#)
        = Jp# (word64ToBigNat (int64ToWord64# i))
      | True
        = Jn# (word64ToBigNat (int64ToWord64# (negateInt64# i)))
    {-# CONSTANT_FOLDED int64ToInteger #-}

    word64ToInteger :: Word64# -> Integer
    word64ToInteger w
      | isTrue# (w `leWord64#` wordToWord64# 0x7FFFFFFF##)
        = S# (int64ToInt# (word64ToInt64# w))
      | True
        = Jp# (word64ToBigNat w)
    {-# CONSTANT_FOLDED word64ToInteger #-}

    integerToInt64 :: Integer -> Int64#
    integerToInt64 (S# i#)  = intToInt64# i#
    integerToInt64 (Jp# bn) = word64ToInt64# (bigNatToWord64 bn)
    integerToInt64 (Jn# bn) = negateInt64# (word64ToInt64# (bigNatToWord64 bn))
    {-# CONSTANT_FOLDED integerToInt64 #-}

    integerToWord64 :: Integer -> Word64#
    integerToWord64 (S# i#)  = int64ToWord64# (intToInt64# i#)
    integerToWord64 (Jp# bn) = bigNatToWord64 bn
    integerToWord64 (Jn# bn)
        = int64ToWord64# (negateInt64# (word64ToInt64# (bigNatToWord64 bn)))
    {-# CONSTANT_FOLDED integerToWord64 #-}

    #if GMP_LIMB_BITS == 32
    word64ToBigNat :: Word64# -> BigNat
    word64ToBigNat w64 = wordToBigNat2 wh# wl#
      where
        wh# = word64ToWord# (uncheckedShiftRL64# w64 32#)
        wl# = word64ToWord# w64

    bigNatToWord64 :: BigNat -> Word64#
    bigNatToWord64 bn
      | isTrue# (sizeofBigNat# bn ># 1#)
        = let wh# = wordToWord64# (indexBigNat# bn 1#)
          in uncheckedShiftL64# wh# 32# `or64#` wl#
      | True = wl#
      where
        wl# = wordToWord64# (bigNatToWord bn)
    #endif
    #endif

    -- End of Int64/Word64 specific primitives
    ----------------------------------------------------------------------------

    -- | Truncates 'Integer' to least-significant 'Int#'
    integerToInt :: Integer -> Int#
    integerToInt (S# i#)  = i#
    integerToInt (Jp# bn) = bigNatToInt bn
    integerToInt (Jn# bn) = negateInt# (bigNatToInt bn)
    {-# CONSTANT_FOLDED integerToInt #-}

    hashInteger :: Integer -> Int#
    hashInteger = integerToInt -- emulating what integer-{simple,gmp} already do

    integerToWord :: Integer -> Word#
    integerToWord (S# i#)  = int2Word# i#
    integerToWord (Jp# bn) = bigNatToWord bn
    integerToWord (Jn# bn) = int2Word# (negateInt# (bigNatToInt bn))
    {-# CONSTANT_FOLDED integerToWord #-}

    wordToInteger :: Word# -> Integer
    wordToInteger w#
      | isTrue# (i# >=# 0#) = S# i#
      | True                = Jp# (wordToBigNat w#)
      where
        i# = word2Int# w#
    {-# CONSTANT_FOLDED wordToInteger #-}

    wordToNegInteger :: Word# -> Integer
    wordToNegInteger w#
      | isTrue# (i# <=# 0#) = S# i#
      | True                = Jn# (wordToBigNat w#)
      where
        i# = negateInt# (word2Int# w#)

    -- we could almost auto-derive Ord if it wasn't for the Jn#-Jn# case
    compareInteger :: Integer -> Integer -> Ordering
    compareInteger (Jn# x)  (Jn# y) = compareBigNat y x
    compareInteger (S#  x)  (S#  y) = compareInt#   x y
    compareInteger (Jp# x)  (Jp# y) = compareBigNat x y
    compareInteger (Jn# _)  _       = LT
    compareInteger (S#  _)  (Jp# _) = LT
    compareInteger (S#  _)  (Jn# _) = GT
    compareInteger (Jp# _)  _       = GT
    {-# CONSTANT_FOLDED compareInteger #-}

    isNegInteger# :: Integer -> Int#
    isNegInteger# (S# i#) = i# <# 0#
    isNegInteger# (Jp# _)  = 0#
    isNegInteger# (Jn# _)  = 1#

    -- | Not-equal predicate.
    neqInteger :: Integer -> Integer -> Bool
    neqInteger x y = isTrue# (neqInteger# x y)

    eqInteger, leInteger, ltInteger, gtInteger, geInteger
      :: Integer -> Integer -> Bool
    eqInteger  x y = isTrue# (eqInteger#  x y)
    leInteger  x y = isTrue# (leInteger#  x y)
    ltInteger  x y = isTrue# (ltInteger#  x y)
    gtInteger  x y = isTrue# (gtInteger#  x y)
    geInteger  x y = isTrue# (geInteger#  x y)

    eqInteger#, neqInteger#, leInteger#, ltInteger#, gtInteger#, geInteger#
      :: Integer -> Integer -> Int#
    eqInteger# (S# x#) (S# y#)   = x# ==# y#
    eqInteger# (Jn# x) (Jn# y)   = eqBigNat# x y
    eqInteger# (Jp# x) (Jp# y)   = eqBigNat# x y
    eqInteger# _       _         = 0#
    {-# CONSTANT_FOLDED eqInteger# #-}

    neqInteger# (S# x#) (S# y#)  = x# /=# y#
    neqInteger# (Jn# x) (Jn# y)  = neqBigNat# x y
    neqInteger# (Jp# x) (Jp# y)  = neqBigNat# x y
    neqInteger# _       _        = 1#
    {-# CONSTANT_FOLDED neqInteger# #-}

    gtInteger# (S# x#) (S# y#)   = x# ># y#
    gtInteger# x y | inline compareInteger x y == GT  = 1#
    gtInteger# _ _                                    = 0#
    {-# CONSTANT_FOLDED gtInteger# #-}

    leInteger# (S# x#) (S# y#)   = x# <=# y#
    leInteger# x y | inline compareInteger x y /= GT  = 1#
    leInteger# _ _                             = 0#
    {-# CONSTANT_FOLDED leInteger# #-}

    ltInteger# (S# x#) (S# y#)   = x# <# y#
    ltInteger# x y | inline compareInteger x y == LT  = 1#
    ltInteger# _ _                             = 0#
    {-# CONSTANT_FOLDED ltInteger# #-}

    geInteger# (S# x#) (S# y#)   = x# >=# y#
    geInteger# x y | inline compareInteger x y /= LT  = 1#
    geInteger# _ _                             = 0#
    {-# CONSTANT_FOLDED geInteger# #-}

    -- | Compute absolute value of an 'Integer'
    absInteger :: Integer -> Integer
    absInteger (Jn# n)                       = Jp# n
    absInteger (S# INT_MINBOUND#)            = Jp# (wordToBigNat ABS_INT_MINBOUND##)
    absInteger (S# i#) | isTrue# (i# <# 0#)  = S# (negateInt# i#)
    absInteger i@(S# _)                      = i
    absInteger i@(Jp# _)                     = i
    {-# CONSTANT_FOLDED absInteger #-}

    -- | Return @-1@, @0@, and @1@ depending on whether argument is
    -- negative, zero, or positive, respectively
    signumInteger :: Integer -> Integer
    signumInteger j = S# (signumInteger# j)
    {-# CONSTANT_FOLDED signumInteger #-}

    -- | Return @-1#@, @0#@, and @1#@ depending on whether argument is
    -- negative, zero, or positive, respectively
    signumInteger# :: Integer -> Int#
    signumInteger# (Jn# _)  = -1#
    signumInteger# (S# i#) = sgnI# i#
    signumInteger# (Jp# _ ) =  1#

    -- | Negate 'Integer'
    negateInteger :: Integer -> Integer
    negateInteger (Jn# n)      = Jp# n
    negateInteger (S# INT_MINBOUND#) = Jp# (wordToBigNat ABS_INT_MINBOUND##)
    negateInteger (S# i#)             = S# (negateInt# i#)
    negateInteger (Jp# bn)
      | isTrue# (eqBigNatWord# bn ABS_INT_MINBOUND##) = S# INT_MINBOUND#
      | True                                        = Jn# bn
    {-# CONSTANT_FOLDED negateInteger #-}

    -- one edge-case issue to take into account is that Int's range is not
    -- symmetric around 0.  I.e. @minBound+maxBound = -1@
    --
    -- Jp# is used iff n > maxBound::Int
    -- Jn# is used iff n < minBound::Int

    -- | Add two 'Integer's
    plusInteger :: Integer -> Integer -> Integer
    plusInteger x    (S# 0#)  = x
    plusInteger (S# 0#) y     = y
    plusInteger (S# x#) (S# y#)
      = case addIntC# x# y# of
        (# z#, 0# #) -> S# z#
        (# 0#, _  #) -> Jn# (wordToBigNat2 1## 0##) -- 2*minBound::Int
        (# z#, _  #)
          | isTrue# (z# ># 0#) -> Jn# (wordToBigNat ( (int2Word# (negateInt# z#))))
          | True               -> Jp# (wordToBigNat ( (int2Word# z#)))
    plusInteger y@(S# _) x = plusInteger x y
    -- no S# as first arg from here on
    plusInteger (Jp# x) (Jp# y) = Jp# (plusBigNat x y)
    plusInteger (Jn# x) (Jn# y) = Jn# (plusBigNat x y)
    plusInteger (Jp# x) (S# y#) -- edge-case: @(maxBound+1) + minBound == 0@
      | isTrue# (y# >=# 0#) = Jp# (plusBigNatWord x (int2Word# y#))
      | True                = bigNatToInteger (minusBigNatWord x (int2Word#
                                                                  (negateInt# y#)))
    plusInteger (Jn# x) (S# y#) -- edge-case: @(minBound-1) + maxBound == -2@
      | isTrue# (y# >=# 0#) = bigNatToNegInteger (minusBigNatWord x (int2Word# y#))
      | True                = Jn# (plusBigNatWord x (int2Word# (negateInt# y#)))
    plusInteger y@(Jn# _) x@(Jp# _) = plusInteger x y
    plusInteger (Jp# x) (Jn# y)
        = case compareBigNat x y of
          LT -> bigNatToNegInteger (minusBigNat y x)
          EQ -> S# 0#
          GT -> bigNatToInteger (minusBigNat x y)
    {-# CONSTANT_FOLDED plusInteger #-}

    -- | Subtract one 'Integer' from another.
    minusInteger :: Integer -> Integer -> Integer
    minusInteger x       (S# 0#)            = x
    minusInteger (S# x#) (S# y#)
      = case subIntC# x# y# of
        (# z#, 0# #) -> S# z#
        (# 0#, _  #) -> Jn# (wordToBigNat2 1## 0##)
        (# z#, _  #)
          | isTrue# (z# ># 0#) -> Jn# (wordToBigNat ( (int2Word# (negateInt# z#))))
          | True               -> Jp# (wordToBigNat ( (int2Word# z#)))
    minusInteger (S# x#) (Jp# y)
      | isTrue# (x# >=# 0#) = bigNatToNegInteger (minusBigNatWord y (int2Word# x#))
      | True                = Jn# (plusBigNatWord y (int2Word# (negateInt# x#)))
    minusInteger (S# x#) (Jn# y)
      | isTrue# (x# >=# 0#) = Jp# (plusBigNatWord y (int2Word# x#))
      | True                = bigNatToInteger (minusBigNatWord y (int2Word#
                                                                  (negateInt# x#)))
    minusInteger (Jp# x) (Jp# y)
        = case compareBigNat x y of
          LT -> bigNatToNegInteger (minusBigNat y x)
          EQ -> S# 0#
          GT -> bigNatToInteger (minusBigNat x y)
    minusInteger (Jp# x) (Jn# y) = Jp# (plusBigNat x y)
    minusInteger (Jn# x) (Jp# y) = Jn# (plusBigNat x y)
    minusInteger (Jn# x) (Jn# y)
        = case compareBigNat x y of
          LT -> bigNatToInteger (minusBigNat y x)
          EQ -> S# 0#
          GT -> bigNatToNegInteger (minusBigNat x y)
    minusInteger (Jp# x) (S# y#)
      | isTrue# (y# >=# 0#) = bigNatToInteger (minusBigNatWord x (int2Word# y#))
      | True                = Jp# (plusBigNatWord x (int2Word# (negateInt# y#)))
    minusInteger (Jn# x) (S# y#)
      | isTrue# (y# >=# 0#) = Jn# (plusBigNatWord x (int2Word# y#))
      | True                = bigNatToNegInteger (minusBigNatWord x
                                                  (int2Word# (negateInt# y#)))
    {-# CONSTANT_FOLDED minusInteger #-}

    -- | Multiply two 'Integer's
    timesInteger :: Integer -> Integer -> Integer
    timesInteger !_      (S# 0#) = S# 0#
    timesInteger (S# 0#) _       = S# 0#
    timesInteger x       (S# 1#) = x
    timesInteger (S# 1#) y       = y
    timesInteger x      (S# -1#) = negateInteger x
    timesInteger (S# -1#) y      = negateInteger y
    timesInteger (S# x#) (S# y#)
      = case mulIntMayOflo# x# y# of
        0# -> S# (x# *# y#)
        _  -> timesInt2Integer x# y#
    timesInteger x@(S# _) y      = timesInteger y x
    -- no S# as first arg from here on
    timesInteger (Jp# x) (Jp# y) = Jp# (timesBigNat x y)
    timesInteger (Jp# x) (Jn# y) = Jn# (timesBigNat x y)
    timesInteger (Jp# x) (S# y#)
      | isTrue# (y# >=# 0#) = Jp# (timesBigNatWord x (int2Word# y#))
      | True       = Jn# (timesBigNatWord x (int2Word# (negateInt# y#)))
    timesInteger (Jn# x) (Jn# y) = Jp# (timesBigNat x y)
    timesInteger (Jn# x) (Jp# y) = Jn# (timesBigNat x y)
    timesInteger (Jn# x) (S# y#)
      | isTrue# (y# >=# 0#) = Jn# (timesBigNatWord x (int2Word# y#))
      | True       = Jp# (timesBigNatWord x (int2Word# (negateInt# y#)))
    {-# CONSTANT_FOLDED timesInteger #-}

    -- | Square 'Integer'
    sqrInteger :: Integer -> Integer
    sqrInteger (S# INT_MINBOUND#) = timesInt2Integer INT_MINBOUND# INT_MINBOUND#
    sqrInteger (S# j#) | isTrue# (absI# j# <=# SQRT_INT_MAXBOUND#) = S# (j# *# j#)
    sqrInteger (S# j#) = timesInt2Integer j# j#
    sqrInteger (Jp# bn) = Jp# (sqrBigNat bn)
    sqrInteger (Jn# bn) = Jp# (sqrBigNat bn)

    -- | Construct 'Integer' from the product of two 'Int#'s
    timesInt2Integer :: Int# -> Int# -> Integer
    timesInt2Integer x# y# = case (# isTrue# (x# >=# 0#), isTrue# (y# >=# 0#) #) of
        (# False, False #) -> case timesWord2# (int2Word# (negateInt# x#))
                                         (int2Word# (negateInt# y#)) of
            (# 0##,l #) -> inline wordToInteger l
            (# h  ,l #) -> Jp# (wordToBigNat2 h l)

        (#  True, False #) -> case timesWord2# (int2Word# x#)
                                         (int2Word# (negateInt# y#)) of
            (# 0##,l #) -> wordToNegInteger l
            (# h  ,l #) -> Jn# (wordToBigNat2 h l)

        (# False,  True #) -> case timesWord2# (int2Word# (negateInt# x#))
                                         (int2Word# y#) of
            (# 0##,l #) -> wordToNegInteger l
            (# h  ,l #) -> Jn# (wordToBigNat2 h l)

        (#  True,  True #) -> case timesWord2# (int2Word# x#)
                                         (int2Word# y#) of
            (# 0##,l #) -> inline wordToInteger l
            (# h  ,l #) -> Jp# (wordToBigNat2 h l)

    bigNatToInteger :: BigNat -> Integer
    bigNatToInteger bn
      | isTrue# ((sizeofBigNat# bn ==# 1#) `andI#` (i# >=# 0#)) = S# i#
      | True                                                    = Jp# bn
      where
        i# = word2Int# (bigNatToWord bn)

    bigNatToNegInteger :: BigNat -> Integer
    bigNatToNegInteger bn
      | isTrue# ((sizeofBigNat# bn ==# 1#) `andI#` (i# <=# 0#)) = S# i#
      | True                                                    = Jn# bn
      where
        i# = negateInt# (word2Int# (bigNatToWord bn))

    -- | Count number of set bits. For negative arguments returns negative
    -- population count of negated argument.
    popCountInteger :: Integer -> Int#
    popCountInteger (S# i#)
      | isTrue# (i# >=# 0#) = popCntI# i#
      | True                = negateInt# (popCntI# (negateInt# i#))
    popCountInteger (Jp# bn)  = popCountBigNat bn
    popCountInteger (Jn# bn)  = negateInt# (popCountBigNat bn)
    {-# CONSTANT_FOLDED popCountInteger #-}

    -- | 'Integer' for which only /n/-th bit is set. Undefined behaviour
    -- for negative /n/ values.
    bitInteger :: Int# -> Integer
    bitInteger i#
      | isTrue# (i# <# (GMP_LIMB_BITS# -# 1#)) = S# (uncheckedIShiftL# 1# i#)
      | True = Jp# (bitBigNat i#)
    {-# CONSTANT_FOLDED bitInteger #-}

    -- | Test if /n/-th bit is set.
    testBitInteger :: Integer -> Int# -> Bool
    testBitInteger !_  n# | isTrue# (n# <# 0#) = False
    testBitInteger (S# i#) n#
      | isTrue# (n# <# GMP_LIMB_BITS#) = isTrue# (((uncheckedIShiftL# 1# n#)
                                                   `andI#` i#) /=# 0#)
      | True                          = isTrue# (i# <# 0#)
    testBitInteger (Jp# bn) n = testBitBigNat bn n
    testBitInteger (Jn# bn) n = testBitNegBigNat bn n
    {-# CONSTANT_FOLDED testBitInteger #-}

    -- | Bitwise @NOT@ operation
    complementInteger :: Integer -> Integer
    complementInteger (S# i#) = S# (notI# i#)
    complementInteger (Jp# bn) = Jn# (plusBigNatWord  bn 1##)
    complementInteger (Jn# bn) = Jp# (minusBigNatWord bn 1##)
    {-# CONSTANT_FOLDED complementInteger #-}

    -- | Arithmetic shift-right operation
    --
    -- Even though the shift-amount is expressed as `Int#`, the result is
    -- undefined for negative shift-amounts.
    shiftRInteger :: Integer -> Int# -> Integer
    shiftRInteger x        0# = x
    shiftRInteger (S# i#)  n# = S# (iShiftRA# i# n#)
      where
        iShiftRA# a b
          | isTrue# (b >=# WORD_SIZE_IN_BITS#) = (a <# 0#) *# (-1#)
          | True                               = a `uncheckedIShiftRA#` b
    shiftRInteger (Jp# bn) n# = bigNatToInteger (shiftRBigNat bn n#)
    shiftRInteger (Jn# bn) n#
        = case bigNatToNegInteger (shiftRNegBigNat bn n#) of
            S# 0# -> S# -1#
            r           -> r
    {-# CONSTANT_FOLDED shiftRInteger #-}

    -- | Shift-left operation
    --
    -- Even though the shift-amount is expressed as `Int#`, the result is
    -- undefined for negative shift-amounts.
    shiftLInteger :: Integer -> Int# -> Integer
    shiftLInteger x       0#  = x
    shiftLInteger (S# 0#) _  = S# 0#
    shiftLInteger (S# 1#) n# = bitInteger n#
    shiftLInteger (S# i#) n#
      | isTrue# (i# >=# 0#)   = bigNatToInteger (shiftLBigNat
                                                 (wordToBigNat (int2Word# i#)) n#)
      | True               = bigNatToNegInteger (shiftLBigNat
                                                 (wordToBigNat (int2Word#
                                                             (negateInt# i#))) n#)
    shiftLInteger (Jp# bn) n# = Jp# (shiftLBigNat bn n#)
    shiftLInteger (Jn# bn) n# = Jn# (shiftLBigNat bn n#)
    {-# CONSTANT_FOLDED shiftLInteger #-}

    -- | Bitwise OR operation
    orInteger :: Integer -> Integer -> Integer
    -- short-cuts
    orInteger  (S# 0#)     y         = y
    orInteger  x           (S# 0#)   = x
    orInteger  (S# -1#)    _         = S# -1#
    orInteger  _           (S# -1#)  = S# -1#
    -- base-cases
    orInteger  (S# x#)     (S# y#)   = S# (orI# x# y#)
    orInteger  (Jp# x)     (Jp# y)   = Jp# (orBigNat x y)
    orInteger  (Jn# x)     (Jn# y)
        = bigNatToNegInteger (plusBigNatWord (andBigNat
                                              (minusBigNatWord x 1##)
                                              (minusBigNatWord y 1##)) 1##)
    orInteger  x@(Jn# _)   y@(Jp# _)  = orInteger y x -- retry with swapped args
    orInteger  (Jp# x)     (Jn# y)
        = bigNatToNegInteger (plusBigNatWord (andnBigNat (minusBigNatWord y 1##) x)
                                             1##)
    -- TODO/FIXpromotion-hack
    orInteger  x@(S# _)   y          = orInteger (unsafePromote x) y
    orInteger  x           y {- S# -}= orInteger x (unsafePromote y)
    {-# CONSTANT_FOLDED orInteger #-}

    -- | Bitwise XOR operation
    xorInteger :: Integer -> Integer -> Integer
    -- short-cuts
    xorInteger (S# 0#)     y          = y
    xorInteger x           (S# 0#)    = x
    -- TODO: (S# -1) cases
    -- base-cases
    xorInteger (S# x#)     (S# y#)    = S# (xorI# x# y#)
    xorInteger (Jp# x)     (Jp# y)    = bigNatToInteger (xorBigNat x y)
    xorInteger (Jn# x)     (Jn# y)
        = bigNatToInteger (xorBigNat (minusBigNatWord x 1##)
                                     (minusBigNatWord y 1##))
    xorInteger x@(Jn# _)   y@(Jp# _)  = xorInteger y x -- retry with swapped args
    xorInteger (Jp# x)     (Jn# y)
        = bigNatToNegInteger (plusBigNatWord (xorBigNat x (minusBigNatWord y 1##))
                                             1##)
    -- TODO/FIXME promotion-hack
    xorInteger x@(S# _)    y          = xorInteger (unsafePromote x) y
    xorInteger x           y {- S# -} = xorInteger x (unsafePromote y)
    {-# CONSTANT_FOLDED xorInteger #-}

    -- | Bitwise AND operation
    andInteger :: Integer -> Integer -> Integer
    -- short-cuts
    andInteger (S# 0#)     !_        = S# 0#
    andInteger _           (S# 0#)   = S# 0#
    andInteger (S# -1#)   y          = y
    andInteger x           (S# -1#)  = x
    -- base-cases
    andInteger (S# x#)     (S# y#)   = S# (andI# x# y#)
    andInteger (Jp# x)     (Jp# y)   = bigNatToInteger (andBigNat x y)
    andInteger (Jn# x)     (Jn# y)
        = bigNatToNegInteger (plusBigNatWord (orBigNat (minusBigNatWord x 1##)
                                                       (minusBigNatWord y 1##)) 1##)
    andInteger x@(Jn# _)   y@(Jp# _)  = andInteger y x
    andInteger (Jp# x)     (Jn# y)
        = bigNatToInteger (andnBigNat x (minusBigNatWord y 1##))
    -- TODO/FIXME promotion-hack
    andInteger x@(S# _)   y          = andInteger (unsafePromote x) y
    andInteger x           y {- S# -}= andInteger x (unsafePromote y)
    {-# CONSTANT_FOLDED andInteger #-}

    -- HACK warning! breaks invariant on purpose
    unsafePromote :: Integer -> Integer
    unsafePromote (S# x#)
        | isTrue# (x# >=# 0#) = Jp# (wordToBigNat (int2Word# x#))
        | True                = Jn# (wordToBigNat (int2Word# (negateInt# x#)))
    unsafePromote x = x

    -- | Simultaneous 'quotInteger' and 'remInteger'.
    --
    -- Divisor must be non-zero otherwise the GHC runtime will terminate
    -- with a division-by-zero fault.
    quotRemInteger :: Integer -> Integer -> (# Integer, Integer #)
    quotRemInteger n       (S# 1#) = (# n, S# 0# #)
    quotRemInteger n      (S# -1#) = let !q = negateInteger n in (# q, (S# 0#) #)
    quotRemInteger !_      (S# 0#) = (# S# (quotInt# 0# 0#),S# (remInt# 0# 0#) #)
    quotRemInteger (S# 0#) _       = (# S# 0#, S# 0# #)
    quotRemInteger (S# n#) (S# d#) = case quotRemInt# n# d# of
        (# q#, r# #) -> (# S# q#, S# r# #)
    quotRemInteger (Jp# n)  (Jp# d)  = case quotRemBigNat n d of
        (# q, r #) -> (# bigNatToInteger q, bigNatToInteger r #)
    quotRemInteger (Jp# n)  (Jn# d)  = case quotRemBigNat n d of
        (# q, r #) -> (# bigNatToNegInteger q, bigNatToInteger r #)
    quotRemInteger (Jn# n)  (Jn# d)  = case quotRemBigNat n d of
        (# q, r #) -> (# bigNatToInteger q, bigNatToNegInteger r #)
    quotRemInteger (Jn# n)  (Jp# d)  = case quotRemBigNat n d of
        (# q, r #) -> (# bigNatToNegInteger q, bigNatToNegInteger r #)
    quotRemInteger (Jp# n)  (S# d#)
      | isTrue# (d# >=# 0#) = case quotRemBigNatWord n (int2Word# d#) of
          (# q, r# #) -> (# bigNatToInteger q, inline wordToInteger r# #)
      | True               = case quotRemBigNatWord n (int2Word# (negateInt# d#)) of
          (# q, r# #) -> (# bigNatToNegInteger q, inline wordToInteger r# #)
    quotRemInteger (Jn# n)  (S# d#)
      | isTrue# (d# >=# 0#) = case quotRemBigNatWord n (int2Word# d#) of
          (# q, r# #) -> (# bigNatToNegInteger q, wordToNegInteger r# #)
      | True               = case quotRemBigNatWord n (int2Word# (negateInt# d#)) of
          (# q, r# #) -> (# bigNatToInteger q, wordToNegInteger r# #)
    quotRemInteger n@(S# _) (Jn# _) = (# S# 0#, n #) -- since @n < d@
    quotRemInteger n@(S# n#) (Jp# d) -- need to account for (S# minBound)
        | isTrue# (n# ># 0#)                                    = (# S# 0#, n #)
        | isTrue# (gtBigNatWord# d (int2Word# (negateInt# n#))) = (# S# 0#, n #)
        | True {- abs(n) == d -}                          = (# S# -1#, S# 0# #)
    {-# CONSTANT_FOLDED quotRemInteger #-}

    quotInteger :: Integer -> Integer -> Integer
    quotInteger n       (S# 1#) = n
    quotInteger n      (S# -1#) = negateInteger n
    quotInteger !_      (S# 0#) = S# (quotInt# 0# 0#)
    quotInteger (S# 0#) _       = S# 0#
    quotInteger (S# n#)  (S# d#) = S# (quotInt# n# d#)
    quotInteger (Jp# n)   (S# d#)
      | isTrue# (d# >=# 0#) = bigNatToInteger    (quotBigNatWord n (int2Word# d#))
      | True                = bigNatToNegInteger (quotBigNatWord n
                                                  (int2Word# (negateInt# d#)))
    quotInteger (Jn# n)   (S# d#)
      | isTrue# (d# >=# 0#) = bigNatToNegInteger (quotBigNatWord n (int2Word# d#))
      | True                = bigNatToInteger    (quotBigNatWord n
                                                  (int2Word# (negateInt# d#)))
    quotInteger (Jp# n) (Jp# d) = bigNatToInteger    (quotBigNat n d)
    quotInteger (Jp# n) (Jn# d) = bigNatToNegInteger (quotBigNat n d)
    quotInteger (Jn# n) (Jp# d) = bigNatToNegInteger (quotBigNat n d)
    quotInteger (Jn# n) (Jn# d) = bigNatToInteger    (quotBigNat n d)
    -- handle remaining non-allocating cases
    quotInteger n d = case inline quotRemInteger n d of (# q, _ #) -> q
    {-# CONSTANT_FOLDED quotInteger #-}

    remInteger :: Integer -> Integer -> Integer
    remInteger !_       (S# 1#) = S# 0#
    remInteger _       (S# -1#) = S# 0#
    remInteger _        (S# 0#) = S# (remInt# 0# 0#)
    remInteger (S# 0#) _        = S# 0#
    remInteger (S# n#) (S# d#) = S# (remInt# n# d#)
    remInteger (Jp# n)  (S# d#)
        = wordToInteger    (remBigNatWord n (int2Word# (absI# d#)))
    remInteger (Jn# n)  (S# d#)
        = wordToNegInteger (remBigNatWord n (int2Word# (absI# d#)))
    remInteger (Jp# n)  (Jp# d)  = bigNatToInteger    (remBigNat n d)
    remInteger (Jp# n)  (Jn# d)  = bigNatToInteger    (remBigNat n d)
    remInteger (Jn# n)  (Jp# d)  = bigNatToNegInteger (remBigNat n d)
    remInteger (Jn# n)  (Jn# d)  = bigNatToNegInteger (remBigNat n d)
    -- handle remaining non-allocating cases
    remInteger n d = case inline quotRemInteger n d of (# _, r #) -> r
    {-# CONSTANT_FOLDED remInteger #-}

    -- | Simultaneous 'divInteger' and 'modInteger'.
    --
    -- Divisor must be non-zero otherwise the GHC runtime will terminate
    -- with a division-by-zero fault.
    divModInteger :: Integer -> Integer -> (# Integer, Integer #)
    divModInteger n d
      | isTrue# (signumInteger# r ==# negateInt# (signumInteger# d))
         = let !q' = plusInteger q (S# -1#) -- TODO: optimize
               !r' = plusInteger r d
           in (# q', r' #)
      | True = qr
      where
        !qr@(# q, r #) = quotRemInteger n d
    {-# CONSTANT_FOLDED divModInteger #-}

    divInteger :: Integer -> Integer -> Integer
    -- same-sign ops can be handled by more efficient 'quotInteger'
    divInteger n d | isTrue# (isNegInteger# n ==# isNegInteger# d) = quotInteger n d
    divInteger n d = case inline divModInteger n d of (# q, _ #) -> q
    {-# CONSTANT_FOLDED divInteger #-}

    modInteger :: Integer -> Integer -> Integer
    -- same-sign ops can be handled by more efficient 'remInteger'
    modInteger n d | isTrue# (isNegInteger# n ==# isNegInteger# d) = remInteger n d
    modInteger n d = case inline divModInteger n d of (# _, r #) -> r
    {-# CONSTANT_FOLDED modInteger #-}

    -- | Compute greatest common divisor.
    gcdInteger :: Integer -> Integer -> Integer
    gcdInteger (S# 0#)        b = absInteger b
    gcdInteger a        (S# 0#) = absInteger a
    gcdInteger (S# 1#)        _ = S# 1#
    gcdInteger (S# -1#)       _ = S# 1#
    gcdInteger _        (S# 1#) = S# 1#
    gcdInteger _       (S# -1#) = S# 1#
    gcdInteger (S# a#) (S# b#)
        = wordToInteger (gcdWord# (int2Word# (absI# a#)) (int2Word# (absI# b#)))
    gcdInteger a@(S# _) b = gcdInteger b a
    gcdInteger (Jn# a) b = gcdInteger (Jp# a) b
    gcdInteger (Jp# a) (Jp# b) = bigNatToInteger (gcdBigNat a b)
    gcdInteger (Jp# a) (Jn# b) = bigNatToInteger (gcdBigNat a b)
    gcdInteger (Jp# a) (S# b#)
        = wordToInteger (gcdBigNatWord a (int2Word# (absI# b#)))
    {-# CONSTANT_FOLDED gcdInteger #-}

    -- | Compute least common multiple.
    lcmInteger :: Integer -> Integer -> Integer
    lcmInteger (S# 0#) !_  = S# 0#
    lcmInteger (S# 1#)  b  = absInteger b
    lcmInteger (S# -1#) b  = absInteger b
    lcmInteger _ (S# 0#)   = S# 0#
    lcmInteger a (S# 1#)   = absInteger a
    lcmInteger a (S# -1#)  = absInteger a
    lcmInteger a b = (aa `quotInteger` (aa `gcdInteger` ab)) `timesInteger` ab
      where
        aa = absInteger a
        ab = absInteger b
    {-# CONSTANT_FOLDED lcmInteger #-}

    -- | Compute greatest common divisor.
    --
    -- __Warning__: result may become negative if (at least) one argument
    -- is 'minBound'
    gcdInt :: Int# -> Int# -> Int#
    gcdInt x# y#
        = word2Int# (gcdWord# (int2Word# (absI# x#)) (int2Word# (absI# y#)))

    -- | Compute greatest common divisor.
    --
    -- @since 1.0.0.0
    gcdWord :: Word# -> Word# -> Word#
    gcdWord = gcdWord#

    ----------------------------------------------------------------------------
    -- BigNat operations

    compareBigNat :: BigNat -> BigNat -> Ordering
    compareBigNat x@(BN# x#) y@(BN# y#)
      | isTrue# (nx# ==# ny#)
          = compareInt# (narrowCInt# (c_mpn_cmp x# y# nx#)) 0#
      | isTrue# (nx# <#  ny#) = LT
      | True                  = GT
      where
        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    compareBigNatWord :: BigNat -> GmpLimb# -> Ordering
    compareBigNatWord bn w#
      | isTrue# (sizeofBigNat# bn ==# 1#) = cmpW# (bigNatToWord bn) w#
      | True                              = GT

    gtBigNatWord# :: BigNat -> GmpLimb# -> Int#
    gtBigNatWord# bn w#
        = (sizeofBigNat# bn ># 1#) `orI#` (bigNatToWord bn `gtWord#` w#)

    eqBigNat :: BigNat -> BigNat -> Bool
    eqBigNat x y = isTrue# (eqBigNat# x y)

    eqBigNat# :: BigNat -> BigNat -> Int#
    eqBigNat# x@(BN# x#) y@(BN# y#)
      | isTrue# (nx# ==# ny#) = c_mpn_cmp x# y# nx# ==# 0#
      | True                  = 0#
      where
        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    neqBigNat# :: BigNat -> BigNat -> Int#
    neqBigNat# x@(BN# x#) y@(BN# y#)
      | isTrue# (nx# ==# ny#) = c_mpn_cmp x# y# nx# /=# 0#
      | True                  = 1#
      where
        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    eqBigNatWord :: BigNat -> GmpLimb# -> Bool
    eqBigNatWord bn w# = isTrue# (eqBigNatWord# bn w#)

    eqBigNatWord# :: BigNat -> GmpLimb# -> Int#
    eqBigNatWord# bn w#
        = (sizeofBigNat# bn ==# 1#) `andI#` (bigNatToWord bn `eqWord#` w#)

    -- | Same as @'indexBigNat#' bn 0\#@
    bigNatToWord :: BigNat -> Word#
    bigNatToWord bn = indexBigNat# bn 0#

    -- | Equivalent to @'word2Int#' . 'bigNatToWord'@
    bigNatToInt :: BigNat -> Int#
    bigNatToInt (BN# ba#) = indexIntArray# ba# 0#

    -- | CAF representing the value @0 :: BigNat@
    zeroBigNat :: BigNat
    zeroBigNat = runS $ do
        mbn <- newBigNat# 1#
        _ <- svoid (writeBigNat# mbn 0# 0##)
        unsafeFreezeBigNat# mbn
    {-# NOINLINE zeroBigNat #-}

    -- | Test if 'BigNat' value is equal to zero.
    isZeroBigNat :: BigNat -> Bool
    isZeroBigNat bn = eqBigNatWord bn 0##

    -- | CAF representing the value @1 :: BigNat@
    oneBigNat :: BigNat
    oneBigNat = runS $ do
        mbn <- newBigNat# 1#
        _ <- svoid (writeBigNat# mbn 0# 1##)
        unsafeFreezeBigNat# mbn
    {-# NOINLINE oneBigNat #-}

    czeroBigNat :: BigNat
    czeroBigNat = runS $ do
        mbn <- newBigNat# 1#
        _ <- svoid (writeBigNat# mbn 0# (not# 0##))
        unsafeFreezeBigNat# mbn
    {-# NOINLINE czeroBigNat #-}

    -- | Special 0-sized bigNat returned in case of arithmetic underflow
    --
    -- This is currently only returned by the following operations:
    --
    --  - 'minusBigNat'
    --  - 'minusBigNatWord'
    --
    -- Other operations such as 'quotBigNat' may return 'nullBigNat' as
    -- well as a dummy/place-holder value instead of 'undefined' since we
    -- can't throw exceptions. But that behaviour should not be relied
    -- upon.
    --
    -- NB: @isValidBigNat# nullBigNat@ is false
    nullBigNat :: BigNat
    nullBigNat = runS (newBigNat# 0# >>= unsafeFreezeBigNat#)
    {-# NOINLINE nullBigNat #-}

    -- | Test for special 0-sized 'BigNat' representing underflows.
    isNullBigNat# :: BigNat -> Int#
    isNullBigNat# (BN# ba#) = sizeofByteArray# ba# ==# 0#

    -- | Construct 1-limb 'BigNat' from 'Word#'
    wordToBigNat :: Word# -> BigNat
    wordToBigNat 0## = zeroBigNat
    wordToBigNat 1## = oneBigNat
    wordToBigNat w#
      | isTrue# (not# w# `eqWord#` 0##) = czeroBigNat
      | True = runS $ do
        mbn <- newBigNat# 1#
        _ <- svoid (writeBigNat# mbn 0# w#)
        unsafeFreezeBigNat# mbn

    -- | Construct BigNat from 2 limbs.
    -- The first argument is the most-significant limb.
    wordToBigNat2 :: Word# -> Word# -> BigNat
    wordToBigNat2 0## lw# = wordToBigNat lw#
    wordToBigNat2 hw# lw# = runS $ do
        mbn <- newBigNat# 2#
        _ <- svoid (writeBigNat# mbn 0# lw#)
        _ <- svoid (writeBigNat# mbn 1# hw#)
        unsafeFreezeBigNat# mbn

    plusBigNat :: BigNat -> BigNat -> BigNat
    plusBigNat x y
      | isTrue# (eqBigNatWord# x 0##) = y
      | isTrue# (eqBigNatWord# y 0##) = x
      | isTrue# (nx# >=# ny#) = go x nx# y ny#
      | True                  = go y ny# x nx#
      where
        go (BN# a#) na# (BN# b#) nb# = runS $ do
            mbn@(MBN# mba#) <- newBigNat# na#
            (W# c#) <- liftIO (c_mpn_add mba# a# na# b# nb#)
            case c# of
                  0## -> unsafeFreezeBigNat# mbn
                  _   -> unsafeSnocFreezeBigNat# mbn c#

        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    plusBigNatWord :: BigNat -> GmpLimb# -> BigNat
    plusBigNatWord x          0## = x
    plusBigNatWord x@(BN# x#) y# = runS $ do
        mbn@(MBN# mba#) <- newBigNat# nx#
        (W# c#) <- liftIO (c_mpn_add_1 mba# x# nx# y#)
        case c# of
            0## -> unsafeFreezeBigNat# mbn
            _   -> unsafeSnocFreezeBigNat# mbn c#
      where
        nx# = sizeofBigNat# x

    -- | Returns 'nullBigNat' (see 'isNullBigNat#') in case of underflow
    minusBigNat :: BigNat -> BigNat -> BigNat
    minusBigNat x@(BN# x#) y@(BN# y#)
      | isZeroBigNat y = x
      | isTrue# (nx# >=# ny#) = runS $ do
        mbn@(MBN# mba#) <- newBigNat# nx#
        (W# b#) <- liftIO (c_mpn_sub mba# x# nx# y# ny#)
        case b# of
            0## -> unsafeRenormFreezeBigNat# mbn
            _   -> return nullBigNat

      | True = nullBigNat
      where
        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    -- | Returns 'nullBigNat' (see 'isNullBigNat#') in case of underflow
    minusBigNatWord :: BigNat -> GmpLimb# -> BigNat
    minusBigNatWord x 0## = x
    minusBigNatWord x@(BN# x#) y# = runS $ do
        mbn@(MBN# mba#) <- newBigNat# nx#
        (W# b#) <- liftIO $ c_mpn_sub_1 mba# x# nx# y#
        case b# of
            0## -> unsafeRenormFreezeBigNat# mbn
            _   -> return nullBigNat
      where
        nx# = sizeofBigNat# x

    timesBigNat :: BigNat -> BigNat -> BigNat
    timesBigNat x y
      | isZeroBigNat x = zeroBigNat
      | isZeroBigNat y = zeroBigNat
      | isTrue# (nx# >=# ny#) = go x nx# y ny#
      | True                  = go y ny# x nx#
      where
        go (BN# a#) na# (BN# b#) nb# = runS $ do
            let n# = nx# +# ny#
            mbn@(MBN# mba#) <- newBigNat# n#
            (W# msl#) <- liftIO (c_mpn_mul mba# a# na# b# nb#)
            case msl# of
                  0## -> unsafeShrinkFreezeBigNat# mbn (n# -# 1#)
                  _   -> unsafeFreezeBigNat# mbn

        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    -- | Square 'BigNat'
    sqrBigNat :: BigNat -> BigNat
    sqrBigNat x
      | isZeroBigNat x = zeroBigNat
      -- TODO: 1-limb BigNats below sqrt(maxBound::GmpLimb)
    sqrBigNat x = timesBigNat x x -- TODO: mpn_sqr

    timesBigNatWord :: BigNat -> GmpLimb# -> BigNat
    timesBigNatWord !_ 0## = zeroBigNat
    timesBigNatWord x 1## = x
    timesBigNatWord x@(BN# x#) y#
      | isTrue# (nx# ==# 1#) =
          let !(# !h#, !l# #) = timesWord2# (bigNatToWord x) y#
          in wordToBigNat2 h# l#
      | True = runS $ do
            mbn@(MBN# mba#) <- newBigNat# nx#
            (W# msl#) <- liftIO (c_mpn_mul_1 mba# x# nx# y#)
            case msl# of
                  0## -> unsafeFreezeBigNat# mbn
                  _   -> unsafeSnocFreezeBigNat# mbn msl#

      where
        nx# = sizeofBigNat# x

    -- | Specialised version of
    --
    -- > bitBigNat = shiftLBigNat (wordToBigNat 1##)
    --
    -- avoiding a few redundant allocations
    bitBigNat :: Int# -> BigNat
    bitBigNat i#
      | isTrue# (i#  <#  0#) = zeroBigNat -- or maybe 'nullBigNat'?
      | isTrue# (i# ==#  0#) = oneBigNat
      | True = runS $ do
          mbn@(MBN# mba#) <- newBigNat# (li# +# 1#)
          -- FIXME: do we really need to zero-init MBAs returned by 'newByteArray#'?
          -- clear all limbs (except for the most-significant limb)
          _ <- svoid (setByteArray# mba# 0# (li# `uncheckedIShiftL#` GMP_LIMB_SHIFT#) 0#)
          -- set single bit in most-significant limb
          _ <- svoid (writeBigNat# mbn li# (uncheckedShiftL# 1## bi#))
          unsafeFreezeBigNat# mbn
      where
        !(# li#, bi# #) = quotRemInt# i# GMP_LIMB_BITS#

    testBitBigNat :: BigNat -> Int# -> Bool
    testBitBigNat bn i#
      | isTrue# (i#  <#  0#) = False
      | isTrue# (li# <# nx#) = isTrue# (testBitWord# (indexBigNat# bn li#) bi#)
      | True                 = False
      where
        !(# li#, bi# #) = quotRemInt# i# GMP_LIMB_BITS#
        nx# = sizeofBigNat# bn

    testBitNegBigNat :: BigNat -> Int# -> Bool
    testBitNegBigNat bn i#
      | isTrue# (i#  <#  0#)  = False
      | isTrue# (li# >=# nx#) = True
      | allZ li# = isTrue# ((testBitWord#
                             (indexBigNat# bn li# `minusWord#` 1##) bi#) ==# 0#)
      | True     = isTrue# ((testBitWord# (indexBigNat# bn li#) bi#) ==# 0#)
      where
        !(# li#, bi# #) = quotRemInt# i# GMP_LIMB_BITS#
        nx# = sizeofBigNat# bn

        allZ 0# = True
        allZ j | isTrue# (indexBigNat# bn (j -# 1#) `eqWord#` 0##) = allZ (j -# 1#)
               | True                 = False

    popCountBigNat :: BigNat -> Int#
    popCountBigNat bn@(BN# ba#) = word2Int# (c_mpn_popcount ba# (sizeofBigNat# bn))

    shiftLBigNat :: BigNat -> Int# -> BigNat
    shiftLBigNat x 0# = x
    shiftLBigNat x _ | isZeroBigNat x = zeroBigNat
    shiftLBigNat x@(BN# xba#) n# = runS $ do
        ymbn@(MBN# ymba#) <- newBigNat# yn#
        W# ymsl <- liftIO (c_mpn_lshift ymba# xba# xn# (int2Word# n#))
        case ymsl of
            0## -> unsafeShrinkFreezeBigNat# ymbn (yn# -# 1#)
            _   -> unsafeFreezeBigNat# ymbn
      where
        xn# = sizeofBigNat# x
        yn# = xn# +# nlimbs# +# (nbits# /=# 0#)
        !(# nlimbs#, nbits# #) = quotRemInt# n# GMP_LIMB_BITS#

    shiftRBigNat :: BigNat -> Int# -> BigNat
    shiftRBigNat x 0# = x
    shiftRBigNat x _ | isZeroBigNat x = zeroBigNat
    shiftRBigNat x@(BN# xba#) n#
      | isTrue# (nlimbs# >=# xn#) = zeroBigNat
      | True = runS $ do
          ymbn@(MBN# ymba#) <- newBigNat# yn#
          W# ymsl <- liftIO (c_mpn_rshift ymba# xba# xn# (int2Word# n#))
          case ymsl of
              0## -> unsafeRenormFreezeBigNat# ymbn -- may shrink more than one
              _   -> unsafeFreezeBigNat# ymbn
      where
        xn# = sizeofBigNat# x
        yn# = xn# -# nlimbs#
        nlimbs# = quotInt# n# GMP_LIMB_BITS#

    shiftRNegBigNat :: BigNat -> Int# -> BigNat
    shiftRNegBigNat x 0# = x
    shiftRNegBigNat x _ | isZeroBigNat x = zeroBigNat
    shiftRNegBigNat x@(BN# xba#) n#
      | isTrue# (nlimbs# >=# xn#) = zeroBigNat
      | True = runS $ do
          ymbn@(MBN# ymba#) <- newBigNat# yn#
          W# ymsl <- liftIO (c_mpn_rshift_2c ymba# xba# xn# (int2Word# n#))
          case ymsl of
              0## -> unsafeRenormFreezeBigNat# ymbn -- may shrink more than one
              _   -> unsafeFreezeBigNat# ymbn
      where
        xn# = sizeofBigNat# x
        yn# = xn# -# nlimbs#
        nlimbs# = quotInt# (n# -# 1#) GMP_LIMB_BITS#

    orBigNat :: BigNat -> BigNat -> BigNat
    orBigNat x@(BN# x#) y@(BN# y#)
      | isZeroBigNat x = y
      | isZeroBigNat y = x
      | isTrue# (nx# >=# ny#) = runS (ior' x# nx# y# ny#)
      | True                  = runS (ior' y# ny# x# nx#)
      where
        ior' a# na# b# nb# = do -- na >= nb
            mbn@(MBN# mba#) <- newBigNat# na#
            _ <- liftIO (c_mpn_ior_n mba# a# b# nb#)
            _ <- case isTrue# (na# ==# nb#) of
                False -> svoid (copyWordArray# a# nb# mba# nb# (na# -# nb#))
                True  -> return ()
            unsafeFreezeBigNat# mbn

        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    xorBigNat :: BigNat -> BigNat -> BigNat
    xorBigNat x@(BN# x#) y@(BN# y#)
      | isZeroBigNat x = y
      | isZeroBigNat y = x
      | isTrue# (nx# >=# ny#) = runS (xor' x# nx# y# ny#)
      | True                  = runS (xor' y# ny# x# nx#)
      where
        xor' a# na# b# nb# = do -- na >= nb
            mbn@(MBN# mba#) <- newBigNat# na#
            _ <- liftIO (c_mpn_xor_n mba# a# b# nb#)
            case isTrue# (na# ==# nb#) of
                False -> do _ <- svoid (copyWordArray# a# nb# mba# nb# (na# -# nb#))
                            unsafeFreezeBigNat# mbn
                True  -> unsafeRenormFreezeBigNat# mbn

        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    -- | aka @\x y -> x .&. (complement y)@
    andnBigNat :: BigNat -> BigNat -> BigNat
    andnBigNat x@(BN# x#) y@(BN# y#)
      | isZeroBigNat x = zeroBigNat
      | isZeroBigNat y = x
      | True = runS $ do
          mbn@(MBN# mba#) <- newBigNat# nx#
          _ <- liftIO (c_mpn_andn_n mba# x# y# n#)
          _ <- case isTrue# (nx# ==# n#) of
                False -> svoid (copyWordArray# x# n# mba# n# (nx# -# n#))
                True  -> return ()
          unsafeRenormFreezeBigNat# mbn
      where
        n# | isTrue# (nx# <# ny#) = nx#
           | True                 = ny#
        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    andBigNat :: BigNat -> BigNat -> BigNat
    andBigNat x@(BN# x#) y@(BN# y#)
      | isZeroBigNat x = zeroBigNat
      | isZeroBigNat y = zeroBigNat
      | True = runS $ do
          mbn@(MBN# mba#) <- newBigNat# n#
          _ <- liftIO (c_mpn_and_n mba# x# y# n#)
          unsafeRenormFreezeBigNat# mbn
      where
        n# | isTrue# (nx# <# ny#) = nx#
           | True                 = ny#
        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    -- | If divisor is zero, @(\# 'nullBigNat', 'nullBigNat' \#)@ is returned
    quotRemBigNat :: BigNat -> BigNat -> (# BigNat,BigNat #)
    quotRemBigNat n@(BN# nba#) d@(BN# dba#)
      | isZeroBigNat d     = (# nullBigNat, nullBigNat #)
      | eqBigNatWord d 1## = (# n, zeroBigNat #)
      | n < d              = (# zeroBigNat, n #)
      | True = case runS go of (!q,!r) -> (# q, r #)
      where
        nn# = sizeofBigNat# n
        dn# = sizeofBigNat# d
        qn# = 1# +# nn# -# dn#
        rn# = dn#

        go = do
          qmbn@(MBN# qmba#) <- newBigNat# qn#
          rmbn@(MBN# rmba#) <- newBigNat# rn#

          _ <- liftIO (c_mpn_tdiv_qr qmba# rmba# 0# nba# nn# dba# dn#)

          q <- unsafeRenormFreezeBigNat# qmbn
          r <- unsafeRenormFreezeBigNat# rmbn
          return (q, r)

    quotBigNat :: BigNat -> BigNat -> BigNat
    quotBigNat n@(BN# nba#) d@(BN# dba#)
      | isZeroBigNat d     = nullBigNat
      | eqBigNatWord d 1## = n
      | n < d              = zeroBigNat
      | True = runS $ do
          let nn# = sizeofBigNat# n
          let dn# = sizeofBigNat# d
          let qn# = 1# +# nn# -# dn#
          qmbn@(MBN# qmba#) <- newBigNat# qn#
          _ <- liftIO (c_mpn_tdiv_q qmba# nba# nn# dba# dn#)
          unsafeRenormFreezeBigNat# qmbn

    remBigNat :: BigNat -> BigNat -> BigNat
    remBigNat n@(BN# nba#) d@(BN# dba#)
      | isZeroBigNat d     = nullBigNat
      | eqBigNatWord d 1## = zeroBigNat
      | n < d              = n
      | True = runS $ do
          let nn# = sizeofBigNat# n
          let dn# = sizeofBigNat# d
          rmbn@(MBN# rmba#) <- newBigNat# dn#
          _ <- liftIO (c_mpn_tdiv_r rmba# nba# nn# dba# dn#)
          unsafeRenormFreezeBigNat# rmbn

    -- | Note: Result of div/0 undefined
    quotRemBigNatWord :: BigNat -> GmpLimb# -> (# BigNat, GmpLimb# #)
    quotRemBigNatWord !_           0## = (# nullBigNat, 0## #)
    quotRemBigNatWord n            1## = (# n,          0## #)
    quotRemBigNatWord n@(BN# nba#) d# = case compareBigNatWord n d# of
        LT -> (# zeroBigNat, bigNatToWord n #)
        EQ -> (# oneBigNat, 0## #)
        GT -> case runS go of (!q,!(W# r#)) -> (# q, r# #) -- TODO: handle word/word
      where
        go = do
          let nn# = sizeofBigNat# n
          qmbn@(MBN# qmba#) <- newBigNat# nn#
          r <- liftIO (c_mpn_divrem_1 qmba# 0# nba# nn# d#)
          q <- unsafeRenormFreezeBigNat# qmbn
          return (q,r)

    quotBigNatWord :: BigNat -> GmpLimb# -> BigNat
    quotBigNatWord n d# = case inline quotRemBigNatWord n d# of (# q, _ #) -> q

    -- | div/0 not checked
    remBigNatWord :: BigNat -> GmpLimb# -> Word#
    remBigNatWord n@(BN# nba#) d# = c_mpn_mod_1 nba# (sizeofBigNat# n) d#

    gcdBigNatWord :: BigNat -> Word# -> Word#
    gcdBigNatWord bn@(BN# ba#) = c_mpn_gcd_1# ba# (sizeofBigNat# bn)

    gcdBigNat :: BigNat -> BigNat -> BigNat
    gcdBigNat x@(BN# x#) y@(BN# y#)
      | isZeroBigNat x = y
      | isZeroBigNat y = x
      | isTrue# (nx# >=# ny#) = runS (gcd' x# nx# y# ny#)
      | True                  = runS (gcd' y# ny# x# nx#)
      where
        gcd' a# na# b# nb# = do -- na >= nb
            mbn@(MBN# mba#) <- newBigNat# nb#
            I# rn'# <- liftIO (c_mpn_gcd# mba# a# na# b# nb#)
            let rn# = narrowGmpSize# rn'#
            case isTrue# (rn# ==# nb#) of
                False -> unsafeShrinkFreezeBigNat# mbn rn#
                True  -> unsafeFreezeBigNat# mbn

        nx# = sizeofBigNat# x
        ny# = sizeofBigNat# y

    -- | Extended euclidean algorithm.
    --
    -- For @/a/@ and @/b/@, compute their greatest common divisor @/g/@
    -- and the coefficient @/s/@ satisfying @/a//s/ + /b//t/ = /g/@.
    --
    -- @since 0.5.1.0
    {-# NOINLINE gcdExtInteger #-}
    gcdExtInteger :: Integer -> Integer -> (# Integer, Integer #)
    gcdExtInteger a b = case gcdExtSBigNat a' b' of
        (# g, s #) -> let !g' = bigNatToInteger  g
                          !s' = sBigNatToInteger s
                      in (# g', s' #)
      where
        a' = integerToSBigNat a
        b' = integerToSBigNat b

    -- internal helper
    gcdExtSBigNat :: SBigNat -> SBigNat -> (# BigNat, SBigNat #)
    gcdExtSBigNat x y = case runS go of (g,s) -> (# g, s #)
      where
        go = do
            g@(MBN# g#) <- newBigNat# gn0#
            s@(MBN# s#) <- newBigNat# (absI# xn#)
            I# ssn_# <- liftIO (integer_gmp_gcdext# s# g# x# xn# y# yn#)
            let ssn# = narrowGmpSize# ssn_#
                sn#  = absI# ssn#
            s' <- unsafeShrinkFreezeBigNat# s sn#
            g' <- unsafeRenormFreezeBigNat# g
            case isTrue# (ssn# >=# 0#) of
                False -> return ( g', NegBN s' )
                True  -> return ( g', PosBN s' )

        !(BN# x#) = absSBigNat x
        !(BN# y#) = absSBigNat y
        xn# = ssizeofSBigNat# x
        yn# = ssizeofSBigNat# y

        gn0# = minI# (absI# xn#) (absI# yn#)

    ----------------------------------------------------------------------------
    -- modular exponentiation

    -- | \"@'powModInteger' /b/ /e/ /m/@\" computes base @/b/@ raised to
    -- exponent @/e/@ modulo @abs(/m/)@.
    --
    -- Negative exponents are supported if an inverse modulo @/m/@
    -- exists.
    --
    -- __Warning__: It's advised to avoid calling this primitive with
    -- negative exponents unless it is guaranteed the inverse exists, as
    -- failure to do so will likely cause program abortion due to a
    -- divide-by-zero fault. See also 'recipModInteger'.
    --
    -- Future versions of @integer_gmp@ may not support negative @/e/@
    -- values anymore.
    --
    -- @since 0.5.1.0
    {-# NOINLINE powModInteger #-}
    powModInteger :: Integer -> Integer -> Integer -> Integer
    powModInteger (S# b#) (S# e#) (S# m#)
      | isTrue# (b# >=# 0#), isTrue# (e# >=# 0#)
      = wordToInteger (powModWord (int2Word# b#) (int2Word# e#)
                                  (int2Word# (absI# m#)))
    powModInteger b e m = case m of
        (S# m#) -> wordToInteger (powModSBigNatWord b' e' (int2Word# (absI# m#)))
        (Jp# m') -> bigNatToInteger (powModSBigNat b' e' m')
        (Jn# m') -> bigNatToInteger (powModSBigNat b' e' m')
      where
        b' = integerToSBigNat b
        e' = integerToSBigNat e

    -- | Version of 'powModInteger' operating on 'BigNat's
    --
    -- @since 1.0.0.0
    powModBigNat :: BigNat -> BigNat -> BigNat -> BigNat
    powModBigNat b e m = inline powModSBigNat (PosBN b) (PosBN e) m

    -- | Version of 'powModInteger' for 'Word#'-sized moduli
    --
    -- @since 1.0.0.0
    powModBigNatWord :: BigNat -> BigNat -> GmpLimb# -> GmpLimb#
    powModBigNatWord b e m# = inline powModSBigNatWord (PosBN b) (PosBN e) m#

    -- | Version of 'powModInteger' operating on 'Word#'s
    --
    -- @since 1.0.0.0
    foreign import ccall unsafe "integer_gmp_powm_word"
      powModWord :: GmpLimb# -> GmpLimb# -> GmpLimb# -> GmpLimb#

    -- internal non-exported helper
    powModSBigNat :: SBigNat -> SBigNat -> BigNat -> BigNat
    powModSBigNat b e m@(BN# m#) = runS $ do
        r@(MBN# r#) <- newBigNat# mn#
        I# rn_# <- liftIO (integer_gmp_powm# r# b# bn# e# en# m# mn#)
        let rn# = narrowGmpSize# rn_#
        case isTrue# (rn# ==# mn#) of
            False -> unsafeShrinkFreezeBigNat# r rn#
            True  -> unsafeFreezeBigNat# r
      where
        !(BN# b#) = absSBigNat b
        !(BN# e#) = absSBigNat e
        bn# = ssizeofSBigNat# b
        en# = ssizeofSBigNat# e
        mn# = sizeofBigNat# m

    foreign import ccall unsafe "integer_gmp_powm"
      integer_gmp_powm# :: MutableByteArray# RealWorld
                           -> ByteArray# -> GmpSize# -> ByteArray# -> GmpSize#
                           -> ByteArray# -> GmpSize# -> IO GmpSize

    -- internal non-exported helper
    powModSBigNatWord :: SBigNat -> SBigNat -> GmpLimb# -> GmpLimb#
    powModSBigNatWord b e m# = integer_gmp_powm1# b# bn# e# en# m#
      where
        !(BN# b#) = absSBigNat b
        !(BN# e#) = absSBigNat e
        bn# = ssizeofSBigNat# b
        en# = ssizeofSBigNat# e

    foreign import ccall unsafe "integer_gmp_powm1"
      integer_gmp_powm1# :: ByteArray# -> GmpSize# -> ByteArray# -> GmpSize#
                            -> GmpLimb# -> GmpLimb#

    -- | \"@'recipModInteger' /x/ /m/@\" computes the inverse of @/x/@ modulo @/m/@. If
    -- the inverse exists, the return value @/y/@ will satisfy @0 < /y/ <
    -- abs(/m/)@, otherwise the result is @0@.
    --
    -- @since 0.5.1.0
    {-# NOINLINE recipModInteger #-}
    recipModInteger :: Integer -> Integer -> Integer
    recipModInteger (S# x#) (S# m#)
      | isTrue# (x# >=# 0#)
      = wordToInteger (recipModWord (int2Word# x#) (int2Word# (absI# m#)))
    recipModInteger x m = bigNatToInteger (recipModSBigNat x' m')
      where
        x' = integerToSBigNat x
        m' = absSBigNat (integerToSBigNat m)

    -- | Version of 'recipModInteger' operating on 'BigNat's
    --
    -- @since 1.0.0.0
    recipModBigNat :: BigNat -> BigNat -> BigNat
    recipModBigNat x m = inline recipModSBigNat (PosBN x) m

    -- | Version of 'recipModInteger' operating on 'Word#'s
    --
    -- @since 1.0.0.0
    foreign import ccall unsafe "integer_gmp_invert_word"
      recipModWord :: GmpLimb# -> GmpLimb# -> GmpLimb#

    -- internal non-exported helper
    recipModSBigNat :: SBigNat -> BigNat -> BigNat
    recipModSBigNat x m@(BN# m#) = runS $ do
        r@(MBN# r#) <- newBigNat# mn#
        I# rn_# <- liftIO (integer_gmp_invert# r# x# xn# m# mn#)
        let rn# = narrowGmpSize# rn_#
        case isTrue# (rn# ==# mn#) of
            False -> unsafeShrinkFreezeBigNat# r rn#
            True  -> unsafeFreezeBigNat# r
      where
        !(BN# x#) = absSBigNat x
        xn# = ssizeofSBigNat# x
        mn# = sizeofBigNat# m

    foreign import ccall unsafe "integer_gmp_invert"
      integer_gmp_invert# :: MutableByteArray# RealWorld
                             -> ByteArray# -> GmpSize#
                             -> ByteArray# -> GmpSize# -> IO GmpSize

    ----------------------------------------------------------------------------
    -- Conversions to/from floating point

    decodeDoubleInteger :: Double# -> (# Integer, Int# #)
    -- decodeDoubleInteger 0.0## = (# S# 0#, 0# #)
    #if WORD_SIZE_IN_BITS == 64
    decodeDoubleInteger x = case decodeDouble_Int64# x of
                              (# m#, e# #) -> (# S# m#, e# #)
    #elif WORD_SIZE_IN_BITS == 32
    decodeDoubleInteger x = case decodeDouble_Int64# x of
                              (# m#, e# #) -> (# int64ToInteger m#, e# #)
    #endif
    {-# CONSTANT_FOLDED decodeDoubleInteger #-}

    -- provided by GHC's RTS
    foreign import ccall unsafe "__int_encodeDouble"
      int_encodeDouble# :: Int# -> Int# -> Double#

    encodeDoubleInteger :: Integer -> Int# -> Double#
    encodeDoubleInteger (S# m#) 0# = int2Double# m#
    encodeDoubleInteger (S# m#) e# = int_encodeDouble# m# e#
    encodeDoubleInteger (Jp# bn@(BN# bn#)) e#
        = c_mpn_get_d bn# (sizeofBigNat# bn) e#
    encodeDoubleInteger (Jn# bn@(BN# bn#)) e#
        = c_mpn_get_d bn# (negateInt# (sizeofBigNat# bn)) e#
    {-# CONSTANT_FOLDED encodeDoubleInteger #-}

    -- double integer_gmp_mpn_get_d (const mp_limb_t sp[], const mp_size_t sn)
    foreign import ccall unsafe "integer_gmp_mpn_get_d"
      c_mpn_get_d :: ByteArray# -> GmpSize# -> Int# -> Double#

    doubleFromInteger :: Integer -> Double#
    doubleFromInteger (S# m#) = int2Double# m#
    doubleFromInteger (Jp# bn@(BN# bn#))
        = c_mpn_get_d bn# (sizeofBigNat# bn) 0#
    doubleFromInteger (Jn# bn@(BN# bn#))
        = c_mpn_get_d bn# (negateInt# (sizeofBigNat# bn)) 0#
    {-# CONSTANT_FOLDED doubleFromInteger #-}

    -- TODO: Not sure if it's worth to write 'Float' optimized versions here
    floatFromInteger :: Integer -> Float#
    floatFromInteger i = double2Float# (doubleFromInteger i)

    encodeFloatInteger :: Integer -> Int# -> Float#
    encodeFloatInteger m e = double2Float# (encodeDoubleInteger m e)

    ----------------------------------------------------------------------------
    -- FFI ccall imports

    foreign import ccall unsafe "integer_gmp_gcd_word"
      gcdWord# :: GmpLimb# -> GmpLimb# -> GmpLimb#

    foreign import ccall unsafe "integer_gmp_mpn_gcd_1"
      c_mpn_gcd_1# :: ByteArray# -> GmpSize# -> GmpLimb# -> GmpLimb#

    foreign import ccall unsafe "integer_gmp_mpn_gcd"
      c_mpn_gcd# :: MutableByteArray# s -> ByteArray# -> GmpSize#
                    -> ByteArray# -> GmpSize# -> IO GmpSize

    foreign import ccall unsafe "integer_gmp_gcdext"
      integer_gmp_gcdext# :: MutableByteArray# s -> MutableByteArray# s
                             -> ByteArray# -> GmpSize#
                             -> ByteArray# -> GmpSize# -> IO GmpSize

    -- mp_limb_t mpn_add_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n,
    --                      mp_limb_t s2limb)
    foreign import ccall unsafe "gmp.h __gmpn_add_1"
      c_mpn_add_1 :: MutableByteArray# s -> ByteArray# -> GmpSize# -> GmpLimb#
                     -> IO GmpLimb

    -- mp_limb_t mpn_sub_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n,
    --                      mp_limb_t s2limb)
    foreign import ccall unsafe "gmp.h __gmpn_sub_1"
      c_mpn_sub_1 :: MutableByteArray# s -> ByteArray# -> GmpSize# -> GmpLimb#
                     -> IO GmpLimb

    -- mp_limb_t mpn_mul_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n,
    --                      mp_limb_t s2limb)
    foreign import ccall unsafe "gmp.h __gmpn_mul_1"
      c_mpn_mul_1 :: MutableByteArray# s -> ByteArray# -> GmpSize# -> GmpLimb#
                     -> IO GmpLimb

    -- mp_limb_t mpn_add (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t s1n,
    --                    const mp_limb_t *s2p, mp_size_t s2n)
    foreign import ccall unsafe "gmp.h __gmpn_add"
      c_mpn_add :: MutableByteArray# s -> ByteArray# -> GmpSize#
                   -> ByteArray# -> GmpSize# -> IO GmpLimb

    -- mp_limb_t mpn_sub (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t s1n,
    --                    const mp_limb_t *s2p, mp_size_t s2n)
    foreign import ccall unsafe "gmp.h __gmpn_sub"
      c_mpn_sub :: MutableByteArray# s -> ByteArray# -> GmpSize# -> ByteArray#
                   -> GmpSize# -> IO GmpLimb

    -- mp_limb_t mpn_mul (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t s1n,
    --                    const mp_limb_t *s2p, mp_size_t s2n)
    foreign import ccall unsafe "gmp.h __gmpn_mul"
      c_mpn_mul :: MutableByteArray# s -> ByteArray# -> GmpSize# -> ByteArray#
                   -> GmpSize# -> IO GmpLimb

    -- int mpn_cmp (const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n)
    foreign import ccall unsafe "gmp.h __gmpn_cmp"
      c_mpn_cmp :: ByteArray# -> ByteArray# -> GmpSize# -> CInt#

    -- void mpn_tdiv_qr (mp_limb_t *qp, mp_limb_t *rp, mp_size_t qxn,
    --                   const mp_limb_t *np, mp_size_t nn,
    --                   const mp_limb_t *dp, mp_size_t dn)
    foreign import ccall unsafe "gmp.h __gmpn_tdiv_qr"
      c_mpn_tdiv_qr :: MutableByteArray# s -> MutableByteArray# s -> GmpSize#
                       -> ByteArray# -> GmpSize# -> ByteArray# -> GmpSize# -> IO ()

    foreign import ccall unsafe "integer_gmp_mpn_tdiv_q"
      c_mpn_tdiv_q :: MutableByteArray# s -> ByteArray# -> GmpSize# -> ByteArray#
                      -> GmpSize# -> IO ()

    foreign import ccall unsafe "integer_gmp_mpn_tdiv_r"
      c_mpn_tdiv_r :: MutableByteArray# s -> ByteArray# -> GmpSize# -> ByteArray#
                      -> GmpSize# -> IO ()

    -- mp_limb_t mpn_divrem_1 (mp_limb_t *r1p, mp_size_t qxn, mp_limb_t *s2p,
    --                         mp_size_t s2n, mp_limb_t s3limb)
    foreign import ccall unsafe "gmp.h __gmpn_divrem_1"
      c_mpn_divrem_1 :: MutableByteArray# s -> GmpSize# -> ByteArray# -> GmpSize#
                        -> GmpLimb# -> IO GmpLimb

    -- mp_limb_t mpn_mod_1 (const mp_limb_t *s1p, mp_size_t s1n, mp_limb_t s2limb)
    foreign import ccall unsafe "gmp.h __gmpn_mod_1"
      c_mpn_mod_1 :: ByteArray# -> GmpSize# -> GmpLimb# -> GmpLimb#

    -- mp_limb_t integer_gmp_mpn_rshift (mp_limb_t rp[], const mp_limb_t sp[],
    --                                   mp_size_t sn, mp_bitcnt_t count)
    foreign import ccall unsafe "integer_gmp_mpn_rshift"
      c_mpn_rshift :: MutableByteArray# s -> ByteArray# -> GmpSize# -> GmpBitCnt#
                      -> IO GmpLimb

    -- mp_limb_t integer_gmp_mpn_rshift (mp_limb_t rp[], const mp_limb_t sp[],
    --                                   mp_size_t sn, mp_bitcnt_t count)
    foreign import ccall unsafe "integer_gmp_mpn_rshift_2c"
      c_mpn_rshift_2c :: MutableByteArray# s -> ByteArray# -> GmpSize# -> GmpBitCnt#
                         -> IO GmpLimb

    -- mp_limb_t integer_gmp_mpn_lshift (mp_limb_t rp[], const mp_limb_t sp[],
    --                                   mp_size_t sn, mp_bitcnt_t count)
    foreign import ccall unsafe "integer_gmp_mpn_lshift"
      c_mpn_lshift :: MutableByteArray# s -> ByteArray# -> GmpSize# -> GmpBitCnt#
                      -> IO GmpLimb

    -- void mpn_and_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p,
    --                 mp_size_t n)
    foreign import ccall unsafe "integer_gmp_mpn_and_n"
      c_mpn_and_n :: MutableByteArray# s -> ByteArray# -> ByteArray# -> GmpSize#
                     -> IO ()

    -- void mpn_andn_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p,
    --                  mp_size_t n)
    foreign import ccall unsafe "integer_gmp_mpn_andn_n"
      c_mpn_andn_n :: MutableByteArray# s -> ByteArray# -> ByteArray# -> GmpSize#
                      -> IO ()

    -- void mpn_ior_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p,
    --                 mp_size_t n)
    foreign import ccall unsafe "integer_gmp_mpn_ior_n"
      c_mpn_ior_n :: MutableByteArray# s -> ByteArray# -> ByteArray# -> GmpSize#
                     -> IO ()

    -- void mpn_xor_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p,
    --                 mp_size_t n)
    foreign import ccall unsafe "integer_gmp_mpn_xor_n"
      c_mpn_xor_n :: MutableByteArray# s -> ByteArray# -> ByteArray# -> GmpSize#
                     -> IO ()

    -- mp_bitcnt_t mpn_popcount (const mp_limb_t *s1p, mp_size_t n)
    foreign import ccall unsafe "gmp.h __gmpn_popcount"
      c_mpn_popcount :: ByteArray# -> GmpSize# -> GmpBitCnt#

    ----------------------------------------------------------------------------
    -- BigNat-wrapped ByteArray#-primops

    -- | Return number of limbs contained in 'BigNat'.
    sizeofBigNat# :: BigNat -> GmpSize#
    sizeofBigNat# (BN# x#)
        = sizeofByteArray# x# `uncheckedIShiftRL#` GMP_LIMB_SHIFT#

    data MutBigNat s = MBN# !(MutableByteArray# s)

    getSizeofMutBigNat# :: MutBigNat s -> State# s -> (# State# s, GmpSize# #)
    --getSizeofMutBigNat# :: MutBigNat s -> S s GmpSize#
    getSizeofMutBigNat# (MBN# x#) s =
        case getSizeofMutableByteArray# x# s of
            (# s', n# #) -> (# s', n# `uncheckedIShiftRL#` GMP_LIMB_SHIFT# #)

    newBigNat# :: GmpSize# -> S s (MutBigNat s)
    newBigNat# limbs# s =
        case newByteArray# (limbs# `uncheckedIShiftL#` GMP_LIMB_SHIFT#) s of
            (# s', mba# #) -> (# s', MBN# mba# #)

    writeBigNat# :: MutBigNat s -> GmpSize# -> GmpLimb# -> State# s -> State# s
    writeBigNat# (MBN# mba#) = writeWordArray# mba#

    -- | Extract /n/-th (0-based) limb in 'BigNat'.
    -- /n/ must be less than size as reported by 'sizeofBigNat#'.
    indexBigNat# :: BigNat -> GmpSize# -> GmpLimb#
    indexBigNat# (BN# ba#) = indexWordArray# ba#

    unsafeFreezeBigNat# :: MutBigNat s -> S s BigNat
    unsafeFreezeBigNat# (MBN# mba#) s = case unsafeFreezeByteArray# mba# s of
                                          (# s', ba# #) -> (# s', BN# ba# #)

    resizeMutBigNat# :: MutBigNat s -> GmpSize# -> S s (MutBigNat s)
    resizeMutBigNat# (MBN# mba0#) nsz# s
      | isTrue# (bsz# ==# n#) = (# s', MBN# mba0# #)
      | True =
        case resizeMutableByteArray# mba0# bsz# s' of
            (# s'', mba# #) -> (# s'', MBN# mba# #)
      where
        bsz# = nsz# `uncheckedIShiftL#` GMP_LIMB_SHIFT#
        !(# s', n# #) = getSizeofMutableByteArray# mba0# s

    shrinkMutBigNat# :: MutBigNat s -> GmpSize# -> State# s -> State# s
    shrinkMutBigNat# (MBN# mba0#) nsz# s
      | isTrue# (bsz# ==# n#) = s' -- no-op
      | True                  = shrinkMutableByteArray# mba0# bsz# s'
      where
        bsz# = nsz# `uncheckedIShiftL#` GMP_LIMB_SHIFT#
        !(# s', n# #) = getSizeofMutableByteArray# mba0# s

    unsafeSnocFreezeBigNat# :: MutBigNat s -> GmpLimb# -> S s BigNat
    unsafeSnocFreezeBigNat# mbn0@(MBN# mba0#) limb# s = go s'
      where
        n#   = nb0# `uncheckedIShiftRL#` GMP_LIMB_SHIFT#
        !(# s', nb0# #) = getSizeofMutableByteArray# mba0# s
        go = do
            (MBN# mba#) <- resizeMutBigNat# mbn0 (n# +# 1#)
            _ <- svoid (writeWordArray# mba# n# limb#)
            unsafeFreezeBigNat# (MBN# mba#)

    -- | May shrink underlyng 'ByteArray#' if needed to satisfy BigNat invariant
    unsafeRenormFreezeBigNat# :: MutBigNat s -> S s BigNat
    unsafeRenormFreezeBigNat# mbn s
      | isTrue# (n0# ==# 0#)  = (# s'', nullBigNat #)
      | isTrue# (n#  ==# 0#)  = (# s'', zeroBigNat #)
      | isTrue# (n#  ==# n0#) = (unsafeFreezeBigNat# mbn) s''
      | True                  = (unsafeShrinkFreezeBigNat# mbn n#) s''
      where
        !(# s', n0# #) = getSizeofMutBigNat# mbn s
        !(# s'', n# #) = normSizeofMutBigNat'# mbn n0# s'

    -- | Shrink MBN
    unsafeShrinkFreezeBigNat# :: MutBigNat s -> GmpSize# -> S s BigNat
    unsafeShrinkFreezeBigNat# x@(MBN# xmba) 1#
        = \s -> case readWordArray# xmba 0# s of
            (# s', w#   #) -> freezeOneLimb w# s'
      where
        freezeOneLimb 0## = return zeroBigNat
        freezeOneLimb 1## = return oneBigNat
        freezeOneLimb w# | isTrue# (not# w# `eqWord#` 0##) = return czeroBigNat
        freezeOneLimb _   = do
            _ <- svoid (shrinkMutBigNat# x 1#)
            unsafeFreezeBigNat# x
    unsafeShrinkFreezeBigNat# x y# = do
        _ <- svoid (shrinkMutBigNat# x y#)
        unsafeFreezeBigNat# x

    copyWordArray# :: ByteArray# -> Int# -> MutableByteArray# s -> Int# -> Int#
                      -> State# s -> State# s
    copyWordArray# src src_ofs dst dst_ofs len
      = copyByteArray# src (src_ofs `uncheckedIShiftL#` GMP_LIMB_SHIFT#)
                       dst (dst_ofs `uncheckedIShiftL#` GMP_LIMB_SHIFT#)
                       (len `uncheckedIShiftL#` GMP_LIMB_SHIFT#)

    -- | Version of 'normSizeofMutBigNat'#' which scans all allocated 'MutBigNat#'
    normSizeofMutBigNat# :: MutBigNat s -> State# s -> (# State# s, Int# #)
    normSizeofMutBigNat# mbn@(MBN# mba) s = normSizeofMutBigNat'# mbn sz# s'
      where
        !(# s', n# #) = getSizeofMutableByteArray# mba s
        sz# = n# `uncheckedIShiftRA#` GMP_LIMB_SHIFT#

    -- | Find most-significant non-zero limb and return its index-position
    -- plus one. Start scanning downward from the initial limb-size
    -- (i.e. start-index plus one) given as second argument.
    --
    -- NB: The 'normSizeofMutBigNat' of 'zeroBigNat' would be @0#@
    normSizeofMutBigNat'# :: MutBigNat s -> GmpSize#
                             -> State# s -> (# State# s, GmpSize# #)
    normSizeofMutBigNat'# (MBN# mba) = go
      where
        go  0# s = (# s, 0# #)
        go i0# s = case readWordArray# mba (i0# -# 1#) s of
            (# s', 0## #) -> go (i0# -# 1#) s'
            (# s', _  #) -> (# s', i0# #)

    -- | Construct 'BigNat' from existing 'ByteArray#' containing /n/
    -- 'GmpLimb's in least-significant-first order.
    --
    -- If possible 'ByteArray#', will be used directly (i.e. shared
    -- /without/ cloning the 'ByteArray#' into a newly allocated one)
    --
    -- Note: size parameter (times @sizeof(GmpLimb)@) must be less or
    -- equal to its 'sizeofByteArray#'.
    byteArrayToBigNat# :: ByteArray# -> GmpSize# -> BigNat
    byteArrayToBigNat# ba# n0#
      | isTrue# (n#  ==# 0#)    = zeroBigNat
      | isTrue# (baszr# ==# 0#) -- i.e. ba# is multiple of limb-size
      , isTrue# (baszq# ==# n#) = (BN# ba#)
      | True = runS $ \s ->
          let !(# s', mbn@(MBN# mba#) #) = newBigNat# n# s
              !(# s'', ba_sz# #) = getSizeofMutableByteArray# mba# s'
              go = do _ <- svoid (copyByteArray# ba# 0# mba# 0# ba_sz# )
                      unsafeFreezeBigNat# mbn
          in go s''
      where
        !(# baszq#, baszr# #) = quotRemInt# (sizeofByteArray# ba#) GMP_LIMB_BYTES#

        n#  = fmssl (n0# -# 1#)

        -- find most significant set limb, return normalized size
        fmssl i#
          | isTrue# (i# <# 0#)                             = 0#
          | isTrue# (neWord# (indexWordArray# ba# i#) 0##) = i# +# 1#
          | True                                           = fmssl (i# -# 1#)

    -- | Read 'Integer' (without sign) from memory location at @/addr/@ in
    -- base-256 representation.
    --
    -- @'importIntegerFromAddr' /addr/ /size/ /msbf/@
    --
    -- See description of 'importIntegerFromByteArray' for more details.
    --
    -- @since 1.0.0.0
    importIntegerFromAddr :: Addr# -> Word# -> Int# -> IO Integer
    importIntegerFromAddr addr len msbf = IO $ do
        bn <- liftIO (importBigNatFromAddr addr len msbf)
        return (bigNatToInteger bn)

    -- | Version of 'importIntegerFromAddr' constructing a 'BigNat'
    importBigNatFromAddr :: Addr# -> Word# -> Int# -> IO BigNat
    importBigNatFromAddr _ 0## _ = IO (\s -> (# s, zeroBigNat #))
    importBigNatFromAddr addr len0 1# = IO $ do -- MSBF
        W# ofs <- liftIO (c_scan_nzbyte_addr addr 0## len0)
        let len = len0 `minusWord#` ofs
            addr' = addr `plusAddr#` (word2Int# ofs)
        importBigNatFromAddr# addr' len 1#
    importBigNatFromAddr addr len0 _ = IO $ do -- LSBF
        W# len <- liftIO (c_rscan_nzbyte_addr addr 0## len0)
        importBigNatFromAddr# addr len 0#

    foreign import ccall unsafe "integer_gmp_scan_nzbyte"
        c_scan_nzbyte_addr :: Addr# -> Word# -> Word# -> IO Word

    foreign import ccall unsafe "integer_gmp_rscan_nzbyte"
        c_rscan_nzbyte_addr :: Addr# -> Word# -> Word# -> IO Word

    -- | Helper for 'importBigNatFromAddr'
    importBigNatFromAddr# :: Addr# -> Word# -> Int# -> S RealWorld BigNat
    importBigNatFromAddr# _ 0## _ = return zeroBigNat
    importBigNatFromAddr# addr len msbf = do
        mbn@(MBN# mba#) <- newBigNat# n#
        () <- liftIO (c_mpn_import_addr mba# addr 0## len msbf)
        unsafeFreezeBigNat# mbn
      where
        -- n = ceiling(len / SIZEOF_HSWORD), i.e. number of limbs required
        n# = (word2Int# len +# (SIZEOF_HSWORD# -# 1#)) `quotInt#` SIZEOF_HSWORD#

    foreign import ccall unsafe "integer_gmp_mpn_import"
        c_mpn_import_addr :: MutableByteArray# RealWorld -> Addr# -> Word# -> Word#
                          -> Int# -> IO ()

    -- | Read 'Integer' (without sign) from byte-array in base-256 representation.
    --
    -- The call
    --
    -- @'importIntegerFromByteArray' /ba/ /offset/ /size/ /msbf/@
    --
    -- reads
    --
    -- * @/size/@ bytes from the 'ByteArray#' @/ba/@ starting at @/offset/@
    --
    -- * with most significant byte first if @/msbf/@ is @1#@ or least
    --   significant byte first if @/msbf/@ is @0#@, and
    --
    -- * returns a new 'Integer'
    --
    -- @since 1.0.0.0
    importIntegerFromByteArray :: ByteArray# -> Word# -> Word# -> Int# -> Integer
    importIntegerFromByteArray ba ofs len msbf
        = bigNatToInteger (importBigNatFromByteArray ba ofs len msbf)

    -- | Version of 'importIntegerFromByteArray' constructing a 'BigNat'
    importBigNatFromByteArray :: ByteArray# -> Word# -> Word# -> Int# -> BigNat
    importBigNatFromByteArray _  _    0##  _  = zeroBigNat
    importBigNatFromByteArray ba ofs0 len0 1# = runS $ do -- MSBF
        W# ofs <- liftIO (c_scan_nzbyte_bytearray ba ofs0 len0)
        let len = (len0 `plusWord#` ofs0) `minusWord#` ofs
        importBigNatFromByteArray# ba ofs len 1#
    importBigNatFromByteArray ba ofs  len0 _  = runS $ do -- LSBF
        W# len <- liftIO (c_rscan_nzbyte_bytearray ba ofs len0)
        importBigNatFromByteArray# ba ofs len 0#

    foreign import ccall unsafe "integer_gmp_scan_nzbyte"
        c_scan_nzbyte_bytearray :: ByteArray# -> Word# -> Word# -> IO Word

    foreign import ccall unsafe "integer_gmp_rscan_nzbyte"
        c_rscan_nzbyte_bytearray :: ByteArray# -> Word# -> Word# -> IO Word

    -- | Helper for 'importBigNatFromByteArray'
    importBigNatFromByteArray# :: ByteArray# -> Word# -> Word# -> Int#
                               -> S RealWorld BigNat
    importBigNatFromByteArray# _ _ 0## _ = return zeroBigNat
    importBigNatFromByteArray# ba ofs len msbf = do
        mbn@(MBN# mba#) <- newBigNat# n#
        () <- liftIO (c_mpn_import_bytearray mba# ba ofs len msbf)
        unsafeFreezeBigNat# mbn
      where
        -- n = ceiling(len / SIZEOF_HSWORD), i.e. number of limbs required
        n# = (word2Int# len +# (SIZEOF_HSWORD# -# 1#)) `quotInt#` SIZEOF_HSWORD#

    foreign import ccall unsafe "integer_gmp_mpn_import"
        c_mpn_import_bytearray :: MutableByteArray# RealWorld -> ByteArray# -> Word#
                               -> Word# -> Int# -> IO ()

    -- | Test whether all internal invariants are satisfied by 'BigNat' value
    --
    -- Returns @1#@ if valid, @0#@ otherwise.
    --
    -- This operation is mostly useful for test-suites and/or code which
    -- constructs 'Integer' values directly.
    isValidBigNat# :: BigNat -> Int#
    isValidBigNat# (BN# ba#)
      = (szq# ># 0#) `andI#` (szr# ==# 0#) `andI#` isNorm#
      where
        isNorm#
          | isTrue# (szq# ># 1#) = (indexWordArray# ba# (szq# -# 1#)) `neWord#` 0##
          | True                 = 1#

        sz# = sizeofByteArray# ba#

        !(# szq#, szr# #) = quotRemInt# sz# GMP_LIMB_BYTES#

    -- | Version of 'nextPrimeInteger' operating on 'BigNat's
    --
    -- @since 1.0.0.0
    nextPrimeBigNat :: BigNat -> BigNat
    nextPrimeBigNat bn@(BN# ba#) = runS $ do
        mbn@(MBN# mba#) <- newBigNat# n#
        (W# c#) <- liftIO (nextPrime# mba# ba# n#)
        case c# of
            0## -> unsafeFreezeBigNat# mbn
            _   -> unsafeSnocFreezeBigNat# mbn c#
      where
        n# = sizeofBigNat# bn

    foreign import ccall unsafe "integer_gmp_next_prime"
      nextPrime# :: MutableByteArray# RealWorld -> ByteArray# -> GmpSize#
                    -> IO GmpLimb

    ----------------------------------------------------------------------------
    -- monadic combinators for low-level state threading

    type S s a = State# s -> (# State# s, a #)

    infixl 1 >>=
    infixl 1 >>
    infixr 0 $

    {-# INLINE ($) #-}
    ($)                     :: (a -> b) -> a -> b
    f $ x                   =  f x

    {-# INLINE (>>=) #-}
    (>>=) :: S s a -> (a -> S s b) -> S s b
    (>>=) m k = \s -> case m s of (# s', a #) -> k a s'

    {-# INLINE (>>) #-}
    (>>) :: S s a -> S s b -> S s b
    (>>) m k = \s -> case m s of (# s', _ #) -> k s'

    {-# INLINE svoid #-}
    svoid :: (State# s -> State# s) -> S s ()
    svoid m0 = \s -> case m0 s of s' -> (# s', () #)

    {-# INLINE return #-}
    return :: a -> S s a
    return a = \s -> (# s, a #)

    {-# INLINE liftIO #-}
    liftIO :: IO a -> S RealWorld a
    liftIO (IO m) = m

    -- NB: equivalent of GHC.IO.unsafeDupablePerformIO, see notes there
    runS :: S RealWorld a -> a
    runS m = case runRW# m of (# _, a #) -> a

    -- stupid hack
    fail :: [Char] -> S s a
    fail s = return (raise# s)

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

    -- | Internal helper type for "signed" 'BigNat's
    --
    -- This is a useful abstraction for operations which support negative
    -- mp_size_t arguments.
    data SBigNat = NegBN !BigNat | PosBN !BigNat

    -- | Absolute value of 'SBigNat'
    absSBigNat :: SBigNat -> BigNat
    absSBigNat (NegBN bn) = bn
    absSBigNat (PosBN bn) = bn

    -- | /Signed/ limb count. Negative sizes denote negative integers
    ssizeofSBigNat# :: SBigNat -> GmpSize#
    ssizeofSBigNat# (NegBN bn) = negateInt# (sizeofBigNat# bn)
    ssizeofSBigNat# (PosBN bn) = sizeofBigNat# bn

    -- | Construct 'SBigNat' from 'Int#' value
    intToSBigNat# :: Int# -> SBigNat
    intToSBigNat# 0#     = PosBN zeroBigNat
    intToSBigNat# 1#     = PosBN oneBigNat
    intToSBigNat# (-1#)  = NegBN oneBigNat
    intToSBigNat# i# | isTrue# (i# ># 0#) = PosBN (wordToBigNat (int2Word# i#))
                     | True   = PosBN (wordToBigNat (int2Word# (negateInt# i#)))

    -- | Convert 'Integer' into 'SBigNat'
    integerToSBigNat :: Integer -> SBigNat
    integerToSBigNat (S#  i#) = intToSBigNat# i#
    integerToSBigNat (Jp# bn) = PosBN bn
    integerToSBigNat (Jn# bn) = NegBN bn

    -- | Convert 'SBigNat' into 'Integer'
    sBigNatToInteger :: SBigNat -> Integer
    sBigNatToInteger (NegBN bn) = bigNatToNegInteger bn
    sBigNatToInteger (PosBN bn) = bigNatToInteger bn

    ----------------------------------------------------------------------------
    -- misc helpers, some of these should rather be primitives exported by ghc-prim

    cmpW# :: Word# -> Word# -> Ordering
    cmpW# x# y#
      | isTrue# (x# `ltWord#` y#) = LT
      | isTrue# (x# `eqWord#` y#) = EQ
      | True                      = GT
    {-# INLINE cmpW# #-}

    bitWord# :: Int# -> Word#
    bitWord# = uncheckedShiftL# 1##
    {-# INLINE bitWord# #-}

    testBitWord# :: Word# -> Int# -> Int#
    testBitWord# w# i# = (bitWord# i# `and#` w#) `neWord#` 0##
    {-# INLINE testBitWord# #-}

    popCntI# :: Int# -> Int#
    popCntI# i# = word2Int# (popCnt# (int2Word# i#))
    {-# INLINE popCntI# #-}

    -- branchless version
    absI# :: Int# -> Int#
    absI# i# = (i# `xorI#` nsign) -# nsign
      where
        -- nsign = negateInt# (i# <# 0#)
        nsign = uncheckedIShiftRA# i# (WORD_SIZE_IN_BITS# -# 1#)

    -- branchless version
    sgnI# :: Int# -> Int#
    sgnI# x# = (x# ># 0#) -# (x# <# 0#)

    cmpI# :: Int# -> Int# -> Int#
    cmpI# x# y# = (x# ># y#) -# (x# <# y#)

    minI# :: Int# -> Int# -> Int#
    minI# x# y# | isTrue# (x# <=# y#) = x#
                | True                = y#