{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE MagicHash #-}
{-# LANGUAGE UnboxedTuples #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  GHC.Natural
-- Copyright   :  (C) 2014 Herbert Valerio Riedel,
--                (C) 2011 Edward Kmett
-- License     :  see libraries/base/LICENSE
--
-- Maintainer  :  libraries@haskell.org
-- Stability   :  internal
-- Portability :  non-portable (GHC Extensions)
--
-- The arbitrary-precision 'Natural' number type.
--
-- __Note__: This is an internal GHC module with an API subject to
-- change.  It's recommended use the "Numeric.Natural" module to import
-- the 'Natural' type.
--
-- @since 4.8.0.0
-----------------------------------------------------------------------------
module GHC.Natural
    ( -- * The 'Natural' number type
      --
      -- | __Warning__: The internal implementation of 'Natural'
      -- (i.e. which constructors are available) depends on the
      -- 'Integer' backend used!
      Natural(..)
    , mkNatural
    , isValidNatural
      -- * Arithmetic
    , plusNatural
    , minusNatural
    , minusNaturalMaybe
    , timesNatural
    , negateNatural
    , signumNatural
    , quotRemNatural
    , quotNatural
    , remNatural
    , gcdNatural
    , lcmNatural
      -- * Bits
    , andNatural
    , orNatural
    , xorNatural
    , bitNatural
    , testBitNatural
    , popCountNatural
    , shiftLNatural
    , shiftRNatural
      -- * Conversions
    , naturalToInteger
    , naturalToWord
    , naturalToInt
    , naturalFromInteger
    , wordToNatural
    , intToNatural
    , naturalToWordMaybe
    , wordToNatural#
    , wordToNaturalBase
      -- * Modular arithmetic
    , powModNatural
    ) where

#include "MachDeps.h"

import GHC.Classes
import GHC.Maybe
import GHC.Types
import GHC.Prim
import {-# SOURCE #-} GHC.Exception.Type (underflowException, divZeroException)
#if defined(MIN_VERSION_integer_gmp)
import GHC.Integer.GMP.Internals
#else
import GHC.Integer
#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 `Natural`-typed expression.
--
-- To this end, the CPP hack below allows to write the pseudo-pragma
--
--   {-# CONSTANT_FOLDED plusNatural #-}
--
-- which is simply expanded into a
--
--   {-# NOINLINE plusNatural #-}
--
--
-- TODO: Note that some functions have commented CONSTANT_FOLDED annotations,
-- that's because the Integer counter-parts of these functions do actually have
-- a builtinRule in PrelRules, where the Natural functions do not. The plan is
-- to eventually also add builtin rules for those function on Natural.
#define CONSTANT_FOLDED NOINLINE

-------------------------------------------------------------------------------
-- Arithmetic underflow
-------------------------------------------------------------------------------

-- We put them here because they are needed relatively early
-- in the libraries before the Exception type has been defined yet.

{-# NOINLINE underflowError #-}
underflowError :: a
underflowError :: a
underflowError = SomeException -> a
forall b a. b -> a
raise# SomeException
underflowException

{-# NOINLINE divZeroError #-}
divZeroError :: a
divZeroError :: a
divZeroError = SomeException -> a
forall b a. b -> a
raise# SomeException
divZeroException

-------------------------------------------------------------------------------
-- Natural type
-------------------------------------------------------------------------------

#if defined(MIN_VERSION_integer_gmp)
-- TODO: if saturated arithmetic is to used, replace 'underflowError' by '0'

-- | Type representing arbitrary-precision non-negative integers.
--
-- >>> 2^100 :: Natural
-- 1267650600228229401496703205376
--
-- Operations whose result would be negative @'Control.Exception.throw'
-- ('Control.Exception.Underflow' :: 'Control.Exception.ArithException')@,
--
-- >>> -1 :: Natural
-- *** Exception: arithmetic underflow
--
-- @since 4.8.0.0
data Natural = NatS#                 GmpLimb# -- ^ in @[0, maxBound::Word]@
             | NatJ# {-# UNPACK #-} !BigNat   -- ^ in @]maxBound::Word, +inf[@
                                              --
                                              -- __Invariant__: 'NatJ#' is used
                                              -- /iff/ value doesn't fit in
                                              -- 'NatS#' constructor.
                               -- NB: Order of constructors *must*
                               -- coincide with 'Ord' relation
             deriving ( Eq  -- ^ @since 4.8.0.0
                      , Ord -- ^ @since 4.8.0.0
                      )


-- | Test whether all internal invariants are satisfied by 'Natural' value
--
-- This operation is mostly useful for test-suites and/or code which
-- constructs 'Integer' values directly.
--
-- @since 4.8.0.0
isValidNatural :: Natural -> Bool
isValidNatural :: Natural -> Bool
isValidNatural (NatS# _)  = Bool
True
isValidNatural (NatJ# bn :: BigNat
bn) = Int# -> Bool
isTrue# (BigNat -> Int#
isValidBigNat# BigNat
bn)
                            Bool -> Bool -> Bool
&& Int# -> Bool
isTrue# (BigNat -> Int#
sizeofBigNat# BigNat
bn Int# -> Int# -> Int#
># 0#)

signumNatural :: Natural -> Natural
signumNatural :: Natural -> Natural
signumNatural (NatS# 0##) = GmpLimb# -> Natural
NatS# 0##
signumNatural _           = GmpLimb# -> Natural
NatS# 1##
-- {-# CONSTANT_FOLDED signumNatural #-}

negateNatural :: Natural -> Natural
negateNatural :: Natural -> Natural
negateNatural (NatS# 0##) = GmpLimb# -> Natural
NatS# 0##
negateNatural _           = Natural
forall a. a
underflowError
-- {-# CONSTANT_FOLDED negateNatural #-}

-- | @since 4.10.0.0
naturalFromInteger :: Integer -> Natural
naturalFromInteger :: Integer -> Natural
naturalFromInteger (S# i# :: Int#
i#)
  | Int# -> Bool
isTrue# (Int#
i# Int# -> Int# -> Int#
>=# 0#)     = GmpLimb# -> Natural
NatS# (Int# -> GmpLimb#
int2Word# Int#
i#)
naturalFromInteger (Jp# bn :: BigNat
bn) = BigNat -> Natural
bigNatToNatural BigNat
bn
naturalFromInteger _        = Natural
forall a. a
underflowError
{-# CONSTANT_FOLDED naturalFromInteger #-}

-- | Compute greatest common divisor.
gcdNatural :: Natural -> Natural -> Natural
gcdNatural :: Natural -> Natural -> Natural
gcdNatural (NatS# 0##) y :: Natural
y       = Natural
y
gcdNatural x :: Natural
x       (NatS# 0##) = Natural
x
gcdNatural (NatS# 1##) _       = GmpLimb# -> Natural
NatS# 1##
gcdNatural _       (NatS# 1##) = GmpLimb# -> Natural
NatS# 1##
gcdNatural (NatJ# x :: BigNat
x) (NatJ# y :: BigNat
y) = BigNat -> Natural
bigNatToNatural (BigNat -> BigNat -> BigNat
gcdBigNat BigNat
x BigNat
y)
gcdNatural (NatJ# x :: BigNat
x) (NatS# y :: GmpLimb#
y) = GmpLimb# -> Natural
NatS# (BigNat -> GmpLimb# -> GmpLimb#
gcdBigNatWord BigNat
x GmpLimb#
y)
gcdNatural (NatS# x :: GmpLimb#
x) (NatJ# y :: BigNat
y) = GmpLimb# -> Natural
NatS# (BigNat -> GmpLimb# -> GmpLimb#
gcdBigNatWord BigNat
y GmpLimb#
x)
gcdNatural (NatS# x :: GmpLimb#
x) (NatS# y :: GmpLimb#
y) = GmpLimb# -> Natural
NatS# (GmpLimb# -> GmpLimb# -> GmpLimb#
gcdWord GmpLimb#
x GmpLimb#
y)

-- | compute least common multiplier.
lcmNatural :: Natural -> Natural -> Natural
lcmNatural :: Natural -> Natural -> Natural
lcmNatural (NatS# 0##) _ = GmpLimb# -> Natural
NatS# 0##
lcmNatural _ (NatS# 0##) = GmpLimb# -> Natural
NatS# 0##
lcmNatural (NatS# 1##) y :: Natural
y = Natural
y
lcmNatural x :: Natural
x (NatS# 1##) = Natural
x
lcmNatural x :: Natural
x y :: Natural
y           = (Natural
x Natural -> Natural -> Natural
`quotNatural` (Natural -> Natural -> Natural
gcdNatural Natural
x Natural
y)) Natural -> Natural -> Natural
`timesNatural` Natural
y

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

quotRemNatural :: Natural -> Natural -> (Natural, Natural)
quotRemNatural :: Natural -> Natural -> (Natural, Natural)
quotRemNatural _ (NatS# 0##) = (Natural, Natural)
forall a. a
divZeroError
quotRemNatural n :: Natural
n (NatS# 1##) = (Natural
n,GmpLimb# -> Natural
NatS# 0##)
quotRemNatural n :: Natural
n@(NatS# _) (NatJ# _) = (GmpLimb# -> Natural
NatS# 0##, Natural
n)
quotRemNatural (NatS# n :: GmpLimb#
n) (NatS# d :: GmpLimb#
d) = case GmpLimb# -> GmpLimb# -> (# GmpLimb#, GmpLimb# #)
quotRemWord# GmpLimb#
n GmpLimb#
d of
    (# q :: GmpLimb#
q, r :: GmpLimb#
r #) -> (GmpLimb# -> Natural
NatS# GmpLimb#
q, GmpLimb# -> Natural
NatS# GmpLimb#
r)
quotRemNatural (NatJ# n :: BigNat
n) (NatS# d :: GmpLimb#
d) = case BigNat -> GmpLimb# -> (# BigNat, GmpLimb# #)
quotRemBigNatWord BigNat
n GmpLimb#
d of
    (# q :: BigNat
q, r :: GmpLimb#
r #) -> (BigNat -> Natural
bigNatToNatural BigNat
q, GmpLimb# -> Natural
NatS# GmpLimb#
r)
quotRemNatural (NatJ# n :: BigNat
n) (NatJ# d :: BigNat
d) = case BigNat -> BigNat -> (# BigNat, BigNat #)
quotRemBigNat BigNat
n BigNat
d of
    (# q :: BigNat
q, r :: BigNat
r #) -> (BigNat -> Natural
bigNatToNatural BigNat
q, BigNat -> Natural
bigNatToNatural BigNat
r)
-- {-# CONSTANT_FOLDED quotRemNatural #-}

quotNatural :: Natural -> Natural -> Natural
quotNatural :: Natural -> Natural -> Natural
quotNatural _       (NatS# 0##) = Natural
forall a. a
divZeroError
quotNatural n :: Natural
n       (NatS# 1##) = Natural
n
quotNatural (NatS# _) (NatJ# _) = GmpLimb# -> Natural
NatS# 0##
quotNatural (NatS# n :: GmpLimb#
n) (NatS# d :: GmpLimb#
d) = GmpLimb# -> Natural
NatS# (GmpLimb# -> GmpLimb# -> GmpLimb#
quotWord# GmpLimb#
n GmpLimb#
d)
quotNatural (NatJ# n :: BigNat
n) (NatS# d :: GmpLimb#
d) = BigNat -> Natural
bigNatToNatural (BigNat -> GmpLimb# -> BigNat
quotBigNatWord BigNat
n GmpLimb#
d)
quotNatural (NatJ# n :: BigNat
n) (NatJ# d :: BigNat
d) = BigNat -> Natural
bigNatToNatural (BigNat -> BigNat -> BigNat
quotBigNat BigNat
n BigNat
d)
-- {-# CONSTANT_FOLDED quotNatural #-}

remNatural :: Natural -> Natural -> Natural
remNatural :: Natural -> Natural -> Natural
remNatural _         (NatS# 0##) = Natural
forall a. a
divZeroError
remNatural _         (NatS# 1##) = GmpLimb# -> Natural
NatS# 0##
remNatural n :: Natural
n@(NatS# _) (NatJ# _) = Natural
n
remNatural   (NatS# n :: GmpLimb#
n) (NatS# d :: GmpLimb#
d) = GmpLimb# -> Natural
NatS# (GmpLimb# -> GmpLimb# -> GmpLimb#
remWord# GmpLimb#
n GmpLimb#
d)
remNatural   (NatJ# n :: BigNat
n) (NatS# d :: GmpLimb#
d) = GmpLimb# -> Natural
NatS# (BigNat -> GmpLimb# -> GmpLimb#
remBigNatWord BigNat
n GmpLimb#
d)
remNatural   (NatJ# n :: BigNat
n) (NatJ# d :: BigNat
d) = BigNat -> Natural
bigNatToNatural (BigNat -> BigNat -> BigNat
remBigNat BigNat
n BigNat
d)
-- {-# CONSTANT_FOLDED remNatural #-}

-- | @since 4.12.0.0
naturalToInteger :: Natural -> Integer
naturalToInteger :: Natural -> Integer
naturalToInteger (NatS# w :: GmpLimb#
w)  = GmpLimb# -> Integer
wordToInteger GmpLimb#
w
naturalToInteger (NatJ# bn :: BigNat
bn) = BigNat -> Integer
Jp# BigNat
bn
{-# CONSTANT_FOLDED naturalToInteger #-}

andNatural :: Natural -> Natural -> Natural
andNatural :: Natural -> Natural -> Natural
andNatural (NatS# n :: GmpLimb#
n) (NatS# m :: GmpLimb#
m) = GmpLimb# -> Natural
NatS# (GmpLimb#
n GmpLimb# -> GmpLimb# -> GmpLimb#
`and#` GmpLimb#
m)
andNatural (NatS# n :: GmpLimb#
n) (NatJ# m :: BigNat
m) = GmpLimb# -> Natural
NatS# (GmpLimb#
n GmpLimb# -> GmpLimb# -> GmpLimb#
`and#` BigNat -> GmpLimb#
bigNatToWord BigNat
m)
andNatural (NatJ# n :: BigNat
n) (NatS# m :: GmpLimb#
m) = GmpLimb# -> Natural
NatS# (BigNat -> GmpLimb#
bigNatToWord BigNat
n GmpLimb# -> GmpLimb# -> GmpLimb#
`and#` GmpLimb#
m)
andNatural (NatJ# n :: BigNat
n) (NatJ# m :: BigNat
m) = BigNat -> Natural
bigNatToNatural (BigNat -> BigNat -> BigNat
andBigNat BigNat
n BigNat
m)
-- {-# CONSTANT_FOLDED andNatural #-}

orNatural :: Natural -> Natural -> Natural
orNatural :: Natural -> Natural -> Natural
orNatural (NatS# n :: GmpLimb#
n) (NatS# m :: GmpLimb#
m) = GmpLimb# -> Natural
NatS# (GmpLimb#
n GmpLimb# -> GmpLimb# -> GmpLimb#
`or#` GmpLimb#
m)
orNatural (NatS# n :: GmpLimb#
n) (NatJ# m :: BigNat
m) = BigNat -> Natural
NatJ# (BigNat -> BigNat -> BigNat
orBigNat (GmpLimb# -> BigNat
wordToBigNat GmpLimb#
n) BigNat
m)
orNatural (NatJ# n :: BigNat
n) (NatS# m :: GmpLimb#
m) = BigNat -> Natural
NatJ# (BigNat -> BigNat -> BigNat
orBigNat BigNat
n (GmpLimb# -> BigNat
wordToBigNat GmpLimb#
m))
orNatural (NatJ# n :: BigNat
n) (NatJ# m :: BigNat
m) = BigNat -> Natural
NatJ# (BigNat -> BigNat -> BigNat
orBigNat BigNat
n BigNat
m)
-- {-# CONSTANT_FOLDED orNatural #-}

xorNatural :: Natural -> Natural -> Natural
xorNatural :: Natural -> Natural -> Natural
xorNatural (NatS# n :: GmpLimb#
n) (NatS# m :: GmpLimb#
m) = GmpLimb# -> Natural
NatS# (GmpLimb#
n GmpLimb# -> GmpLimb# -> GmpLimb#
`xor#` GmpLimb#
m)
xorNatural (NatS# n :: GmpLimb#
n) (NatJ# m :: BigNat
m) = BigNat -> Natural
NatJ# (BigNat -> BigNat -> BigNat
xorBigNat (GmpLimb# -> BigNat
wordToBigNat GmpLimb#
n) BigNat
m)
xorNatural (NatJ# n :: BigNat
n) (NatS# m :: GmpLimb#
m) = BigNat -> Natural
NatJ# (BigNat -> BigNat -> BigNat
xorBigNat BigNat
n (GmpLimb# -> BigNat
wordToBigNat GmpLimb#
m))
xorNatural (NatJ# n :: BigNat
n) (NatJ# m :: BigNat
m) = BigNat -> Natural
bigNatToNatural (BigNat -> BigNat -> BigNat
xorBigNat BigNat
n BigNat
m)
-- {-# CONSTANT_FOLDED xorNatural #-}

bitNatural :: Int# -> Natural
bitNatural :: Int# -> Natural
bitNatural i# :: Int#
i#
  | Int# -> Bool
isTrue# (Int#
i# Int# -> Int# -> Int#
<# WORD_SIZE_IN_BITS#) = NatS# (1## `uncheckedShiftL#` i#)
  | Bool
True                               = BigNat -> Natural
NatJ# (Int# -> BigNat
bitBigNat Int#
i#)
-- {-# CONSTANT_FOLDED bitNatural #-}

testBitNatural :: Natural -> Int -> Bool
testBitNatural :: Natural -> Int -> Bool
testBitNatural (NatS# w :: GmpLimb#
w) (I# i# :: Int#
i#)
  | Int# -> Bool
isTrue# (Int#
i# Int# -> Int# -> Int#
<# WORD_SIZE_IN_BITS#) =
      Int# -> Bool
isTrue# ((GmpLimb#
w GmpLimb# -> GmpLimb# -> GmpLimb#
`and#` (1## GmpLimb# -> Int# -> GmpLimb#
`uncheckedShiftL#` Int#
i#)) GmpLimb# -> GmpLimb# -> Int#
`neWord#` 0##)
  | Bool
True                               = Bool
False
testBitNatural (NatJ# bn :: BigNat
bn) (I# i# :: Int#
i#)      = BigNat -> Int# -> Bool
testBitBigNat BigNat
bn Int#
i#
-- {-# CONSTANT_FOLDED testBitNatural #-}

popCountNatural :: Natural -> Int
popCountNatural :: Natural -> Int
popCountNatural (NatS# w :: GmpLimb#
w)  = Int# -> Int
I# (GmpLimb# -> Int#
word2Int# (GmpLimb# -> GmpLimb#
popCnt# GmpLimb#
w))
popCountNatural (NatJ# bn :: BigNat
bn) = Int# -> Int
I# (BigNat -> Int#
popCountBigNat BigNat
bn)
-- {-# CONSTANT_FOLDED popCountNatural #-}

shiftLNatural :: Natural -> Int -> Natural
shiftLNatural :: Natural -> Int -> Natural
shiftLNatural n :: Natural
n           (I# 0#) = Natural
n
shiftLNatural (NatS# 0##) _       = GmpLimb# -> Natural
NatS# 0##
shiftLNatural (NatS# 1##) (I# i# :: Int#
i#) = Int# -> Natural
bitNatural Int#
i#
shiftLNatural (NatS# w :: GmpLimb#
w) (I# i# :: Int#
i#)
    = BigNat -> Natural
bigNatToNatural (BigNat -> Int# -> BigNat
shiftLBigNat (GmpLimb# -> BigNat
wordToBigNat GmpLimb#
w) Int#
i#)
shiftLNatural (NatJ# bn :: BigNat
bn) (I# i# :: Int#
i#)
    = BigNat -> Natural
bigNatToNatural (BigNat -> Int# -> BigNat
shiftLBigNat BigNat
bn Int#
i#)
-- {-# CONSTANT_FOLDED shiftLNatural #-}

shiftRNatural :: Natural -> Int -> Natural
shiftRNatural :: Natural -> Int -> Natural
shiftRNatural n :: Natural
n          (I# 0#) = Natural
n
shiftRNatural (NatS# w :: GmpLimb#
w)  (I# i# :: Int#
i#)
      | Int# -> Bool
isTrue# (Int#
i# Int# -> Int# -> Int#
>=# WORD_SIZE_IN_BITS#) = NatS# 0##
      | Bool
True = GmpLimb# -> Natural
NatS# (GmpLimb#
w GmpLimb# -> Int# -> GmpLimb#
`uncheckedShiftRL#` Int#
i#)
shiftRNatural (NatJ# bn :: BigNat
bn) (I# i# :: Int#
i#) = BigNat -> Natural
bigNatToNatural (BigNat -> Int# -> BigNat
shiftRBigNat BigNat
bn Int#
i#)
-- {-# CONSTANT_FOLDED shiftRNatural #-}

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

-- | 'Natural' Addition
plusNatural :: Natural -> Natural -> Natural
plusNatural :: Natural -> Natural -> Natural
plusNatural (NatS# 0##) y :: Natural
y         = Natural
y
plusNatural x :: Natural
x         (NatS# 0##) = Natural
x
plusNatural (NatS# x :: GmpLimb#
x) (NatS# y :: GmpLimb#
y)
    = case GmpLimb# -> GmpLimb# -> (# GmpLimb#, GmpLimb# #)
plusWord2# GmpLimb#
x GmpLimb#
y of
       (# 0##, l :: GmpLimb#
l #) -> GmpLimb# -> Natural
NatS# GmpLimb#
l
       (# h :: GmpLimb#
h,   l :: GmpLimb#
l #) -> BigNat -> Natural
NatJ# (GmpLimb# -> GmpLimb# -> BigNat
wordToBigNat2 GmpLimb#
h GmpLimb#
l)
plusNatural (NatS# x :: GmpLimb#
x) (NatJ# y :: BigNat
y) = BigNat -> Natural
NatJ# (BigNat -> GmpLimb# -> BigNat
plusBigNatWord BigNat
y GmpLimb#
x)
plusNatural (NatJ# x :: BigNat
x) (NatS# y :: GmpLimb#
y) = BigNat -> Natural
NatJ# (BigNat -> GmpLimb# -> BigNat
plusBigNatWord BigNat
x GmpLimb#
y)
plusNatural (NatJ# x :: BigNat
x) (NatJ# y :: BigNat
y) = BigNat -> Natural
NatJ# (BigNat -> BigNat -> BigNat
plusBigNat     BigNat
x BigNat
y)
{-# CONSTANT_FOLDED plusNatural #-}

-- | 'Natural' multiplication
timesNatural :: Natural -> Natural -> Natural
timesNatural :: Natural -> Natural -> Natural
timesNatural _         (NatS# 0##) = GmpLimb# -> Natural
NatS# 0##
timesNatural (NatS# 0##) _         = GmpLimb# -> Natural
NatS# 0##
timesNatural x :: Natural
x         (NatS# 1##) = Natural
x
timesNatural (NatS# 1##) y :: Natural
y         = Natural
y
timesNatural (NatS# x :: GmpLimb#
x) (NatS# y :: GmpLimb#
y) = case GmpLimb# -> GmpLimb# -> (# GmpLimb#, GmpLimb# #)
timesWord2# GmpLimb#
x GmpLimb#
y of
    (# 0##, 0## #) -> GmpLimb# -> Natural
NatS# 0##
    (# 0##, xy :: GmpLimb#
xy  #) -> GmpLimb# -> Natural
NatS# GmpLimb#
xy
    (# h :: GmpLimb#
h  , l :: GmpLimb#
l   #) -> BigNat -> Natural
NatJ# (GmpLimb# -> GmpLimb# -> BigNat
wordToBigNat2 GmpLimb#
h GmpLimb#
l)
timesNatural (NatS# x :: GmpLimb#
x) (NatJ# y :: BigNat
y) = BigNat -> Natural
NatJ# (BigNat -> GmpLimb# -> BigNat
timesBigNatWord BigNat
y GmpLimb#
x)
timesNatural (NatJ# x :: BigNat
x) (NatS# y :: GmpLimb#
y) = BigNat -> Natural
NatJ# (BigNat -> GmpLimb# -> BigNat
timesBigNatWord BigNat
x GmpLimb#
y)
timesNatural (NatJ# x :: BigNat
x) (NatJ# y :: BigNat
y) = BigNat -> Natural
NatJ# (BigNat -> BigNat -> BigNat
timesBigNat     BigNat
x BigNat
y)
{-# CONSTANT_FOLDED timesNatural #-}

-- | 'Natural' subtraction. May @'Control.Exception.throw'
-- 'Control.Exception.Underflow'@.
minusNatural :: Natural -> Natural -> Natural
minusNatural :: Natural -> Natural -> Natural
minusNatural x :: Natural
x         (NatS# 0##) = Natural
x
minusNatural (NatS# x :: GmpLimb#
x) (NatS# y :: GmpLimb#
y) = case GmpLimb# -> GmpLimb# -> (# GmpLimb#, Int# #)
subWordC# GmpLimb#
x GmpLimb#
y of
    (# l :: GmpLimb#
l, 0# #) -> GmpLimb# -> Natural
NatS# GmpLimb#
l
    _           -> Natural
forall a. a
underflowError
minusNatural (NatS# _) (NatJ# _) = Natural
forall a. a
underflowError
minusNatural (NatJ# x :: BigNat
x) (NatS# y :: GmpLimb#
y)
    = BigNat -> Natural
bigNatToNatural (BigNat -> GmpLimb# -> BigNat
minusBigNatWord BigNat
x GmpLimb#
y)
minusNatural (NatJ# x :: BigNat
x) (NatJ# y :: BigNat
y)
    = BigNat -> Natural
bigNatToNatural (BigNat -> BigNat -> BigNat
minusBigNat     BigNat
x BigNat
y)
{-# CONSTANT_FOLDED minusNatural #-}

-- | 'Natural' subtraction. Returns 'Nothing's for non-positive results.
--
-- @since 4.8.0.0
minusNaturalMaybe :: Natural -> Natural -> Maybe Natural
minusNaturalMaybe :: Natural -> Natural -> Maybe Natural
minusNaturalMaybe x :: Natural
x         (NatS# 0##) = Natural -> Maybe Natural
forall a. a -> Maybe a
Just Natural
x
minusNaturalMaybe (NatS# x :: GmpLimb#
x) (NatS# y :: GmpLimb#
y) = case GmpLimb# -> GmpLimb# -> (# GmpLimb#, Int# #)
subWordC# GmpLimb#
x GmpLimb#
y of
    (# l :: GmpLimb#
l, 0# #) -> Natural -> Maybe Natural
forall a. a -> Maybe a
Just (GmpLimb# -> Natural
NatS# GmpLimb#
l)
    _           -> Maybe Natural
forall a. Maybe a
Nothing
minusNaturalMaybe (NatS# _) (NatJ# _) = Maybe Natural
forall a. Maybe a
Nothing
minusNaturalMaybe (NatJ# x :: BigNat
x) (NatS# y :: GmpLimb#
y)
    = Natural -> Maybe Natural
forall a. a -> Maybe a
Just (BigNat -> Natural
bigNatToNatural (BigNat -> GmpLimb# -> BigNat
minusBigNatWord BigNat
x GmpLimb#
y))
minusNaturalMaybe (NatJ# x :: BigNat
x) (NatJ# y :: BigNat
y)
  | Int# -> Bool
isTrue# (BigNat -> Int#
isNullBigNat# BigNat
res) = Maybe Natural
forall a. Maybe a
Nothing
  | Bool
True                        = Natural -> Maybe Natural
forall a. a -> Maybe a
Just (BigNat -> Natural
bigNatToNatural BigNat
res)
  where
    res :: BigNat
res = BigNat -> BigNat -> BigNat
minusBigNat BigNat
x BigNat
y

-- | Convert 'BigNat' to 'Natural'.
-- Throws 'Control.Exception.Underflow' if passed a 'nullBigNat'.
bigNatToNatural :: BigNat -> Natural
bigNatToNatural :: BigNat -> Natural
bigNatToNatural bn :: BigNat
bn
  | Int# -> Bool
isTrue# (BigNat -> Int#
sizeofBigNat# BigNat
bn Int# -> Int# -> Int#
==# 1#) = GmpLimb# -> Natural
NatS# (BigNat -> GmpLimb#
bigNatToWord BigNat
bn)
  | Int# -> Bool
isTrue# (BigNat -> Int#
isNullBigNat# BigNat
bn)        = Natural
forall a. a
underflowError
  | Bool
True                              = BigNat -> Natural
NatJ# BigNat
bn

naturalToBigNat :: Natural -> BigNat
naturalToBigNat :: Natural -> BigNat
naturalToBigNat (NatS# w# :: GmpLimb#
w#) = GmpLimb# -> BigNat
wordToBigNat GmpLimb#
w#
naturalToBigNat (NatJ# bn :: BigNat
bn) = BigNat
bn

naturalToWord :: Natural -> Word
naturalToWord :: Natural -> Word
naturalToWord (NatS# w# :: GmpLimb#
w#) = GmpLimb# -> Word
W# GmpLimb#
w#
naturalToWord (NatJ# bn :: BigNat
bn) = GmpLimb# -> Word
W# (BigNat -> GmpLimb#
bigNatToWord BigNat
bn)

naturalToInt :: Natural -> Int
naturalToInt :: Natural -> Int
naturalToInt (NatS# w# :: GmpLimb#
w#) = Int# -> Int
I# (GmpLimb# -> Int#
word2Int# GmpLimb#
w#)
naturalToInt (NatJ# bn :: BigNat
bn) = Int# -> Int
I# (BigNat -> Int#
bigNatToInt BigNat
bn)

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

-- | Convert a Word# into a Natural
--
-- Built-in rule ensures that applications of this function to literal Word# are
-- lifted into Natural literals.
wordToNatural# :: Word# -> Natural
wordToNatural# :: GmpLimb# -> Natural
wordToNatural# w# :: GmpLimb#
w# = GmpLimb# -> Natural
NatS# GmpLimb#
w#
{-# CONSTANT_FOLDED wordToNatural# #-}

-- | Convert a Word# into a Natural
--
-- In base we can't use wordToNatural# as built-in rules transform some of them
-- into Natural literals. Use this function instead.
wordToNaturalBase :: Word# -> Natural
wordToNaturalBase :: GmpLimb# -> Natural
wordToNaturalBase w# :: GmpLimb#
w# = GmpLimb# -> Natural
NatS# GmpLimb#
w#

#else /* !defined(MIN_VERSION_integer_gmp) */
----------------------------------------------------------------------------
-- Use wrapped 'Integer' as fallback; taken from Edward Kmett's nats package

-- | Type representing arbitrary-precision non-negative integers.
--
-- Operations whose result would be negative @'Control.Exception.throw'
-- ('Control.Exception.Underflow' :: 'Control.Exception.ArithException')@.
--
-- @since 4.8.0.0
newtype Natural = Natural Integer -- ^ __Invariant__: non-negative 'Integer'
                  deriving (Eq,Ord)


-- | Test whether all internal invariants are satisfied by 'Natural' value
--
-- This operation is mostly useful for test-suites and/or code which
-- constructs 'Natural' values directly.
--
-- @since 4.8.0.0
isValidNatural :: Natural -> Bool
isValidNatural (Natural i) = i >= wordToInteger 0##

-- | Convert a 'Word#' into a 'Natural'
--
-- Built-in rule ensures that applications of this function to literal 'Word#'
-- are lifted into 'Natural' literals.
wordToNatural# :: Word# -> Natural
wordToNatural# w## = Natural (wordToInteger w##)
{-# CONSTANT_FOLDED wordToNatural# #-}

-- | Convert a 'Word#' into a Natural
--
-- In base we can't use wordToNatural# as built-in rules transform some of them
-- into Natural literals. Use this function instead.
wordToNaturalBase :: Word# -> Natural
wordToNaturalBase w## = Natural (wordToInteger w##)

-- | @since 4.10.0.0
naturalFromInteger :: Integer -> Natural
naturalFromInteger n
  | n >= wordToInteger 0## = Natural n
  | True                   = underflowError
{-# INLINE naturalFromInteger #-}


-- | Compute greatest common divisor.
gcdNatural :: Natural -> Natural -> Natural
gcdNatural (Natural n) (Natural m) = Natural (n `gcdInteger` m)

-- | Compute lowest common multiple.
lcmNatural :: Natural -> Natural -> Natural
lcmNatural (Natural n) (Natural m) = Natural (n `lcmInteger` m)

-- | 'Natural' subtraction. Returns 'Nothing's for non-positive results.
--
-- @since 4.8.0.0
minusNaturalMaybe :: Natural -> Natural -> Maybe Natural
minusNaturalMaybe (Natural x) (Natural y)
  | x >= y  = Just (Natural (x `minusInteger` y))
  | True    = Nothing

shiftLNatural :: Natural -> Int -> Natural
shiftLNatural (Natural n) (I# i) = Natural (n `shiftLInteger` i)
-- {-# CONSTANT_FOLDED shiftLNatural #-}

shiftRNatural :: Natural -> Int -> Natural
shiftRNatural (Natural n) (I# i) = Natural (n `shiftRInteger` i)
-- {-# CONSTANT_FOLDED shiftRNatural #-}

plusNatural :: Natural -> Natural -> Natural
plusNatural (Natural x) (Natural y) = Natural (x `plusInteger` y)
{-# CONSTANT_FOLDED plusNatural #-}

minusNatural :: Natural -> Natural -> Natural
minusNatural (Natural x) (Natural y)
  = if z `ltInteger` wordToInteger 0## then underflowError else Natural z
  where z = x `minusInteger` y
{-# CONSTANT_FOLDED minusNatural #-}

timesNatural :: Natural -> Natural -> Natural
timesNatural (Natural x) (Natural y) = Natural (x `timesInteger` y)
{-# CONSTANT_FOLDED timesNatural #-}

orNatural :: Natural -> Natural -> Natural
orNatural (Natural x) (Natural y) = Natural (x `orInteger` y)
-- {-# CONSTANT_FOLDED orNatural #-}

xorNatural :: Natural -> Natural -> Natural
xorNatural (Natural x) (Natural y) = Natural (x `xorInteger` y)
-- {-# CONSTANT_FOLDED xorNatural #-}

andNatural :: Natural -> Natural -> Natural
andNatural (Natural x) (Natural y) = Natural (x `andInteger` y)
-- {-# CONSTANT_FOLDED andNatural #-}

naturalToInt :: Natural -> Int
naturalToInt (Natural i) = I# (integerToInt i)

naturalToWord :: Natural -> Word
naturalToWord (Natural i) = W# (integerToWord i)

naturalToInteger :: Natural -> Integer
naturalToInteger (Natural i) = i
{-# CONSTANT_FOLDED naturalToInteger #-}

testBitNatural :: Natural -> Int -> Bool
testBitNatural (Natural n) (I# i) = testBitInteger n i
-- {-# CONSTANT_FOLDED testBitNatural #-}

popCountNatural :: Natural -> Int
popCountNatural (Natural n) = I# (popCountInteger n)

bitNatural :: Int# -> Natural
bitNatural i#
  | isTrue# (i# <# WORD_SIZE_IN_BITS#) = wordToNaturalBase (1## `uncheckedShiftL#` i#)
  | True                               = Natural (1 `shiftLInteger` i#)
-- {-# CONSTANT_FOLDED bitNatural #-}

quotNatural :: Natural -> Natural -> Natural
quotNatural n@(Natural x) (Natural y)
   | y == wordToInteger 0## = divZeroError
   | y == wordToInteger 1## = n
   | True                   = Natural (x `quotInteger` y)
-- {-# CONSTANT_FOLDED quotNatural #-}

remNatural :: Natural -> Natural -> Natural
remNatural (Natural x) (Natural y)
   | y == wordToInteger 0## = divZeroError
   | y == wordToInteger 1## = wordToNaturalBase 0##
   | True                   = Natural (x `remInteger` y)
-- {-# CONSTANT_FOLDED remNatural #-}

quotRemNatural :: Natural -> Natural -> (Natural, Natural)
quotRemNatural n@(Natural x) (Natural y)
   | y == wordToInteger 0## = divZeroError
   | y == wordToInteger 1## = (n,wordToNaturalBase 0##)
   | True                   = case quotRemInteger x y of
      (# k, r #) -> (Natural k, Natural r)
-- {-# CONSTANT_FOLDED quotRemNatural #-}

signumNatural :: Natural -> Natural
signumNatural (Natural x)
   | x == wordToInteger 0## = wordToNaturalBase 0##
   | True                   = wordToNaturalBase 1##
-- {-# CONSTANT_FOLDED signumNatural #-}

negateNatural :: Natural -> Natural
negateNatural (Natural x)
   | x == wordToInteger 0## = wordToNaturalBase 0##
   | True                   = underflowError
-- {-# CONSTANT_FOLDED negateNatural #-}

#endif

-- | Construct 'Natural' from 'Word' value.
--
-- @since 4.8.0.0
wordToNatural :: Word -> Natural
wordToNatural :: Word -> Natural
wordToNatural (W# w# :: GmpLimb#
w#) = GmpLimb# -> Natural
wordToNatural# GmpLimb#
w#

-- | Try downcasting 'Natural' to 'Word' value.
-- Returns 'Nothing' if value doesn't fit in 'Word'.
--
-- @since 4.8.0.0
naturalToWordMaybe :: Natural -> Maybe Word
#if defined(MIN_VERSION_integer_gmp)
naturalToWordMaybe :: Natural -> Maybe Word
naturalToWordMaybe (NatS# w# :: GmpLimb#
w#) = Word -> Maybe Word
forall a. a -> Maybe a
Just (GmpLimb# -> Word
W# GmpLimb#
w#)
naturalToWordMaybe (NatJ# _)  = Maybe Word
forall a. Maybe a
Nothing
#else
naturalToWordMaybe (Natural i)
  | i < maxw  = Just (W# (integerToWord i))
  | True      = Nothing
  where
    maxw = 1 `shiftLInteger` WORD_SIZE_IN_BITS#
#endif

-- | \"@'powModNatural' /b/ /e/ /m/@\" computes base @/b/@ raised to
-- exponent @/e/@ modulo @/m/@.
--
-- @since 4.8.0.0
powModNatural :: Natural -> Natural -> Natural -> Natural
#if defined(MIN_VERSION_integer_gmp)
powModNatural :: Natural -> Natural -> Natural -> Natural
powModNatural _           _           (NatS# 0##) = Natural
forall a. a
divZeroError
powModNatural _           _           (NatS# 1##) = GmpLimb# -> Natural
NatS# 0##
powModNatural _           (NatS# 0##) _           = GmpLimb# -> Natural
NatS# 1##
powModNatural (NatS# 0##) _           _           = GmpLimb# -> Natural
NatS# 0##
powModNatural (NatS# 1##) _           _           = GmpLimb# -> Natural
NatS# 1##
powModNatural (NatS# b :: GmpLimb#
b)   (NatS# e :: GmpLimb#
e)   (NatS# m :: GmpLimb#
m)   = GmpLimb# -> Natural
NatS# (GmpLimb# -> GmpLimb# -> GmpLimb# -> GmpLimb#
powModWord GmpLimb#
b GmpLimb#
e GmpLimb#
m)
powModNatural b :: Natural
b           e :: Natural
e           (NatS# m :: GmpLimb#
m)
  = GmpLimb# -> Natural
NatS# (BigNat -> BigNat -> GmpLimb# -> GmpLimb#
powModBigNatWord (Natural -> BigNat
naturalToBigNat Natural
b) (Natural -> BigNat
naturalToBigNat Natural
e) GmpLimb#
m)
powModNatural b :: Natural
b           e :: Natural
e           (NatJ# m :: BigNat
m)
  = BigNat -> Natural
bigNatToNatural (BigNat -> BigNat -> BigNat -> BigNat
powModBigNat (Natural -> BigNat
naturalToBigNat Natural
b) (Natural -> BigNat
naturalToBigNat Natural
e) BigNat
m)
#else
-- Portable reference fallback implementation
powModNatural (Natural b0) (Natural e0) (Natural m)
   | m  == wordToInteger 0## = divZeroError
   | m  == wordToInteger 1## = wordToNaturalBase 0##
   | e0 == wordToInteger 0## = wordToNaturalBase 1##
   | b0 == wordToInteger 0## = wordToNaturalBase 0##
   | b0 == wordToInteger 1## = wordToNaturalBase 1##
   | True    = go b0 e0 (wordToInteger 1##)
  where
    go !b e !r
      | e `testBitInteger` 0#  = go b' e' ((r `timesInteger` b) `modInteger` m)
      | e == wordToInteger 0## = naturalFromInteger r
      | True                   = go b' e' r
      where
        b' = (b `timesInteger` b) `modInteger` m
        e' = e `shiftRInteger` 1# -- slightly faster than "e `div` 2"
#endif


-- | Construct 'Natural' value from list of 'Word's.
--
-- This function is used by GHC for constructing 'Natural' literals.
mkNatural :: [Word]  -- ^ value expressed in 32 bit chunks, least
                     --   significant first
          -> Natural
mkNatural :: [Word] -> Natural
mkNatural [] = GmpLimb# -> Natural
wordToNaturalBase 0##
mkNatural (W# i :: GmpLimb#
i : is' :: [Word]
is') = GmpLimb# -> Natural
wordToNaturalBase (GmpLimb#
i GmpLimb# -> GmpLimb# -> GmpLimb#
`and#` 0xffffffff##) Natural -> Natural -> Natural
`orNatural`
                         Natural -> Int -> Natural
shiftLNatural ([Word] -> Natural
mkNatural [Word]
is') 32
{-# CONSTANT_FOLDED mkNatural #-}

-- | Convert 'Int' to 'Natural'.
-- Throws 'Control.Exception.Underflow' when passed a negative 'Int'.
intToNatural :: Int -> Natural
intToNatural :: Int -> Natural
intToNatural (I# i# :: Int#
i#)
  | Int# -> Bool
isTrue# (Int#
i# Int# -> Int# -> Int#
<# 0#) = Natural
forall a. a
underflowError
  | Bool
True               = GmpLimb# -> Natural
wordToNaturalBase (Int# -> GmpLimb#
int2Word# Int#
i#)