{-# LANGUAGE CPP #-}
{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE MagicHash #-}
{-# LANGUAGE UnboxedTuples #-}
{-# LANGUAGE NegativeLiterals #-}
{-# LANGUAGE BinaryLiterals #-}
{-# LANGUAGE BlockArguments #-}
{-# LANGUAGE LambdaCase #-}

-- |
-- Module      :  GHC.Num.Integer
-- Copyright   :  (c) Sylvain Henry 2019,
--                (c) Herbert Valerio Riedel 2014
-- License     :  BSD3
--
-- Maintainer  :  sylvain@haskus.fr
-- Stability   :  provisional
-- Portability :  non-portable (GHC Extensions)
--
-- The 'Integer' type.

module GHC.Num.Integer where

#include "MachDeps.h"
#include "WordSize.h"

import GHC.Prim
import GHC.Types
import GHC.Classes
import GHC.Magic
import GHC.Num.Primitives
import GHC.Num.BigNat
import GHC.Num.Natural
import qualified GHC.Num.Backend as Backend

default ()

-- | Arbitrary precision integers. In contrast with fixed-size integral types
-- such as 'Int', the 'Integer' type represents the entire infinite range of
-- integers.
--
-- Integers are stored in a kind of sign-magnitude form, hence do not expect
-- two's complement form when using bit operations.
--
-- If the value is small (fit into an 'Int'), 'IS' constructor is used.
-- Otherwise 'IP' and 'IN' constructors are used to store a 'BigNat'
-- representing respectively the positive or the negative value magnitude.
--
-- Invariant: 'IP' and 'IN' are used iff value doesn't fit in 'IS'
data Integer
   = IS !Int#    -- ^ iff value in @[minBound::'Int', maxBound::'Int']@ range
   | IP !BigNat# -- ^ iff value in @]maxBound::'Int', +inf[@ range
   | IN !BigNat# -- ^ iff value in @]-inf, minBound::'Int'[@ range


-- | Check Integer invariants
integerCheck# :: Integer -> Bool#
integerCheck# (IS  _) = 1#
integerCheck# (IP bn) = bigNatCheck# bn &&# (bn `bigNatGtWord#` INT_MAXBOUND##)
integerCheck# (IN bn) = bigNatCheck# bn &&# (bn `bigNatGtWord#` ABS_INT_MINBOUND##)

-- | Check Integer invariants
integerCheck :: Integer -> Bool
integerCheck i = isTrue# (integerCheck# i)

-- | Integer Zero
integerZero :: Integer
integerZero = IS 0#

-- | Integer One
integerOne :: Integer
integerOne = IS 1#

---------------------------------------------------------------------
-- Conversions
---------------------------------------------------------------------

-- | Create a positive Integer from a BigNat
integerFromBigNat# :: BigNat# -> Integer
integerFromBigNat# !bn
   | bigNatIsZero bn
   = integerZero

   | isTrue# (bn `bigNatLeWord#` INT_MAXBOUND##)
   = IS (word2Int# (bigNatIndex# bn 0#))

   | True
   = IP bn

-- | Create a negative Integer from a BigNat
integerFromBigNatNeg# :: BigNat# -> Integer
integerFromBigNatNeg# !bn
   | bigNatIsZero bn
   = integerZero

   | 1# <- bigNatSize# bn
   , i <- negateInt# (word2Int# (bigNatIndex# bn 0#))
   , isTrue# (i <=# 0#)
   = IS i

   | True
   = IN bn

-- | Create an Integer from a sign-bit and a BigNat
integerFromBigNatSign# :: Int# -> BigNat# -> Integer
integerFromBigNatSign# !sign !bn
   | 0# <- sign
   = integerFromBigNat# bn

   | True
   = integerFromBigNatNeg# bn

-- | Convert an Integer into a sign-bit and a BigNat
integerToBigNatSign# :: Integer -> (# Int#, BigNat# #)
integerToBigNatSign# = \case
   IS x
      | isTrue# (x >=# 0#)
      -> (# 0#, bigNatFromWord# (int2Word# x) #)
      | True
      -> (# 1#, bigNatFromWord# (int2Word# (negateInt# x)) #)
   IP x -> (# 0#, x #)
   IN x -> (# 1#, x #)

-- | Convert an Integer into a BigNat.
--
-- Return 0 for negative Integers.
integerToBigNatClamp# :: Integer -> BigNat#
integerToBigNatClamp# (IP x) = x
integerToBigNatClamp# (IS x)
   | isTrue# (x >=# 0#)     = bigNatFromWord# (int2Word# x)
integerToBigNatClamp# _     = bigNatZero# (# #)

-- | Create an Integer from an Int#
integerFromInt# :: Int# -> Integer
integerFromInt# i = IS i

-- | Create an Integer from an Int
integerFromInt :: Int -> Integer
integerFromInt (I# i) = IS i

-- | Truncates 'Integer' to least-significant 'Int#'
integerToInt# :: Integer -> Int#
{-# NOINLINE integerToInt# #-}
integerToInt# (IS i) = i
integerToInt# (IP b) = word2Int# (bigNatToWord# b)
integerToInt# (IN b) = negateInt# (word2Int# (bigNatToWord# b))

-- | Truncates 'Integer' to least-significant 'Int#'
integerToInt :: Integer -> Int
integerToInt i = I# (integerToInt# i)

-- | Convert a Word# into an Integer
integerFromWord# :: Word# -> Integer
{-# NOINLINE integerFromWord# #-}
integerFromWord# w
   | i <- word2Int# w
   , isTrue# (i >=# 0#)
   = IS i

   | True
   = IP (bigNatFromWord# w)

-- | Convert a Word into an Integer
integerFromWord :: Word -> Integer
integerFromWord (W# w) = integerFromWord# w

-- | Create a negative Integer with the given Word magnitude
integerFromWordNeg# :: Word# -> Integer
integerFromWordNeg# w
  | isTrue# (w `leWord#` ABS_INT_MINBOUND##)
  = IS (negateInt# (word2Int# w))

  | True
  = IN (bigNatFromWord# w)

-- | Create an Integer from a sign and a Word magnitude
integerFromWordSign# :: Int# -> Word# -> Integer
integerFromWordSign# 0# w = integerFromWord# w
integerFromWordSign# _  w = integerFromWordNeg# w

-- | Truncate an Integer into a Word
integerToWord# :: Integer -> Word#
{-# NOINLINE integerToWord# #-}
integerToWord# (IS i)  = int2Word# i
integerToWord# (IP bn) = bigNatToWord# bn
integerToWord# (IN bn) = int2Word# (negateInt# (word2Int# (bigNatToWord# bn)))

-- | Truncate an Integer into a Word
integerToWord :: Integer -> Word
integerToWord !i = W# (integerToWord# i)

-- | Convert a Natural into an Integer
integerFromNatural :: Natural -> Integer
{-# NOINLINE integerFromNatural #-}
integerFromNatural (NS x) = integerFromWord# x
integerFromNatural (NB x) = IP x

-- | Convert a list of Word into an Integer
integerFromWordList :: Bool -> [Word] -> Integer
integerFromWordList True  ws = integerFromBigNatNeg# (bigNatFromWordList ws)
integerFromWordList False ws = integerFromBigNat#    (bigNatFromWordList ws)

-- | Convert an Integer into a Natural
--
-- Return 0 for negative Integers.
integerToNaturalClamp :: Integer -> Natural
{-# NOINLINE integerToNaturalClamp #-}
integerToNaturalClamp (IS x)
   | isTrue# (x <# 0#) = naturalZero
   | True              = naturalFromWord# (int2Word# x)
integerToNaturalClamp (IP x) = naturalFromBigNat# x
integerToNaturalClamp (IN _) = naturalZero

-- | Convert an Integer into a Natural
--
-- Return absolute value
integerToNatural :: Integer -> Natural
{-# NOINLINE integerToNatural #-}
integerToNatural (IS x) = naturalFromWord# (wordFromAbsInt# x)
integerToNatural (IP x) = naturalFromBigNat# x
integerToNatural (IN x) = naturalFromBigNat# x

-- | Convert an Integer into a Natural
--
-- Throw an Underflow exception if input is negative.
integerToNaturalThrow :: Integer -> Natural
{-# NOINLINE integerToNaturalThrow #-}
integerToNaturalThrow (IS x)
  | isTrue# (x <# 0#) = raiseUnderflow
  | True              = naturalFromWord# (int2Word# x)
integerToNaturalThrow (IP x) = naturalFromBigNat# x
integerToNaturalThrow (IN _) = raiseUnderflow

---------------------------------------------------------------------
-- Predicates
---------------------------------------------------------------------

-- | Negative predicate
integerIsNegative# :: Integer -> Bool#
integerIsNegative# (IS i#) = i# <# 0#
integerIsNegative# (IP _)  = 0#
integerIsNegative# (IN _)  = 1#

-- | Negative predicate
integerIsNegative :: Integer -> Bool
integerIsNegative !i = isTrue# (integerIsNegative# i)

-- | Zero predicate
integerIsZero :: Integer -> Bool
integerIsZero (IS 0#) = True
integerIsZero _       = False

-- | One predicate
integerIsOne :: Integer -> Bool
integerIsOne (IS 1#) = True
integerIsOne _       = False

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

-- | Equal predicate.
integerEq :: Integer -> Integer -> Bool
integerEq !x !y = isTrue# (integerEq# x y)

-- | Lower-or-equal predicate.
integerLe :: Integer -> Integer -> Bool
integerLe !x !y = isTrue# (integerLe# x y)

-- | Lower predicate.
integerLt :: Integer -> Integer -> Bool
integerLt !x !y = isTrue# (integerLt# x y)

-- | Greater predicate.
integerGt :: Integer -> Integer -> Bool
integerGt !x !y = isTrue# (integerGt# x y)

-- | Greater-or-equal predicate.
integerGe :: Integer -> Integer -> Bool
integerGe !x !y = isTrue# (integerGe# x y)

-- | Equal predicate.
integerEq# :: Integer -> Integer -> Bool#
integerEq# (IS x) (IS y) = x ==# y
integerEq# (IN x) (IN y) = bigNatEq# x y
integerEq# (IP x) (IP y) = bigNatEq# x y
integerEq# _       _     = 0#

-- | Not-equal predicate.
integerNe# :: Integer -> Integer -> Bool#
integerNe# (IS x) (IS y) = x /=# y
integerNe# (IN x) (IN y) = bigNatNe# x y
integerNe# (IP x) (IP y) = bigNatNe# x y
integerNe# _       _     = 1#

-- | Greater predicate.
integerGt# :: Integer -> Integer -> Bool#
integerGt# (IS x) (IS y)                  = x ># y
integerGt# x y | GT <- integerCompare x y = 1#
integerGt# _ _                            = 0#

-- | Lower-or-equal predicate.
integerLe# :: Integer -> Integer -> Bool#
integerLe# (IS x) (IS y)                  = x <=# y
integerLe# x y | GT <- integerCompare x y = 0#
integerLe# _ _                            = 1#

-- | Lower predicate.
integerLt# :: Integer -> Integer -> Bool#
integerLt# (IS x) (IS y)                  = x <# y
integerLt# x y | LT <- integerCompare x y = 1#
integerLt# _ _                            = 0#

-- | Greater-or-equal predicate.
integerGe# :: Integer -> Integer -> Bool#
integerGe# (IS x) (IS y)                  = x >=# y
integerGe# x y | LT <- integerCompare x y = 0#
integerGe# _ _                            = 1#

instance Eq Integer where
   (==) = integerEq
   (/=) = integerNe

-- | Compare two Integer
integerCompare :: Integer -> Integer -> Ordering
{-# INLINEABLE integerCompare #-}
integerCompare (IS x) (IS y) = compareInt# x y
integerCompare (IP x) (IP y) = bigNatCompare x y
integerCompare (IN x) (IN y) = bigNatCompare y x
integerCompare (IS _) (IP _) = LT
integerCompare (IS _) (IN _) = GT
integerCompare (IP _) (IS _) = GT
integerCompare (IN _) (IS _) = LT
integerCompare (IP _) (IN _) = GT
integerCompare (IN _) (IP _) = LT

instance Ord Integer where
   compare = integerCompare
   (<)     = integerLt
   (<=)    = integerLe
   (>)     = integerGt
   (>=)    = integerGe

---------------------------------------------------------------------
-- Operations
---------------------------------------------------------------------

-- | Subtract one 'Integer' from another.
integerSub :: Integer -> Integer -> Integer
{-# NOINLINE integerSub #-}
integerSub !x      (IS 0#) = x
integerSub (IS x#) (IS y#)
  = case subIntC# x# y# of
    (# z#, 0# #) -> IS z#
    (# 0#, _  #) -> IN (bigNatFromWord2# 1## 0##)
    (# z#, _  #)
      | isTrue# (z# ># 0#)
      -> IN (bigNatFromWord# ( (int2Word# (negateInt# z#))))
      | True
      -> IP (bigNatFromWord# ( (int2Word# z#)))
integerSub (IS x#) (IP y)
  | isTrue# (x# >=# 0#)
  = integerFromBigNatNeg# (bigNatSubWordUnsafe# y (int2Word# x#))
  | True
  = IN (bigNatAddWord# y (int2Word# (negateInt# x#)))
integerSub (IS x#) (IN y)
  | isTrue# (x# >=# 0#)
  = IP (bigNatAddWord# y (int2Word# x#))
  | True
  = integerFromBigNat# (bigNatSubWordUnsafe# y (int2Word# (negateInt# x#)))
integerSub (IP x) (IP y)
  = case bigNatCompare x y of
    LT -> integerFromBigNatNeg# (bigNatSubUnsafe y x)
    EQ -> IS 0#
    GT -> integerFromBigNat# (bigNatSubUnsafe x y)
integerSub (IP x) (IN y) = IP (bigNatAdd x y)
integerSub (IN x) (IP y) = IN (bigNatAdd x y)
integerSub (IN x) (IN y)
  = case bigNatCompare x y of
    LT -> integerFromBigNat# (bigNatSubUnsafe y x)
    EQ -> IS 0#
    GT -> integerFromBigNatNeg# (bigNatSubUnsafe x y)
integerSub (IP x) (IS y#)
  | isTrue# (y# >=# 0#)
  = integerFromBigNat# (bigNatSubWordUnsafe# x (int2Word# y#))
  | True
  = IP (bigNatAddWord# x (int2Word# (negateInt# y#)))
integerSub (IN x) (IS y#)
  | isTrue# (y# >=# 0#)
  = IN (bigNatAddWord# x (int2Word# y#))
  | True
  = integerFromBigNatNeg# (bigNatSubWordUnsafe# x (int2Word# (negateInt# y#)))

-- | Add two 'Integer's
integerAdd :: Integer -> Integer -> Integer
{-# NOINLINE integerAdd #-}
integerAdd !x      (IS 0#) = x
integerAdd (IS 0#) y       = y
integerAdd (IS x#) (IS y#)
  = case addIntC# x# y# of
    (# z#, 0# #) -> IS z#
    (# 0#, _  #) -> IN (bigNatFromWord2# 1## 0##) -- 2*minBound::Int
    (# z#, _  #)
      | isTrue# (z# ># 0#) -> IN (bigNatFromWord# ( (int2Word# (negateInt# z#))))
      | True               -> IP (bigNatFromWord# ( (int2Word# z#)))
integerAdd y@(IS _) x = integerAdd x y
integerAdd (IP x) (IP y) = IP (bigNatAdd x y)
integerAdd (IN x) (IN y) = IN (bigNatAdd x y)
integerAdd (IP x) (IS y#) -- edge-case: @(maxBound+1) + minBound == 0@
  | isTrue# (y# >=# 0#) = IP (bigNatAddWord# x (int2Word# y#))
  | True                = integerFromBigNat# (bigNatSubWordUnsafe# x (int2Word#
                                                              (negateInt# y#)))
integerAdd (IN x) (IS y#) -- edge-case: @(minBound-1) + maxBound == -2@
  | isTrue# (y# >=# 0#) = integerFromBigNatNeg# (bigNatSubWordUnsafe# x (int2Word# y#))
  | True                = IN (bigNatAddWord# x (int2Word# (negateInt# y#)))
integerAdd y@(IN _) x@(IP _) = integerAdd x y
integerAdd (IP x) (IN y)
    = case bigNatCompare x y of
      LT -> integerFromBigNatNeg# (bigNatSubUnsafe y x)
      EQ -> IS 0#
      GT -> integerFromBigNat# (bigNatSubUnsafe x y)

-- | Multiply two 'Integer's
integerMul :: Integer -> Integer -> Integer
{-# NOINLINE integerMul #-}
integerMul !_       (IS 0#)  = IS 0#
integerMul (IS 0#)  _        = IS 0#
integerMul x        (IS 1#)  = x
integerMul (IS 1#)  y        = y
integerMul x        (IS -1#) = integerNegate x
integerMul (IS -1#) y        = integerNegate y
#if __GLASGOW_HASKELL__ < 811
integerMul (IS x)   (IS y)   = case mulIntMayOflo# x y of
   0# -> IS (x *# y)
   _  -> case (# isTrue# (x >=# 0#), isTrue# (y >=# 0#) #) of
      (# False, False #) -> case timesWord2# (int2Word# (negateInt# x))
                                       (int2Word# (negateInt# y)) of
          (# 0##,l #) -> integerFromWord# l
          (# h  ,l #) -> IP (bigNatFromWord2# h l)

      (#  True, False #) -> case timesWord2# (int2Word# x)
                                       (int2Word# (negateInt# y)) of
          (# 0##,l #) -> integerFromWordNeg# l
          (# h  ,l #) -> IN (bigNatFromWord2# h l)

      (# False,  True #) -> case timesWord2# (int2Word# (negateInt# x))
                                       (int2Word# y) of
          (# 0##,l #) -> integerFromWordNeg# l
          (# h  ,l #) -> IN (bigNatFromWord2# h l)

      (#  True,  True #) -> case timesWord2# (int2Word# x)
                                       (int2Word# y) of
          (# 0##,l #) -> integerFromWord# l
          (# h  ,l #) -> IP (bigNatFromWord2# h l)
#else
integerMul (IS x)   (IS y)   = case timesInt2# x y of
   (# 0#, _h, l #) -> IS l
   (# _ ,  h, l #)
      | isTrue# (h >=# 0#)
      -> IP (bigNatFromWord2# (int2Word# h) (int2Word# l))
      | True
      -> let
          -- two's complement of a two-word negative Int:
          --   l' = complement l + 1
          --   h' = complement h + carry
          !(# l',c #) = addWordC# (not# (int2Word# l)) 1##
          !h'         = int2Word# c `plusWord#` not# (int2Word# h)
         in IN (bigNatFromWord2# h' l')
#endif
integerMul x@(IS _) y    = integerMul y x
integerMul (IP x) (IP y) = IP (bigNatMul x y)
integerMul (IP x) (IN y) = IN (bigNatMul x y)
integerMul (IP x) (IS y)
  | isTrue# (y >=# 0#)   = IP (bigNatMulWord# x (int2Word# y))
  | True                 = IN (bigNatMulWord# x (int2Word# (negateInt# y)))
integerMul (IN x) (IN y) = IP (bigNatMul x y)
integerMul (IN x) (IP y) = IN (bigNatMul x y)
integerMul (IN x) (IS y)
  | isTrue# (y >=# 0#)   = IN (bigNatMulWord# x (int2Word# y))
  | True                 = IP (bigNatMulWord# x (int2Word# (negateInt# y)))

-- | Negate 'Integer'.
--
-- One edge-case issue to take into account is that Int's range is not
-- symmetric around 0.  I.e. @minBound+maxBound = -1@
--
-- IP is used iff n > maxBound::Int
-- IN is used iff n < minBound::Int
integerNegate :: Integer -> Integer
{-# NOINLINE integerNegate #-}
integerNegate (IN b)             = IP b
integerNegate (IS INT_MINBOUND#) = IP (bigNatFromWord# ABS_INT_MINBOUND##)
integerNegate (IS i)             = IS (negateInt# i)
integerNegate (IP b)
  | isTrue# (bigNatEqWord# b ABS_INT_MINBOUND##) = IS INT_MINBOUND#
  | True                                         = IN b

{-# RULES
"integerNegate/integerNegate" forall x. integerNegate (integerNegate x) = x
#-}

-- | Compute absolute value of an 'Integer'
integerAbs :: Integer -> Integer
{-# NOINLINE integerAbs #-}
integerAbs   (IN i)     = IP i
integerAbs n@(IP _)     = n
integerAbs n@(IS i)
   | isTrue# (i >=# 0#) = n
   | INT_MINBOUND# <- i = IP (bigNatFromWord# ABS_INT_MINBOUND##)
   | True               = IS (negateInt# i)


-- | Return @-1@, @0@, and @1@ depending on whether argument is
-- negative, zero, or positive, respectively
integerSignum :: Integer -> Integer
integerSignum !j = IS (integerSignum# j)

-- | Return @-1#@, @0#@, and @1#@ depending on whether argument is
-- negative, zero, or positive, respectively
integerSignum# :: Integer -> Int#
integerSignum# (IN _)  = -1#
integerSignum# (IS i#) = sgnI# i#
integerSignum# (IP _ ) =  1#

-- | Count number of set bits. For negative arguments returns
-- the negated population count of the absolute value.
integerPopCount# :: Integer -> Int#
{-# NOINLINE integerPopCount# #-}
integerPopCount# (IS i)
   | isTrue# (i >=# 0#)  = word2Int# (popCntI# i)
   | True                = negateInt# (word2Int# (popCntI# (negateInt# i)))
integerPopCount# (IP bn) = word2Int# (bigNatPopCount# bn)
integerPopCount# (IN bn) = negateInt# (word2Int# (bigNatPopCount# bn))

-- | Positive 'Integer' for which only /n/-th bit is set
integerBit# :: Word# -> Integer
{-# NOINLINE integerBit# #-}
integerBit# i
  | isTrue# (i `ltWord#` (WORD_SIZE_IN_BITS## `minusWord#` 1##))
  = IS (uncheckedIShiftL# 1# (word2Int# i))

  | True = IP (bigNatBit# i)

-- | 'Integer' for which only /n/-th bit is set
integerBit :: Word -> Integer
integerBit (W# i) = integerBit# i

-- | Test if /n/-th bit is set.
--
-- Fake 2's complement for negative values (might be slow)
integerTestBit# :: Integer -> Word# -> Bool#
{-# NOINLINE integerTestBit# #-}
integerTestBit# (IS x) i
   | isTrue# (i `ltWord#` WORD_SIZE_IN_BITS##)
   = testBitI# x i
   | True
   = x <# 0#
integerTestBit# (IP x) i = bigNatTestBit# x i
integerTestBit# (IN x) i
   | isTrue# (iw >=# n)
   = 1#
   -- if all the limbs j with j < iw are null, then we have to consider the
   -- carry of the 2's complement conversion. Otherwise we just have to return
   -- the inverse of the bit test
   | allZ iw = testBitW# (xi `minusWord#` 1##) ib ==# 0#
   | True    = testBitW# xi ib ==# 0#
   where
      !xi  = bigNatIndex# x iw
      !n   = bigNatSize# x
      !iw  = word2Int# (i `uncheckedShiftRL#` WORD_SIZE_BITS_SHIFT#)
      !ib  = i `and#` WORD_SIZE_BITS_MASK##

      allZ 0# = True
      allZ j | isTrue# (bigNatIndex# x (j -# 1#) `eqWord#` 0##) = allZ (j -# 1#)
             | True                 = False

-- | Test if /n/-th bit is set. For negative Integers it tests the n-th bit of
-- the negated argument.
--
-- Fake 2's complement for negative values (might be slow)
integerTestBit :: Integer -> Word -> Bool
integerTestBit !i (W# n) = isTrue# (integerTestBit# i n)

-- | Shift-right operation
--
-- Fake 2's complement for negative values (might be slow)
integerShiftR# :: Integer -> Word# -> Integer
{-# NOINLINE integerShiftR# #-}
integerShiftR# !x      0## = x
integerShiftR# (IS i)  n   = IS (iShiftRA# i (word2Int# n))
  where
    iShiftRA# a b
      | isTrue# (b >=# WORD_SIZE_IN_BITS#) = (a <# 0#) *# (-1#)
      | True                               = a `uncheckedIShiftRA#` b
integerShiftR# (IP bn) n   = integerFromBigNat# (bigNatShiftR# bn n)
integerShiftR# (IN bn) n   =
   case integerFromBigNatNeg# (bigNatShiftRNeg# bn n) of
      IS 0# -> IS -1#
      r     -> r

-- | Shift-right operation
--
-- Fake 2's complement for negative values (might be slow)
integerShiftR :: Integer -> Word -> Integer
integerShiftR !x (W# w) = integerShiftR# x w

-- | Shift-left operation
integerShiftL# :: Integer -> Word# -> Integer
{-# NOINLINE integerShiftL# #-}
integerShiftL# !x      0## = x
integerShiftL# (IS 0#) _   = IS 0#
integerShiftL# (IS 1#) n   = integerBit# n
integerShiftL# (IS i)  n
  | isTrue# (i >=# 0#) = integerFromBigNat#    (bigNatShiftL# (bigNatFromWord# (int2Word# i)) n)
  | True               = integerFromBigNatNeg# (bigNatShiftL# (bigNatFromWord# (int2Word# (negateInt# i))) n)
integerShiftL# (IP bn) n   = IP (bigNatShiftL# bn n)
integerShiftL# (IN bn) n   = IN (bigNatShiftL# bn n)

-- | Shift-left operation
--
-- Remember that bits are stored in sign-magnitude form, hence the behavior of
-- negative Integers is different from negative Int's behavior.
integerShiftL :: Integer -> Word -> Integer
integerShiftL !x (W# w) = integerShiftL# x w

-- | Bitwise OR operation
--
-- Fake 2's complement for negative values (might be slow)
integerOr :: Integer -> Integer -> Integer
{-# NOINLINE integerOr #-}
integerOr a b = case a of
   IS  0# -> b
   IS -1# -> IS -1#
   IS  x  -> case b of
               IS  0# -> a
               IS -1# -> IS -1#
               IS  y  -> IS (orI# x y)
               IP  y
                  | isTrue# (x >=# 0#) -> IP (bigNatOrWord# y (int2Word# x))
                  | True               -> integerFromBigNatNeg#
                                             (bigNatAddWord#
                                                (bigNatAndNot -- use De Morgan's laws
                                                   (bigNatFromWord#
                                                      (int2Word# (negateInt# x) `minusWord#` 1##))
                                                   y)
                                                1##)
               IN y
                  | isTrue# (x >=# 0#) -> integerFromBigNatNeg#
                                             (bigNatAddWord#
                                                (bigNatAndNotWord# -- use De Morgan's laws
                                                   (bigNatSubWordUnsafe# y 1##)
                                                   (int2Word# x))
                                                1##)
                  | True               -> integerFromBigNatNeg#
                                             (bigNatAddWord#
                                                (bigNatAndWord#  -- use De Morgan's laws
                                                   (bigNatSubWordUnsafe# y 1##)
                                                   (int2Word# (negateInt# x) `minusWord#` 1##))
                                                1##)
   IP  x  -> case b of
               IS _ -> integerOr b a
               IP y -> IP (bigNatOr x y)
               IN y -> integerFromBigNatNeg#
                        (bigNatAddWord#
                           (bigNatAndNot -- use De Morgan's laws
                              (bigNatSubWordUnsafe# y 1##)
                              x)
                           1##)
   IN  x  -> case b of
               IS _ -> integerOr b a
               IN y -> integerFromBigNatNeg#
                        (bigNatAddWord#
                           (bigNatAnd  -- use De Morgan's laws
                              (bigNatSubWordUnsafe# x 1##)
                              (bigNatSubWordUnsafe# y 1##))
                           1##)
               IP y -> integerFromBigNatNeg#
                        (bigNatAddWord#
                           (bigNatAndNot -- use De Morgan's laws
                              (bigNatSubWordUnsafe# x 1##)
                              y)
                           1##)


-- | Bitwise XOR operation
--
-- Fake 2's complement for negative values (might be slow)
integerXor :: Integer -> Integer -> Integer
{-# NOINLINE integerXor #-}
integerXor a b = case a of
   IS  0# -> b
   IS -1# -> integerComplement b
   IS x   -> case b of
               IS  0# -> a
               IS -1# -> integerComplement a
               IS y   -> IS (xorI# x y)
               IP y
                  | isTrue# (x >=# 0#) -> integerFromBigNat# (bigNatXorWord# y (int2Word# x))
                  | True               -> integerFromBigNatNeg#
                                             (bigNatAddWord#
                                                (bigNatXorWord#
                                                   y
                                                   (int2Word# (negateInt# x) `minusWord#` 1##))
                                                1##)
               IN y
                  | isTrue# (x >=# 0#) -> integerFromBigNatNeg#
                                             (bigNatAddWord#
                                                (bigNatXorWord#
                                                   (bigNatSubWordUnsafe# y 1##)
                                                   (int2Word# x))
                                                1##)
                  | True               -> integerFromBigNat#
                                             (bigNatXorWord# -- xor (not x) (not y) = xor x y
                                                (bigNatSubWordUnsafe# y 1##)
                                                (int2Word# (negateInt# x) `minusWord#` 1##))
   IP x   -> case b of
               IS _ -> integerXor b a
               IP y -> integerFromBigNat# (bigNatXor x y)
               IN y -> integerFromBigNatNeg#
                        (bigNatAddWord#
                           (bigNatXor
                              x
                              (bigNatSubWordUnsafe# y 1##))
                           1##)
   IN x   -> case b of
               IS _ -> integerXor b a
               IN y -> integerFromBigNat#
                        (bigNatXor -- xor (not x) (not y) = xor x y
                           (bigNatSubWordUnsafe# x 1##)
                           (bigNatSubWordUnsafe# y 1##))
               IP y -> integerFromBigNatNeg#
                        (bigNatAddWord#
                           (bigNatXor
                              y
                              (bigNatSubWordUnsafe# x 1##))
                           1##)



-- | Bitwise AND operation
--
-- Fake 2's complement for negative values (might be slow)
integerAnd :: Integer -> Integer -> Integer
{-# NOINLINE integerAnd #-}
integerAnd a b = case a of
   IS 0#  -> IS 0#
   IS -1# -> b
   IS x   -> case b of
               IS  0# -> IS 0#
               IS -1# -> a
               IS y   -> IS (andI# x y)
               IP y   -> integerFromBigNat# (bigNatAndInt# y x)
               IN y
                  | isTrue# (x >=# 0#) -> integerFromWord# (int2Word# x `andNot#` (indexWordArray# y 0# `minusWord#` 1##))
                  | True               -> integerFromBigNatNeg#
                                             (bigNatAddWord#
                                                (bigNatOrWord#  -- use De Morgan's laws
                                                   (bigNatSubWordUnsafe# y 1##)
                                                   (wordFromAbsInt# x `minusWord#` 1##))
                                                1##)
   IP x   -> case b of
               IS _ -> integerAnd b a
               IP y -> integerFromBigNat# (bigNatAnd x y)
               IN y -> integerFromBigNat# (bigNatAndNot x (bigNatSubWordUnsafe# y 1##))
   IN x   -> case b of
               IS _ -> integerAnd b a
               IN y -> integerFromBigNatNeg#
                        (bigNatAddWord#
                           (bigNatOr  -- use De Morgan's laws
                              (bigNatSubWordUnsafe# x 1##)
                              (bigNatSubWordUnsafe# y 1##))
                           1##)
               IP y -> integerFromBigNat# (bigNatAndNot y (bigNatSubWordUnsafe# x 1##))



-- | Binary complement of the
integerComplement :: Integer -> Integer
{-# NOINLINE integerComplement #-}
integerComplement (IS x) = IS (notI# x)
integerComplement (IP x) = IN (bigNatAddWord# x 1##)
integerComplement (IN x) = IP (bigNatSubWordUnsafe# x 1##)


-- | Simultaneous 'integerQuot' and 'integerRem'.
--
-- Divisor must be non-zero otherwise the GHC runtime will terminate
-- with a division-by-zero fault.
integerQuotRem# :: Integer -> Integer -> (# Integer, Integer #)
{-# NOINLINE integerQuotRem# #-}
integerQuotRem# !n      (IS 1#) = (# n, IS 0# #)
integerQuotRem# !n     (IS -1#) = let !q = integerNegate n in (# q, (IS 0#) #)
integerQuotRem# !_      (IS 0#) = case raiseDivZero of
                                    !_ -> (# IS 0#, IS 0# #)
                                    -- see Note [ghc-bignum exceptions] in GHC.Num.Primitives
integerQuotRem# (IS 0#) _       = (# IS 0#, IS 0# #)
integerQuotRem# (IS n#) (IS d#) = case quotRemInt# n# d# of
    (# q#, r# #) -> (# IS q#, IS r# #)
integerQuotRem# (IP n)  (IP d)  = case bigNatQuotRem# n d of
    (# q, r #) -> (# integerFromBigNat# q, integerFromBigNat# r #)
integerQuotRem# (IP n)  (IN d)  = case bigNatQuotRem# n d of
    (# q, r #) -> (# integerFromBigNatNeg# q, integerFromBigNat# r #)
integerQuotRem# (IN n)  (IN d)  = case bigNatQuotRem# n d of
    (# q, r #) -> (# integerFromBigNat# q, integerFromBigNatNeg# r #)
integerQuotRem# (IN n)  (IP d)  = case bigNatQuotRem# n d of
    (# q, r #) -> (# integerFromBigNatNeg# q, integerFromBigNatNeg# r #)
integerQuotRem# (IP n)  (IS d#)
  | isTrue# (d# >=# 0#) = case bigNatQuotRemWord# n (int2Word# d#) of
      (# q, r# #) -> (# integerFromBigNat# q, integerFromWord# r# #)
  | True                = case bigNatQuotRemWord# n (int2Word# (negateInt# d#)) of
      (# q, r# #) -> (# integerFromBigNatNeg# q, integerFromWord# r# #)
integerQuotRem# (IN n)  (IS d#)
  | isTrue# (d# >=# 0#) = case bigNatQuotRemWord# n (int2Word# d#) of
      (# q, r# #) -> (# integerFromBigNatNeg# q, integerFromWordNeg# r# #)
  | True                = case bigNatQuotRemWord# n (int2Word# (negateInt# d#)) of
      (# q, r# #) -> (# integerFromBigNat# q, integerFromWordNeg# r# #)
integerQuotRem# n@(IS _) (IN _) = (# IS 0#, n #) -- since @n < d@
integerQuotRem# n@(IS n#) (IP d) -- need to account for (IS minBound)
    | isTrue# (n# ># 0#)                                    = (# IS 0#, n #)
    | isTrue# (bigNatGtWord# d (int2Word# (negateInt# n#))) = (# IS 0#, n #)
    | True {- abs(n) == d -}                                = (# IS -1#, IS 0# #)

-- | Simultaneous 'integerQuot' and 'integerRem'.
--
-- Divisor must be non-zero otherwise the GHC runtime will terminate
-- with a division-by-zero fault.
integerQuotRem :: Integer -> Integer -> (Integer, Integer)
integerQuotRem !x !y = case integerQuotRem# x y of
   (# q, r #) -> (q, r)


integerQuot :: Integer -> Integer -> Integer
{-# NOINLINE integerQuot #-}
integerQuot !n      (IS 1#)  = n
integerQuot !n      (IS -1#) = integerNegate n
integerQuot !_      (IS 0#)  = raiseDivZero
integerQuot (IS 0#) _        = IS 0#
integerQuot (IS n#) (IS d#)  = IS (quotInt# n# d#)
integerQuot (IP n)  (IS d#)
  | isTrue# (d# >=# 0#) = integerFromBigNat#    (bigNatQuotWord# n (int2Word# d#))
  | True                = integerFromBigNatNeg# (bigNatQuotWord# n
                                              (int2Word# (negateInt# d#)))
integerQuot (IN n)   (IS d#)
  | isTrue# (d# >=# 0#) = integerFromBigNatNeg# (bigNatQuotWord# n (int2Word# d#))
  | True                = integerFromBigNat#    (bigNatQuotWord# n
                                              (int2Word# (negateInt# d#)))
integerQuot (IP n) (IP d) = integerFromBigNat#    (bigNatQuot n d)
integerQuot (IP n) (IN d) = integerFromBigNatNeg# (bigNatQuot n d)
integerQuot (IN n) (IP d) = integerFromBigNatNeg# (bigNatQuot n d)
integerQuot (IN n) (IN d) = integerFromBigNat#    (bigNatQuot n d)
integerQuot n d = case integerQuotRem# n d of (# q, _ #) -> q

integerRem :: Integer -> Integer -> Integer
{-# NOINLINE integerRem #-}
integerRem !_       (IS 1#) = IS 0#
integerRem _       (IS -1#) = IS 0#
integerRem _        (IS 0#) = IS (remInt# 0# 0#)
integerRem (IS 0#) _        = IS 0#
integerRem (IS n#) (IS d#) = IS (remInt# n# d#)
integerRem (IP n)  (IS d#)
    = integerFromWord#    (bigNatRemWord# n (int2Word# (absI# d#)))
integerRem (IN n)  (IS d#)
    = integerFromWordNeg# (bigNatRemWord# n (int2Word# (absI# d#)))
integerRem (IP n)  (IP d)  = integerFromBigNat#    (bigNatRem n d)
integerRem (IP n)  (IN d)  = integerFromBigNat#    (bigNatRem n d)
integerRem (IN n)  (IP d)  = integerFromBigNatNeg# (bigNatRem n d)
integerRem (IN n)  (IN d)  = integerFromBigNatNeg# (bigNatRem n d)
integerRem n d = case integerQuotRem# n d of (# _, r #) -> r


-- | Simultaneous 'integerDiv' and 'integerMod'.
--
-- Divisor must be non-zero otherwise the GHC runtime will terminate
-- with a division-by-zero fault.
integerDivMod# :: Integer -> Integer -> (# Integer, Integer #)
{-# NOINLINE integerDivMod# #-}
integerDivMod# !n !d
  | isTrue# (integerSignum# r ==# negateInt# (integerSignum# d))
     = let !q' = integerSub q (IS 1#)
           !r' = integerAdd r d
       in (# q', r' #)
  | True = qr
  where
    !qr@(# q, r #) = integerQuotRem# n d

-- | Simultaneous 'integerDiv' and 'integerMod'.
--
-- Divisor must be non-zero otherwise the GHC runtime will terminate
-- with a division-by-zero fault.
integerDivMod :: Integer -> Integer -> (Integer, Integer)
integerDivMod !n !d = case integerDivMod# n d of
   (# q,r #) -> (q,r)


integerDiv :: Integer -> Integer -> Integer
{-# NOINLINE integerDiv #-}
integerDiv !n !d
   -- same-sign ops can be handled by more efficient 'integerQuot'
   | isTrue# (integerIsNegative# n ==# integerIsNegative# d) = integerQuot n d
   | True = case integerDivMod# n d of (# q, _ #) -> q


integerMod :: Integer -> Integer -> Integer
{-# NOINLINE integerMod #-}
integerMod !n !d
   -- same-sign ops can be handled by more efficient 'integerRem'
   | isTrue# (integerIsNegative# n ==# integerIsNegative# d) = integerRem n d
   | True = case integerDivMod# n d of (# _, r #) -> r

-- | Compute greatest common divisor.
integerGcd :: Integer -> Integer -> Integer
{-# NOINLINE integerGcd #-}
integerGcd (IS 0#)  !b       = integerAbs b
integerGcd a        (IS 0#)  = integerAbs a
integerGcd (IS 1#)  _        = IS 1#
integerGcd (IS -1#) _        = IS 1#
integerGcd _        (IS 1#)  = IS 1#
integerGcd _        (IS -1#) = IS 1#
integerGcd (IS a)   (IS b)   = integerFromWord# (gcdWord#
                                 (int2Word# (absI# a))
                                 (int2Word# (absI# b)))
integerGcd a@(IS _) b        = integerGcd b a
integerGcd (IN a)   b        = integerGcd (IP a) b
integerGcd (IP a)   (IP b)   = integerFromBigNat# (bigNatGcd a b)
integerGcd (IP a)   (IN b)   = integerFromBigNat# (bigNatGcd a b)
integerGcd (IP a)   (IS b)   = integerFromWord# (bigNatGcdWord# a (int2Word# (absI# b)))

-- | Compute least common multiple.
integerLcm :: Integer -> Integer -> Integer
{-# NOINLINE integerLcm #-}
integerLcm (IS 0#) !_  = IS 0#
integerLcm (IS 1#)  b  = integerAbs b
integerLcm (IS -1#) b  = integerAbs b
integerLcm _ (IS 0#)   = IS 0#
integerLcm a (IS 1#)   = integerAbs a
integerLcm a (IS -1#)  = integerAbs a
integerLcm a b         = (aa `integerQuot` (aa `integerGcd` ab)) `integerMul` ab
  where                   -- TODO: use extended GCD to get a's factor directly
    aa = integerAbs a
    ab = integerAbs b

-- | Square a Integer
integerSqr :: Integer -> Integer
integerSqr !a = integerMul a a


-- | Base 2 logarithm (floor)
--
-- For numbers <= 0, return 0
integerLog2# :: Integer -> Word#
integerLog2# (IS i)
   | isTrue# (i <=# 0#) = 0##
   | True               = wordLog2# (int2Word# i)
integerLog2# (IN _)     = 0##
integerLog2# (IP b)     = bigNatLog2# b

-- | Base 2 logarithm (floor)
--
-- For numbers <= 0, return 0
integerLog2 :: Integer -> Word
integerLog2 !i = W# (integerLog2# i)

-- | Logarithm (floor) for an arbitrary base
--
-- For numbers <= 0, return 0
integerLogBaseWord# :: Word# -> Integer -> Word#
integerLogBaseWord# base !i
   | integerIsNegative i = 0##
   | True                = naturalLogBaseWord# base (integerToNatural i)

-- | Logarithm (floor) for an arbitrary base
--
-- For numbers <= 0, return 0
integerLogBaseWord :: Word -> Integer -> Word
integerLogBaseWord (W# base) !i = W# (integerLogBaseWord# base i)

-- | Logarithm (floor) for an arbitrary base
--
-- For numbers <= 0, return 0
integerLogBase# :: Integer -> Integer -> Word#
integerLogBase# !base !i
   | integerIsNegative i = 0##
   | True                = naturalLogBase# (integerToNatural base)
                                           (integerToNatural i)

-- | Logarithm (floor) for an arbitrary base
--
-- For numbers <= 0, return 0
integerLogBase :: Integer -> Integer -> Word
integerLogBase !base !i = W# (integerLogBase# base i)

-- | Indicate if the value is a power of two and which one
integerIsPowerOf2# :: Integer -> (# (# #) | Word# #)
integerIsPowerOf2# (IS i)
   | isTrue# (i <=# 0#) = (# (# #) | #)
   | True               = wordIsPowerOf2# (int2Word# i)
integerIsPowerOf2# (IN _) = (# (# #) | #)
integerIsPowerOf2# (IP w) = bigNatIsPowerOf2# w

-- | Convert an Int64# into an Integer
integerFromInt64# :: Int64# -> Integer
{-# NOINLINE integerFromInt64# #-}
integerFromInt64# i
  | isTrue# ((i `leInt64#` intToInt64#  INT_MAXBOUND#)
      &&# (i `geInt64#` intToInt64# INT_MINBOUND#))
  = IS (int64ToInt# i)

  | isTrue# (i `geInt64#` intToInt64# 0#)
  = IP (bigNatFromWord64# (int64ToWord64# i))

  | True
  = IN (bigNatFromWord64# (int64ToWord64# (negateInt64# i)))

-- | Convert a Word64# into an Integer
integerFromWord64# :: Word64# -> Integer
{-# NOINLINE integerFromWord64# #-}
integerFromWord64# !w
  | isTrue# (w `leWord64#` wordToWord64# INT_MAXBOUND##)
  = IS (int64ToInt# (word64ToInt64# w))
  | True
  = IP (bigNatFromWord64# w)

-- | Convert an Integer into an Int64#
integerToInt64# :: Integer -> Int64#
{-# NOINLINE integerToInt64# #-}
integerToInt64# (IS i) = intToInt64# i
integerToInt64# (IP b) = word64ToInt64# (bigNatToWord64# b)
integerToInt64# (IN b) = negateInt64# (word64ToInt64# (bigNatToWord64# b))

-- | Convert an Integer into a Word64#
integerToWord64# :: Integer -> Word64#
{-# NOINLINE integerToWord64# #-}
integerToWord64# (IS i) = int64ToWord64# (intToInt64# i)
integerToWord64# (IP b) = bigNatToWord64# b
integerToWord64# (IN b) = int64ToWord64# (negateInt64# (word64ToInt64# (bigNatToWord64# b)))

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

-- | Decode a Double# into (# Integer mantissa, Int# exponent #)
integerDecodeDouble# :: Double# -> (# Integer, Int# #)
{-# INLINE integerDecodeDouble# #-} -- decodeDouble_Int64# is constant-folded
                                    -- in GHC.Core.Opt.ConstantFold
integerDecodeDouble# !x = case decodeDouble_Int64# x of
                            (# m, e #) -> (# integerFromInt64# m, e #)

-- | Encode (# Integer mantissa, Int# exponent #) into a Double#
integerEncodeDouble# :: Integer -> Int# -> Double#
{-# NOINLINE integerEncodeDouble# #-}
integerEncodeDouble# (IS i) 0# = int2Double# i
integerEncodeDouble# (IS i) e  = intEncodeDouble# i e
integerEncodeDouble# (IP b) e  = bigNatEncodeDouble# b e
integerEncodeDouble# (IN b) e  = negateDouble# (bigNatEncodeDouble# b e)

-- | Encode (Integer mantissa, Int exponent) into a Double
integerEncodeDouble :: Integer -> Int -> Double
integerEncodeDouble !m (I# e)  = D# (integerEncodeDouble# m e)

-- | Encode (# Integer mantissa, Int# exponent #) into a Float#
--
-- TODO: Not sure if it's worth to write 'Float' optimized versions here
integerEncodeFloat# :: Integer -> Int# -> Float#
{-# NOINLINE integerEncodeFloat# #-}
integerEncodeFloat# !m e  = double2Float# (integerEncodeDouble# m e)

-- | Compute the number of digits of the Integer (without the sign) in the given base.
--
-- `base` must be > 1
integerSizeInBase# :: Word# -> Integer -> Word#
integerSizeInBase# base (IS i) = wordSizeInBase# base (int2Word# (absI# i))
integerSizeInBase# base (IP n) = bigNatSizeInBase# base n
integerSizeInBase# base (IN n) = bigNatSizeInBase# base n

-- | Write an 'Integer' (without sign) to @/addr/@ in base-256 representation
-- and return the number of bytes written.
--
-- The endianness is selected with the Bool# parameter: write most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
integerToAddr# :: Integer -> Addr# -> Bool# -> State# s -> (# State# s, Word# #)
integerToAddr# (IS i) = wordToAddr# (int2Word# (absI# i))
integerToAddr# (IP n) = bigNatToAddr# n
integerToAddr# (IN n) = bigNatToAddr# n

-- | Write an 'Integer' (without sign) to @/addr/@ in base-256 representation
-- and return the number of bytes written.
--
-- The endianness is selected with the Bool# parameter: write most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
integerToAddr :: Integer -> Addr# -> Bool# -> IO Word
integerToAddr a addr e = IO \s -> case integerToAddr# a addr e s of
   (# s', w #) -> (# s', W# w #)

-- | Read an 'Integer' (without sign) in base-256 representation from an Addr#.
--
-- The size is given in bytes.
--
-- The endianness is selected with the Bool# parameter: most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
--
-- Null higher limbs are automatically trimed.
integerFromAddr# :: Word# -> Addr# -> Bool# -> State# s -> (# State# s, Integer #)
integerFromAddr# sz addr e s =
   case bigNatFromAddr# sz addr e s of
      (# s', n #) -> (# s', integerFromBigNat# n #)

-- | Read an 'Integer' (without sign) in base-256 representation from an Addr#.
--
-- The size is given in bytes.
--
-- The endianness is selected with the Bool# parameter: most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
--
-- Null higher limbs are automatically trimed.
integerFromAddr :: Word# -> Addr# -> Bool# -> IO Integer
integerFromAddr sz addr e = IO (integerFromAddr# sz addr e)



-- | Write an 'Integer' (without sign) in base-256 representation and return the
-- number of bytes written.
--
-- The endianness is selected with the Bool# parameter: most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
integerToMutableByteArray# :: Integer -> MutableByteArray# s -> Word# -> Bool# -> State# s -> (# State# s, Word# #)
integerToMutableByteArray# (IS i) = wordToMutableByteArray# (int2Word# (absI# i))
integerToMutableByteArray# (IP a) = bigNatToMutableByteArray# a
integerToMutableByteArray# (IN a) = bigNatToMutableByteArray# a

-- | Write an 'Integer' (without sign) in base-256 representation and return the
-- number of bytes written.
--
-- The endianness is selected with the Bool# parameter: most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
integerToMutableByteArray :: Integer -> MutableByteArray# RealWorld -> Word# -> Bool# -> IO Word
integerToMutableByteArray i mba w e = IO \s -> case integerToMutableByteArray# i mba w e s of
   (# s', r #) -> (# s', W# r #)

-- | Read an 'Integer' (without sign) in base-256 representation from a ByteArray#.
--
-- The size is given in bytes.
--
-- The endianness is selected with the Bool# parameter: most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
--
-- Null higher limbs are automatically trimed.
integerFromByteArray# :: Word# -> ByteArray# -> Word# -> Bool# -> State# s -> (# State# s, Integer #)
integerFromByteArray# sz ba off e s = case bigNatFromByteArray# sz ba off e s of
   (# s', a #) -> (# s', integerFromBigNat# a #)

-- | Read an 'Integer' (without sign) in base-256 representation from a ByteArray#.
--
-- The size is given in bytes.
--
-- The endianness is selected with the Bool# parameter: most significant
-- byte first (big-endian) if @1#@ or least significant byte first
-- (little-endian) if @0#@.
--
-- Null higher limbs are automatically trimed.
integerFromByteArray :: Word# -> ByteArray# -> Word# -> Bool# -> Integer
integerFromByteArray sz ba off e = case runRW# (integerFromByteArray# sz ba off e) of
   (# _, i #) -> i


-- | Get the extended GCD of two integers.
--
-- `integerGcde# a b` returns (# g,x,y #) where
--    * ax + by = g = |gcd a b|
integerGcde#
   :: Integer
   -> Integer
   -> (# Integer, Integer, Integer #)
integerGcde# a b
   | integerIsZero a && integerIsZero b    =     (# integerZero, integerZero, integerZero #)
   | integerIsZero a                       = fix (# b          , integerZero, integerOne #)
   | integerIsZero b                       = fix (# a          , integerOne,  integerZero #)
   | integerAbs a `integerEq` integerAbs b = fix (# b          , integerZero, integerOne #)
   | True                                  = Backend.integer_gcde a b
   where
      -- returned "g" must be positive
      fix (# g, x, y #)
         | integerIsNegative g = (# integerNegate g, integerNegate x, integerNegate y #)
         | True                = (# g,x,y #)

-- | Get the extended GCD of two integers.
--
-- `integerGcde a b` returns (g,x,y) where
--    * ax + by = g = |gcd a b|
integerGcde
   :: Integer
   -> Integer
   -> ( Integer, Integer, Integer)
integerGcde a b = case integerGcde# a b of
   (# g,x,y #) -> (g,x,y)


-- | Computes the modular inverse.
--
-- I.e. y = integerRecipMod# x m
--        = x^(-1) `mod` m
--
-- with 0 < y < |m|
--
integerRecipMod#
   :: Integer
   -> Natural
   -> (# Natural | () #)
integerRecipMod# x m
   | integerIsZero x = (# | () #)
   | naturalIsZero m = (# | () #)
   | naturalIsOne  m = (# | () #)
   | True            = Backend.integer_recip_mod x m


-- | Computes the modular exponentiation.
--
-- I.e. y = integer_powmod b e m
--        = b^e `mod` m
--
-- with 0 <= y < abs m
--
-- If e is negative, we use `integerRecipMod#` to try to find a modular
-- multiplicative inverse (which may not exist).
integerPowMod# :: Integer -> Integer -> Natural -> (# Natural | () #)
integerPowMod# !b !e !m
   | naturalIsZero m     = (# | () #)
   | naturalIsOne  m     = (# naturalZero | #)
   | integerIsZero e     = (# naturalOne  | #)
   | integerIsZero b     = (# naturalZero | #)
   | integerIsOne  b     = (# naturalOne  | #)
     -- when the exponent is negative, try to find the modular multiplicative
     -- inverse and use it instead
   | integerIsNegative e = case integerRecipMod# b m of
      (#    | () #) -> (# | () #)
      (# b' |    #) -> integerPowMod#
                        (integerFromNatural b')
                        (integerNegate e)
                        m

     -- e > 0 by cases above
   | True = (# Backend.integer_powmod b (integerToNatural e) m | #)


{-
Note [Optimising conversions between numeric types]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Converting between numeric types is very common in Haskell codes.  Suppose that
we have N inter-convertible numeric types (Word, Word8, Word32, Int, etc.).

- We don't want to have to use one conversion function per pair of types as that
would require N^2 functions: wordToWord8, wordToInt, word8ToWord32...

- The following kind of class would allow us to have a single conversion
function but at the price of N^2 instances and of the use of
MultiParamTypeClasses extension.

    class Convert a b where
      convert :: a -> b

So what we do instead is that we use the Integer type (signed, unbounded) as a
passthrough type to perform every conversion. Hence we only need to define two
functions per numeric type:

  class Integral a where
    toInteger :: a -> Integer

  class Num a where
    fromInteger :: Integer -> a

These classes have a single parameter and can be derived automatically (e.g. for
newtypes). So we don't even have to define 2*N instances. For example, all the
instances for the types in Foreign.C.Types (CChar, CShort, CInt, CUInt, etc.)
are automatically derived from the instances for Word, Int, Word8, Word16, etc.

Finally we can define a generic conversion function:

  -- in the Prelude
  fromIntegral :: (Integral a, Num b) => a -> b
  fromIntegral = fromInteger . toInteger

Efficient conversions
~~~~~~~~~~~~~~~~~~~~~

An issue with this approach is that performance might be terrible. E.g.
converting an Int into a Word, which is a no-op at the machine level, becomes
costly when performed via `fromIntegral` or any similar function because an
intermediate Integer has to be allocated in the heap to perform the conversion.

A solution is to bless one particular `fromIntegral`-like function and to use
rewrite rules to replace it with a more efficient function when both types are
known. This is what was done in the past, see next section. We use another
approach nowadays:

Notice that the set of primitive operations to convert from and to Integer and
Natural is pretty small:

  - Natural <-> Word#/BigNat#
  - Integer <-> Int#/Word#/Natural/BigNat# (+ Int64#/Word64# on 32-bit arch)

For example, we have the following primitives:
  - integerToWord#   :: Integer -> Word#
  - integerFromWord# :: Word# -> Integer
  - integerToInt#    :: Integer -> Int#
  - ...

Compared to optimising `fromIntegral :: (Integral a, Num b) => a -> b` where `a`
and `b` are arbitrary, we only have to write rewrite rules for the concrete
types that can be converted from and to Natural/Integer. All the other ones
necessarily pass through these concrete types!

For example we have the following rules:
    integerToWord# (integerFromWord# x) ===> x
    integerToInt# (integerFromWord# x)  ===> word2Int# x

But we don't need rules to handle conversion from/to e.g. Word32# because there
is no Word32#-to-Integer primitive: Word32# must be converted into something
else first (e.g. Word#) for which we have rules.

We rely on inlining of fromInteger/toInteger and on other transformations (e.g.
float-in) to make these rules likely to fire. It seems to work well in practice.

Example 1: converting an Int into a Word

  fromIntegral @Int @Word x

  ===> {inline fromIntegral}
  fromInteger @Word (toInteger @Int x)

  ===> {inline fromInteger and toInteger}
  W# (integerToWord# (case x of { I# x# -> IS x# }))

  ===> {float-in}
  case x of { I# x# -> W# (integerToWord# (IS x#)) }

  ===> {rewrite rule for "integerToWord# . IS"}
  case x of { I# x# -> W# (int2Word# x#) }


Example 2: converting an Int8 into a Word32

  fromIntegral @Int8 @Word32 x

  ===> {inline fromIntegral}
  fromInteger @Word32 (toInteger @Int8 x)

  ===> {inline fromInteger and toInteger}
  W32# (wordToWord32# (integerToWord# (case x of { I8# x# -> IS (int8ToInt# x#) })))

  ===> {float-in}
  case x of { I8# x# -> W32# (wordToWord32# (integerToWord# (IS (int8ToInt# x#)))) }

  ===> {rewrite rule for "integerToWord# . IS"}
  case x of { I8# x# -> W32# (wordToWord32# (int2Word# (int8ToInt# x#))) }

  Notice that in the resulting expression the value passes through types Int#
  and Word# with native machine word size: it is first sign-extended from Int8#
  to Int#, then cast into Word#, and finally truncated into Word32#. These are
  all very cheap operations that are performed in registers without allocating
  anything in the heap.



Historical fromIntegral optimisations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the past, `fromIntegral` function in the Prelude was special because many
rewrite rules were mentioning it explicitly. For example to replace a call to
`fromIntegral :: Int -> Word`, which allocates an intermediate Integer, with a
call to `intToWord`, which is a no-op when compiled into machine code. Nowadays
`fromIntegral` isn't a special function anymore and we just INLINE it (see above).

- first `fromIntegral` was specialized (SPECIALIZE pragma). However it would
need N^2 pragmas to cover every case and it wouldn't cover user defined numeric
types which don't belong to base.

- `-fwarn-identities` enables a warning to detect useless conversions via
fromIntegral (since 0656c72a8f):

  > fromIntegral (1 :: Int) :: Int

  <interactive>:3:21: warning: [-Widentities]
      Call of fromIntegral :: Int -> Int
        can probably be omitted


- many rules were added (e.g. in e0c787c10f) to perform float-in transformations
explicitly (to allow more fromIntegral rules to fire) and to replace some
fromIntegral calls with faster operations:

    "fromIntegral/Int8->Int8" fromIntegral = id :: Int8 -> Int8
    "fromIntegral/a->Int8"    fromIntegral = \x -> case fromIntegral x of I# x# -> I8# (intToInt8# x#)
    "fromIntegral/Int8->a"    fromIntegral = \(I8# x#) -> fromIntegral (I# x#)

It worked but there were still some issues with this approach:

1. These rules only work for `fromIntegral`. If we wanted to define our own
   similar function (e.g. using other type-classes), we would also have to redefine
   all the rules to get similar performance.

2. `fromIntegral` had to be marked `NOINLINE [1]`:
    - NOINLINE to allow rules to match
    - [1] to allow inlining in later phases to avoid incurring a function call
      overhead for such a trivial operation

   Users of the function had to be careful because a simple helper without an
   INLINE pragma like:

      toInt :: Integral a => a -> Int
      toInt = fromIntegral

   had the following unfolding:

      toInt = integerToInt . toInteger

   which doesn't mention `fromIntegral` anymore. Hence `fromIntegral` rules
   wouldn't fire for codes using `toInt` while they would if they had used
   `fromIntegral` directly!
   For this reason, a bunch of rules for bignum primitives as we have now were
   already present to handle these cases.

3. These rewrite rules were tedious to write and error-prone (cf #19345).

For these reasons, it is simpler to not consider fromIntegral special at all and
to only rely on rewrite rules for bignum functions.

-}

-- See Note [Optimising conversions between numeric types]
{-# RULES
"Word# -> Natural -> Integer"
  forall x. integerFromNatural (NS x) = integerFromWord# x

"BigNat# -> Natural -> Integer"
  forall x. integerFromNatural (NB x) = IP x

"Int# -> Integer -> Int#"
  forall x. integerToInt# (IS x) = x

"Word# -> Integer -> Word#"
  forall x. integerToWord# (integerFromWord# x) = x

"Natural -> Integer -> Natural (wrap)"
  forall x. integerToNatural (integerFromNatural x) = x

"Natural -> Integer -> Natural (throw)"
  forall x. integerToNaturalThrow (integerFromNatural x) = x

"Natural -> Integer -> Natural (clamp)"
  forall x. integerToNaturalClamp (integerFromNatural x) = x

"Natural -> Integer -> Word#"
  forall x. integerToWord# (integerFromNatural x) = naturalToWord# x

"Int# -> Integer -> Word#"
  forall x. integerToWord# (IS x) = int2Word# x

"Word# -> Integer -> Int#"
  forall x. integerToInt# (integerFromWord# x) = word2Int# x

"Word# -> Integer -> Natural (wrap)"
  forall x. integerToNatural (integerFromWord# x) = NS x

"Word# -> Integer -> Natural (throw)"
  forall x. integerToNaturalThrow (integerFromWord# x) = NS x

"Word# -> Integer -> Natural (clamp)"
  forall x. integerToNaturalClamp (integerFromWord# x) = NS x

#-}

{-# RULES

"Int64# -> Integer -> Int64#"
  forall x. integerToInt64# (integerFromInt64# x) = x

"Word64# -> Integer -> Word64#"
  forall x. integerToWord64# (integerFromWord64# x) = x

"Int64# -> Integer -> Word64#"
  forall x. integerToWord64# (integerFromInt64# x) = int64ToWord64# x

"Word64# -> Integer -> Int64#"
  forall x. integerToInt64# (integerFromWord64# x) = word64ToInt64# x

"Word# -> Integer -> Word64#"
  forall x. integerToWord64# (integerFromWord# x) = wordToWord64# x

"Word64# -> Integer -> Word#"
  forall x. integerToWord# (integerFromWord64# x) = word64ToWord# x

"Int# -> Integer -> Int64#"
  forall x. integerToInt64# (IS x) = intToInt64# x

"Int64# -> Integer -> Int#"
  forall x. integerToInt# (integerFromInt64# x) = int64ToInt# x

"Int# -> Integer -> Word64#"
  forall x. integerToWord64# (IS x) = int64ToWord64# (intToInt64# x)

"Int64# -> Integer -> Word#"
  forall x. integerToWord# (integerFromInt64# x) = int2Word# (int64ToInt# x)

"Word# -> Integer -> Int64#"
  forall x. integerToInt64# (integerFromWord# x) = word64ToInt64# (wordToWord64# x)

"Word64# -> Integer -> Int#"
  forall x. integerToInt# (integerFromWord64# x) = word2Int# (word64ToWord# x)

#-}
