BSD-3-Clause licensed by Andrew Lelechenko, James Cook
Maintained by Andrew Lelechenko
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Module documentation for

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A newtype over Bool with a better Vector instance: 8x less memory, up to 1000x faster.

The vector package represents unboxed arrays of Bool spending 1 byte (8 bits) per boolean. This library provides a newtype wrapper Bit and a custom instance of unboxed Vector, which packs bits densely, achieving 8x less memory footprint. The performance stays mostly the same; the most significant degradation happens for random writes (up to 10% slower). On the other hand, for certain bulk bit operations Vector Bit is up to 1000x faster than Vector Bool.

Thread safety

  • Data.Bit is faster, but writes and flips are thread-unsafe. This is because naive updates are not atomic: read the whole word from memory, then modify a bit, then write the whole word back.
  • Data.Bit.ThreadSafe is slower (up to 20%), but writes and flips are thread-safe.

Quick start

Consider the following (very naive) implementation of the sieve of Eratosthenes. It returns a vector with True at prime indices and False at composite indices.

import Control.Monad
import Control.Monad.ST
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as MU

eratosthenes :: U.Vector Bool
eratosthenes = runST $ do
  let len = 100
  sieve <- MU.replicate len True
  MU.write sieve 0 False
  MU.write sieve 1 False
  forM_ [2 .. floor (sqrt (fromIntegral len))] $ \p -> do
    isPrime <- sieve p
    when isPrime $
      forM_ [2 * p, 3 * p .. len - 1] $ \i ->
        MU.write sieve i False
  U.unsafeFreeze sieve

We can switch from Bool to Bit just by adding newtype constructors:

import Data.Bit

import Control.Monad
import Control.Monad.ST
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as MU

eratosthenes :: U.Vector Bit
eratosthenes = runST $ do
  let len = 100
  sieve <- MU.replicate len (Bit True)
  MU.write sieve 0 (Bit False)
  MU.write sieve 1 (Bit False)
  forM_ [2 .. floor (sqrt (fromIntegral len))] $ \p -> do
    Bit isPrime <- sieve p
    when isPrime $
      forM_ [2 * p, 3 * p .. len - 1] $ \i ->
        MU.write sieve i (Bit False)
  U.unsafeFreeze sieve

Bit-based implementation requires 8x less memory to store the vector. For large sizes it allows to crunch more data in RAM without swapping. For smaller arrays it helps to fit into CPU caches.

> listBits eratosthenes

There are several high-level helpers, digesting bits in bulk, which makes them up to 64x faster than respective counterparts for Vector Bool. One can query population count (popcount) of a vector (giving us the prime-counting function):

> countBits eratosthenes

And vice versa, query an address of the n-th set bit (which corresponds to the n-th prime number here):

> nthBitIndex (Bit True) 10 eratosthenes
Just 29

One may notice that the order of the inner traversal by i does not matter and get tempted to run it in several parallel threads. In this case it is vital to switch from Data.Bit to Data.Bit.ThreadSafe, because the former is thread-unsafe with regards to writes. There is a moderate performance penalty (up to 20%) for using the thread-safe interface.


Bit vectors can be used as a blazingly fast representation of sets as long as their elements are Enumeratable and sufficiently dense, leaving IntSet far behind.

For example, consider three possible representations of a set of Word16:

  • As an IntSet with a readily available union function.
  • As a 64k-long unboxed Vector Bool, implementing union as zipWith (||).
  • As a 64k-long unboxed Vector Bit, implementing union as zipBits (.|.).

When flag libgmp is enabled, according to our benchmarks (see bench folder) the union of Vector Bit evaluates 24x-36x faster than the union of not-too-sparse IntSet and stunningly outperforms Vector Bool 500x-1000x.

Binary polynomials

Binary polynomials are polynomials with coefficients modulo 2. Their applications include coding theory and cryptography. While one can successfully implement them with poly package, operating on UPoly Bit, this package provides even faster arithmetic routines exposed via F2Poly data type and its instances.

> :set -XBinaryLiterals
> -- (1 + x) (1 + x + x^2) = 1 + x^3 (mod 2)
> 0b11 * 0b111 :: F2Poly
F2Poly {unF2Poly = [1,0,0,1]}

Use fromInteger / toInteger to convert binary polynomials from Integer to F2Poly and back.

Package flags

Similar packages

  • bv and bv-little do not offer mutable vectors.

  • array is memory-efficient for Bool, but lacks a handy Vector interface and is not thread-safe.

Additional resources

  • Bit vectors without compromises, Haskell Love, 31.07.2020: slides, video.


  • Fix setBit, clearBit, complementBit to preserve vector’s length.
  • Fix various issues on big-endian architectures.
  • Fix Cabal 3.7+ incompatibility.

  • Export BitVec and BitMVec constructors.

  • Fix a grave bug in bitIndex.
  • Remove integer-gmp flag.
  • Make libgmp flag disabled by default. Users are strongly encouraged to enable it whenever possible.
  • Add mapBits and mapInPlace functions.
  • Add cloneToByteString and cloneFromByteString functions.

  • Add Bits (Vector Bit) instance.
  • Add castFromWords8, castToWords8, cloneToWords8 to facilitate interoperation with ByteString.

  • Fix out-of-bounds writes in mutable interface.
  • Improve thread-safety of mutable interface.
  • Add extended GCD for F2Poly.
  • Change Show instance of F2Poly.

  • Fix more bugs in F2Poly multiplication.

  • Fix bugs in F2Poly multiplication.
  • Performance improvements.

  • Implement arithmetic of binary polynomials.
  • Add invertBits and reverseBits functions.
  • Add Num, Real, Integral, Fractional and NFData instances.
  • Performance improvements.

  • Performance improvements.

  • Redesign API from the scratch.
  • Add a thread-safe implementation.
  • Add nthBitIndex function.

  • Fix Read instance.

  • Remove hand-written Num, Real, Integral, Bits instances.
  • Derive Bits and FiniteBits instances.
  • Expose Bit constructor directly and remove fromBool function.
  • Rename toBool to unBit.

  • Fix bugs in MVector and Vector instances of Bit.
  • Speed up MVector and Vector instances of Bit.