Data.Bit

This module exposes an interface with thread-unsafe writes and flips. Consider using Data.Bit.ThreadSafe, which is thread-safe, but slower (up to 20%).

Documentation

newtype Bit

A newtype wrapper with a custom instance of Data.Vector.Unboxed, which packs booleans as efficient as possible (8 values per byte). Vectors of Bit use 8x less memory than vectors of Bool (which stores one value per byte). but random writes are up to 10% slower.

Constructors

Bit

Fields

• unBit :: Bool

Instances

Bounded Bit

Defined in Data.Bit.Internal

Methods

minBound :: Bit

maxBound :: Bit

Enum Bit

Defined in Data.Bit.Internal

Methods

succ :: Bit -> Bit

pred :: Bit -> Bit

toEnum :: Int -> Bit

fromEnum :: Bit -> Int

enumFrom :: Bit -> [Bit]

enumFromThen :: Bit -> Bit -> [Bit]

enumFromTo :: Bit -> Bit -> [Bit]

enumFromThenTo :: Bit -> Bit -> Bit -> [Bit]

Eq Bit

Defined in Data.Bit.Internal

Methods

(==) :: Bit -> Bit -> Bool

(/=) :: Bit -> Bit -> Bool

Fractional Bit

Defined in Data.Bit.Internal

Methods

(/) :: Bit -> Bit -> Bit

recip :: Bit -> Bit

fromRational :: Rational -> Bit

Integral Bit

Defined in Data.Bit.Internal

Methods

quot :: Bit -> Bit -> Bit

rem :: Bit -> Bit -> Bit

div :: Bit -> Bit -> Bit

mod :: Bit -> Bit -> Bit

quotRem :: Bit -> Bit -> (Bit, Bit)

divMod :: Bit -> Bit -> (Bit, Bit)

toInteger :: Bit -> Integer

Num Bit

There is only one lawful Num instance possible with + = xor and fromInteger = Bit . odd.

Defined in Data.Bit.Internal

Methods

(+) :: Bit -> Bit -> Bit

(-) :: Bit -> Bit -> Bit

(*) :: Bit -> Bit -> Bit

negate :: Bit -> Bit

abs :: Bit -> Bit

signum :: Bit -> Bit

fromInteger :: Integer -> Bit

Ord Bit

Defined in Data.Bit.Internal

Methods

compare :: Bit -> Bit -> Ordering

(<) :: Bit -> Bit -> Bool

(<=) :: Bit -> Bit -> Bool

(>) :: Bit -> Bit -> Bool

(>=) :: Bit -> Bit -> Bool

max :: Bit -> Bit -> Bit

min :: Bit -> Bit -> Bit

Read Bit

Defined in Data.Bit.Internal

Methods

readsPrec :: Int -> ReadS Bit

readList :: ReadS [Bit]

readPrec :: ReadPrec Bit

readListPrec :: ReadPrec [Bit]

Real Bit

Defined in Data.Bit.Internal

Methods

toRational :: Bit -> Rational

Show Bit

Defined in Data.Bit.Internal

Methods

showsPrec :: Int -> Bit -> ShowS

show :: Bit -> String

showList :: [Bit] -> ShowS

Generic Bit

Defined in Data.Bit.Internal

Associated Types

type Rep Bit :: Type -> Type

Methods

from :: Bit -> Rep Bit x

to :: Rep Bit x -> Bit

Bits Bit

Defined in Data.Bit.Internal

Methods

(.&.) :: Bit -> Bit -> Bit

(.|.) :: Bit -> Bit -> Bit

xor :: Bit -> Bit -> Bit

complement :: Bit -> Bit

shift :: Bit -> Int -> Bit

rotate :: Bit -> Int -> Bit

zeroBits :: Bit

bit :: Int -> Bit

setBit :: Bit -> Int -> Bit

clearBit :: Bit -> Int -> Bit

complementBit :: Bit -> Int -> Bit

testBit :: Bit -> Int -> Bool

bitSizeMaybe :: Bit -> Maybe Int

bitSize :: Bit -> Int

isSigned :: Bit -> Bool

shiftL :: Bit -> Int -> Bit

unsafeShiftL :: Bit -> Int -> Bit

shiftR :: Bit -> Int -> Bit

unsafeShiftR :: Bit -> Int -> Bit

rotateL :: Bit -> Int -> Bit

rotateR :: Bit -> Int -> Bit

popCount :: Bit -> Int

FiniteBits Bit

Defined in Data.Bit.Internal

Methods

finiteBitSize :: Bit -> Int

countLeadingZeros :: Bit -> Int

countTrailingZeros :: Bit -> Int

NFData Bit

Defined in Data.Bit.Internal

Methods

rnf :: Bit -> ()

Unbox Bit

Defined in Data.Bit.Internal

Vector Vector Bit

Defined in Data.Bit.Internal

Methods

basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) Bit -> m (Vector Bit)

basicUnsafeThaw :: PrimMonad m => Vector Bit -> m (Mutable Vector (PrimState m) Bit)

basicLength :: Vector Bit -> Int

basicUnsafeSlice :: Int -> Int -> Vector Bit -> Vector Bit

basicUnsafeIndexM :: Monad m => Vector Bit -> Int -> m Bit

basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) Bit -> Vector Bit -> m ()

elemseq :: Vector Bit -> Bit -> b -> b

MVector MVector Bit

Defined in Data.Bit.Internal

Methods

basicLength :: MVector s Bit -> Int

basicUnsafeSlice :: Int -> Int -> MVector s Bit -> MVector s Bit

basicOverlaps :: MVector s Bit -> MVector s Bit -> Bool

basicUnsafeNew :: PrimMonad m => Int -> m (MVector (PrimState m) Bit)

basicInitialize :: PrimMonad m => MVector (PrimState m) Bit -> m ()

basicUnsafeReplicate :: PrimMonad m => Int -> Bit -> m (MVector (PrimState m) Bit)

basicUnsafeRead :: PrimMonad m => MVector (PrimState m) Bit -> Int -> m Bit

basicUnsafeWrite :: PrimMonad m => MVector (PrimState m) Bit -> Int -> Bit -> m ()

basicClear :: PrimMonad m => MVector (PrimState m) Bit -> m ()

basicSet :: PrimMonad m => MVector (PrimState m) Bit -> Bit -> m ()

basicUnsafeCopy :: PrimMonad m => MVector (PrimState m) Bit -> MVector (PrimState m) Bit -> m ()

basicUnsafeMove :: PrimMonad m => MVector (PrimState m) Bit -> MVector (PrimState m) Bit -> m ()

basicUnsafeGrow :: PrimMonad m => MVector (PrimState m) Bit -> Int -> m (MVector (PrimState m) Bit)

Bits (Vector Bit)

Defined in Data.Bit.Immutable

Methods

(.&.) :: Vector Bit -> Vector Bit -> Vector Bit

(.|.) :: Vector Bit -> Vector Bit -> Vector Bit

xor :: Vector Bit -> Vector Bit -> Vector Bit

complement :: Vector Bit -> Vector Bit

shift :: Vector Bit -> Int -> Vector Bit

rotate :: Vector Bit -> Int -> Vector Bit

zeroBits :: Vector Bit

bit :: Int -> Vector Bit

setBit :: Vector Bit -> Int -> Vector Bit

clearBit :: Vector Bit -> Int -> Vector Bit

complementBit :: Vector Bit -> Int -> Vector Bit

testBit :: Vector Bit -> Int -> Bool

bitSizeMaybe :: Vector Bit -> Maybe Int

bitSize :: Vector Bit -> Int

isSigned :: Vector Bit -> Bool

shiftL :: Vector Bit -> Int -> Vector Bit

unsafeShiftL :: Vector Bit -> Int -> Vector Bit

shiftR :: Vector Bit -> Int -> Vector Bit

unsafeShiftR :: Vector Bit -> Int -> Vector Bit

rotateL :: Vector Bit -> Int -> Vector Bit

rotateR :: Vector Bit -> Int -> Vector Bit

popCount :: Vector Bit -> Int

type Rep Bit

Defined in Data.Bit.Internal

type Rep Bit = D1 ('MetaData "Bit" "Data.Bit.Internal" "bitvec-1.1.1.0-AhfNuieIoBoKwVuNuMi18y" 'True) (C1 ('MetaCons "Bit" 'PrefixI 'True) (S1 ('MetaSel ('Just "unBit") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 Bool)))

data Vector Bit

Defined in Data.Bit.Internal

data Vector Bit = BitVec !Int !Int !ByteArray

data MVector s Bit

Defined in Data.Bit.Internal

data MVector s Bit = BitMVec !Int !Int !(MutableByteArray s)

unsafeFlipBit :: PrimMonad m => MVector (PrimState m) Bit -> Int -> m ()

Flip the bit at the given position. No bounds checks are performed. Equivalent to flip unsafeModify complement, but up to 2x faster.

In general there is no reason to unsafeModify bit vectors: either you modify it with id (which is id altogether) or with complement (which is unsafeFlipBit).

>>> :set -XOverloadedLists
>>> Data.Vector.Unboxed.modify (`unsafeFlipBit` 2) [1,1,1,1]
[1,1,0,1]

flipBit :: PrimMonad m => MVector (PrimState m) Bit -> Int -> m ()

Flip the bit at the given position. Equivalent to flip modify complement, but up to 2x faster.

In general there is no reason to modify bit vectors: either you modify it with id (which is id altogether) or with complement (which is flipBit).

>>> :set -XOverloadedLists
>>> Data.Vector.Unboxed.modify (`flipBit` 2) [1,1,1,1]
[1,1,0,1]

Immutable conversions

castFromWords :: Vector Word -> Vector Bit

Cast an unboxed vector of words to an unboxed vector of bits. Cf. castFromWordsM.

>>> :set -XOverloadedLists
>>> castFromWords [123]
[1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

castToWords :: Vector Bit -> Maybe (Vector Word)

Try to cast an unboxed vector of bits to an unboxed vector of words. It succeeds if a vector of bits is aligned. Use cloneToWords otherwise. Cf. castToWordsM.

castToWords (castFromWords v) == Just v

cloneToWords :: Vector Bit -> Vector Word

Clone an unboxed vector of bits to a new unboxed vector of words. If the bits don't completely fill the words, the last word will be zero-padded. Cf. cloneToWordsM.

>>> :set -XOverloadedLists
>>> cloneToWords [1,1,0,1,1,1,1]
[123]

castFromWords8 :: Vector Word8 -> Vector Bit

Cast a unboxed vector of Word8 to an unboxed vector of bits.

>>> :set -XOverloadedLists
>>> castFromWords8 [123]
[1,1,0,1,1,1,1,0]

castToWords8 :: Vector Bit -> Maybe (Vector Word8)

Try to cast an unboxed vector of bits to an unboxed vector of Word8. It succeeds if a vector of bits is aligned. Use cloneToWords8 otherwise.

castToWords8 (castFromWords8 v) == Just v

cloneToWords8 :: Vector Bit -> Vector Word8

Clone an unboxed vector of bits to a new unboxed vector of Word8. If the bits don't completely fill the bytes, the last Word8 will be zero-padded.

>>> :set -XOverloadedLists
>>> cloneToWords8 [1,1,0,1,1,1,1]
[123]

cloneFromByteString :: ByteString -> Vector Bit

Clone a ByteString to a new unboxed vector of bits.

>>> :set -XOverloadedStrings
>>> cloneFromByteString "abc"
[1,0,0,0,0,1,1,0,0,1,0,0,0,1,1,0,1,1,0,0,0,1,1,0]

cloneToByteString :: Vector Bit -> ByteString

Clone an unboxed vector of bits to a new ByteString. If the bits don't completely fill the bytes, the last character will be zero-padded.

>>> :set -XOverloadedLists
>>> cloneToByteString [1,0,0,0,0,1,1,0,0,1,0,0,0,1,1,0,1,1,0,0,0,1]
"ab#"

Immutable operations

zipBits :: (forall a. Bits a => a -> a -> a) -> Vector Bit -> Vector Bit -> Vector Bit

Zip two vectors with the given function. Similar to zipWith, but up to 1000x (!) faster.

For sufficiently dense sets, represented as bitmaps, zipBits is up to 32x faster than union, intersection, etc.

Users are strongly encouraged to enable flag libgmp for the ultimate performance of zipBits.

>>> :set -XOverloadedLists
>>> import Data.Bits
>>> zipBits (.&.) [1,1,0] [0,1,1] -- intersection
[0,1,0]
>>> zipBits (.|.) [1,1,0] [0,1,1] -- union
[1,1,1]
>>> zipBits (\x y -> x .&. complement y) [1,1,0] [0,1,1] -- difference
[1,0,0]
>>> zipBits xor [1,1,0] [0,1,1] -- symmetric difference
[1,0,1]

mapBits :: (forall a. Bits a => a -> a) -> Vector Bit -> Vector Bit

Map a vectors with the given function. Similar to map, but faster.

>>> :set -XOverloadedLists
>>> import Data.Bits
>>> mapBits complement [0,1,1]
[1,0,0]

invertBits :: Vector Bit -> Vector Bit

Invert (flip) all bits.

Users are strongly encouraged to enable flag libgmp for the ultimate performance of invertBits.

>>> :set -XOverloadedLists
>>> invertBits [0,1,0,1,0]
[1,0,1,0,1]

reverseBits :: Vector Bit -> Vector Bit

Reverse the order of bits.

>>> :set -XOverloadedLists
>>> reverseBits [1,1,0,1,0]
[0,1,0,1,1]

Consider using vector-rotcev package to reverse vectors in O(1) time.

bitIndex :: Bit -> Vector Bit -> Maybe Int

Return the index of the first bit in the vector with the specified value, if any. Similar to elemIndex, but up to 64x faster.

>>> :set -XOverloadedLists
>>> bitIndex 1 [0,0,1,0,1]
Just 2
>>> bitIndex 1 [0,0,0,0,0]
Nothing

bitIndex bit == nthBitIndex bit 1

One can also use it to reduce a vector with disjunction or conjunction:

import Data.Maybe
isAnyBitSet   = isJust    . bitIndex 1
areAllBitsSet = isNothing . bitIndex 0

nthBitIndex :: Bit -> Int -> Vector Bit -> Maybe Int

Return the index of the n-th bit in the vector with the specified value, if any. Here n is 1-based and the index is 0-based. Non-positive n results in an error.

>>> :set -XOverloadedLists
>>> nthBitIndex 1 2 [0,1,0,1,1,1,0] -- 2nd occurence of 1
Just 3
>>> nthBitIndex 1 5 [0,1,0,1,1,1,0] -- 5th occurence of 1
Nothing

One can use nthBitIndex to implement to implement select{0,1} queries for succinct dictionaries.

countBits :: Vector Bit -> Int

Return the number of set bits in a vector (population count, popcount).

Users are strongly encouraged to enable flag libgmp for the ultimate performance of countBits.

>>> :set -XOverloadedLists
>>> countBits [1,1,0,1,0,1]
4

One can combine countBits with take to implement rank{0,1} queries for succinct dictionaries.

listBits :: Vector Bit -> [Int]

Return 0-based indices of set bits in a vector.

>>> :set -XOverloadedLists
>>> listBits [1,1,0,1,0,1]
[0,1,3,5]

selectBits :: Vector Bit -> Vector Bit -> Vector Bit

For each set bit of the first argument, deposit the corresponding bit of the second argument to the result. Similar to the parallel deposit instruction (PDEP).

>>> :set -XOverloadedLists
>>> selectBits [0,1,0,1,1] [1,1,0,0,1]
[1,0,1]

Here is a reference (but slow) implementation:

import qualified Data.Vector.Unboxed as U
selectBits mask ws == U.map snd (U.filter (unBit . fst) (U.zip mask ws))

excludeBits :: Vector Bit -> Vector Bit -> Vector Bit

For each unset bit of the first argument, deposit the corresponding bit of the second argument to the result.

>>> :set -XOverloadedLists
>>> excludeBits [0,1,0,1,1] [1,1,0,0,1]
[1,0]

Here is a reference (but slow) implementation:

import qualified Data.Vector.Unboxed as U
excludeBits mask ws == U.map snd (U.filter (not . unBit . fst) (U.zip mask ws))

Mutable conversions

castFromWordsM :: MVector s Word -> MVector s Bit

Cast a vector of words to a vector of bits. Cf. castFromWords.

castToWordsM :: MVector s Bit -> Maybe (MVector s Word)

Try to cast a vector of bits to a vector of words. It succeeds if a vector of bits is aligned. Use cloneToWordsM otherwise. Cf. castToWords.

cloneToWordsM :: PrimMonad m => MVector (PrimState m) Bit -> m (MVector (PrimState m) Word)

Clone a vector of bits to a new unboxed vector of words. If the bits don't completely fill the words, the last word will be zero-padded. Cf. cloneToWords.

Mutable operations

zipInPlace :: forall m. PrimMonad m => (forall a. Bits a => a -> a -> a) -> Vector Bit -> MVector (PrimState m) Bit -> m ()

Zip two vectors with the given function. rewriting contents of the second argument. Cf. zipBits.

>>> :set -XOverloadedLists
>>> import Data.Bits
>>> Data.Vector.Unboxed.modify (zipInPlace (.&.) [1,1,0]) [0,1,1]
[0,1,0]

Warning: if the immutable vector is shorter than the mutable one, it is a caller's responsibility to trim the result:

>>> :set -XOverloadedLists
>>> import Data.Bits
>>> Data.Vector.Unboxed.modify (zipInPlace (.&.) [1,1,0]) [0,1,1,1,1,1]
[0,1,0,1,1,1] -- note trailing garbage

mapInPlace :: PrimMonad m => (forall a. Bits a => a -> a) -> MVector (PrimState m) Bit -> m ()

Apply a function to a mutable vector bitwise, rewriting its contents. Cf. mapBits.

>>> :set -XOverloadedLists
>>> import Data.Bits
>>> Data.Vector.Unboxed.modify (mapInPlace complement) [0,1,1]
[1,0,0]

invertInPlace :: PrimMonad m => MVector (PrimState m) Bit -> m ()

Invert (flip) all bits in-place.

>>> :set -XOverloadedLists
>>> Data.Vector.Unboxed.modify invertInPlace [0,1,0,1,0]
[1,0,1,0,1]

reverseInPlace :: PrimMonad m => MVector (PrimState m) Bit -> m ()

Reverse the order of bits in-place.

>>> :set -XOverloadedLists
>>> Data.Vector.Unboxed.modify reverseInPlace [1,1,0,1,0]
[0,1,0,1,1]

Consider using vector-rotcev package to reverse vectors in O(1) time.

selectBitsInPlace :: PrimMonad m => Vector Bit -> MVector (PrimState m) Bit -> m Int

Same as selectBits, but deposit selected bits in-place. Returns a number of selected bits. It is caller's responsibility to trim the result to this number.

>>> :set -XOverloadedLists
>>> import Control.Monad.ST (runST)
>>> import qualified Data.Vector.Unboxed as U
>>> runST $ do { vec <- U.unsafeThaw [1,1,0,0,1]; n <- selectBitsInPlace [0,1,0,1,1] vec; U.take n <$> U.unsafeFreeze vec }
[1,0,1]

excludeBitsInPlace :: PrimMonad m => Vector Bit -> MVector (PrimState m) Bit -> m Int

Same as excludeBits, but deposit excluded bits in-place. Returns a number of excluded bits. It is caller's responsibility to trim the result to this number.

>>> :set -XOverloadedLists
>>> import Control.Monad.ST (runST)
>>> import qualified Data.Vector.Unboxed as U
>>> runST $ do { vec <- U.unsafeThaw [1,1,0,0,1]; n <- excludeBitsInPlace [0,1,0,1,1] vec; U.take n <$> U.unsafeFreeze vec }
[1,0]

Binary polynomials

data F2Poly

Binary polynomials of one variable, backed by an unboxed Vector Bit.

Polynomials are stored normalized, without leading zero coefficients.

Ord instance does not make much sense mathematically, it is defined only for the sake of Set, Map, etc.

>>> :set -XBinaryLiterals
>>> -- (1 + x) (1 + x + x^2) = 1 + x^3 (mod 2)
>>> 0b11 * 0b111 :: F2Poly
0b1001

Instances

Enum F2Poly

Defined in Data.Bit.F2Poly

Methods

succ :: F2Poly -> F2Poly

pred :: F2Poly -> F2Poly

toEnum :: Int -> F2Poly

fromEnum :: F2Poly -> Int

enumFrom :: F2Poly -> [F2Poly]

enumFromThen :: F2Poly -> F2Poly -> [F2Poly]

enumFromTo :: F2Poly -> F2Poly -> [F2Poly]

enumFromThenTo :: F2Poly -> F2Poly -> F2Poly -> [F2Poly]

Eq F2Poly

Defined in Data.Bit.F2Poly

Methods

(==) :: F2Poly -> F2Poly -> Bool

(/=) :: F2Poly -> F2Poly -> Bool

Integral F2Poly

toInteger converts a binary polynomial, encoded as F2Poly, to Integer encoding.

Defined in Data.Bit.F2Poly

Methods

quot :: F2Poly -> F2Poly -> F2Poly

rem :: F2Poly -> F2Poly -> F2Poly

div :: F2Poly -> F2Poly -> F2Poly

mod :: F2Poly -> F2Poly -> F2Poly

quotRem :: F2Poly -> F2Poly -> (F2Poly, F2Poly)

divMod :: F2Poly -> F2Poly -> (F2Poly, F2Poly)

toInteger :: F2Poly -> Integer

Num F2Poly

Addition and multiplication are evaluated modulo 2.

abs = id and signum = const 1.

fromInteger converts a binary polynomial, encoded as Integer, to F2Poly encoding.

Defined in Data.Bit.F2Poly

Methods

(+) :: F2Poly -> F2Poly -> F2Poly

(-) :: F2Poly -> F2Poly -> F2Poly

(*) :: F2Poly -> F2Poly -> F2Poly

negate :: F2Poly -> F2Poly

abs :: F2Poly -> F2Poly

signum :: F2Poly -> F2Poly

fromInteger :: Integer -> F2Poly

Ord F2Poly

Defined in Data.Bit.F2Poly

Methods

compare :: F2Poly -> F2Poly -> Ordering

(<) :: F2Poly -> F2Poly -> Bool

(<=) :: F2Poly -> F2Poly -> Bool

(>) :: F2Poly -> F2Poly -> Bool

(>=) :: F2Poly -> F2Poly -> Bool

max :: F2Poly -> F2Poly -> F2Poly

min :: F2Poly -> F2Poly -> F2Poly

Real F2Poly

Defined in Data.Bit.F2Poly

Methods

toRational :: F2Poly -> Rational

Show F2Poly

Defined in Data.Bit.F2Poly

Methods

showsPrec :: Int -> F2Poly -> ShowS

show :: F2Poly -> String

showList :: [F2Poly] -> ShowS

Generic F2Poly

Defined in Data.Bit.F2Poly

Associated Types

type Rep F2Poly :: Type -> Type

Methods

from :: F2Poly -> Rep F2Poly x

to :: Rep F2Poly x -> F2Poly

NFData F2Poly

Defined in Data.Bit.F2Poly

Methods

rnf :: F2Poly -> ()

type Rep F2Poly

Defined in Data.Bit.F2Poly

type Rep F2Poly = D1 ('MetaData "F2Poly" "Data.Bit.F2Poly" "bitvec-1.1.1.0-AhfNuieIoBoKwVuNuMi18y" 'True) (C1 ('MetaCons "F2Poly" 'PrefixI 'True) (S1 ('MetaSel ('Just "unF2Poly") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Vector Bit))))

unF2Poly :: F2Poly -> Vector Bit

Convert F2Poly to a vector of coefficients (first element corresponds to a constant term).

>>> :set -XBinaryLiterals
>>> unF2Poly 0b1101
[1,0,1,1]

toF2Poly :: Vector Bit -> F2Poly

Make F2Poly from a list of coefficients (first element corresponds to a constant term).

>>> :set -XOverloadedLists
>>> toF2Poly [1,0,1,1,0,0]
0b1101

gcdExt :: F2Poly -> F2Poly -> (F2Poly, F2Poly)

Execute the extended Euclidean algorithm. For polynomials a and b, compute their unique greatest common divisor g and the unique coefficient polynomial s satisfying as + bt = g.

>>> :set -XBinaryLiterals
>>> gcdExt 0b101 0b0101
(0b101,0b0)
>>> gcdExt 0b11 0b111
(0b1,0b10)