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computationalalgebra
Wellkinded computational algebra library, currently supporting Groebner basis.
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README
mod
Modular arithmetic, promoting moduli to the type level, with an emphasis on performance. Originally a part of arithmoi package.
> :set XDataKinds
> 4 + 5 :: Mod 7
(2 `modulo` 7)
> 4  5 :: Mod 7
(6 `modulo` 7)
> 4 * 5 :: Mod 7
(6 `modulo` 7)
> 4 / 5 :: Mod 7
(5 `modulo` 7)
> 4 ^ 5 :: Mod 7
(2 `modulo` 7)
Competitors
There are other Haskell packages, employing the very same idea of moduli on the type level,
namely modular
and modulararithmetic
. One can also use finitetypelits
,
which covers some elementary modular arithmetic as well.
Unfortunately, all of them fall behind
in terms of performance. Here is a brief comparison:
Discipline  mod 
modular 
modulararithmetic 
finitetypelits 

Addition  Fast  Slow  Slow  Slow 
Small (*) 
Fast  Slow  Slow  Slow 
Inversion  Fast  N/A  Slow  N/A 
Power  Fast  Slow  Slow  N/A 
Overflows  Safe  Safe  Unsafe  Safe 
Addition. All competing implementations of the modular addition involve divisions, while
mod
completely avoids this costly operation. It makes difference even for small numbers; e. g.,sum [1..10^7]
becomes 5x faster. For larger integers the speed up is even more significant, because the computational complexity of division is not linear.Small
(*)
. When a modulo fits a machine word (which is quite a common case on 64bit architectures),mod
implements the modular multiplication as a couple of CPU instructions and neither allocates intermediate arbitraryprecision values, nor callslibgmp
at all. For computations likeproduct [1..10^7]
this gives a 3x boost to performance in comparison to other libraries.Inversion. This package relies on
libgmp
for modular inversions. Even for small arguments it is about 5x faster than the native implementation of modular inversion inmodulararithmetic
.Power. This package relies on
libgmp
for modular exponentiation. Even for small arguments it is about 2x faster than competitors.Overflows. At first glance
modulararithmetic
is more flexible thanmod
, because it allows to specify the underlying representation of a modular residue, e. g.,Mod Integer 100
,Mod Int 100
,Mod Word8 100
. We argue that this is a dangerous freedom, vulnerable to overflows. For instance,20 ^ 2 :: Mod Word8 100
returns44
instead of expected0
. Even less expected is that50 :: Mod Word8 300
appears to be6
(remember that typelevel numbers are alwaysNatural
).
Citius, altius, fortius!
If you are looking for an ultimate performance
and your moduli fit into Word
,
try Data.Mod.Word
,
which is a dropin replacement of Data.Mod
,
but offers almost 3x faster addition,
2x faster multiplication and much less allocations.
Benchmarks
Here are some relative benchmarks,
which can be reproduced by running cabal bench
.
Discipline  Data.Mod.Word 
Data.Mod 
modular 
modulararithmetic 
finitetypelits 

Sum  0.4x  1x  4.2x  4.9x  2.9x 
Product  0.5x  1x  3.2x  4.4x  2.8x 
Inversion  0.8x  1x  N/A  6.1x  N/A 
Power  0.9x  1x  5.4x  1.8x  N/A 
What's next?
This package was cut out of arithmoi
to provide a modular arithmetic
with a light dependency footprint. This goal certainly limits the scope of API
to the bare minimum. If you need more advanced tools
(the Chinese remainder theorem, cyclic groups, modular equations, etc.)
please refer to Math.NumberTheory.Moduli.