mixedtypesnum
This package provides a version of Prelude where
unary and binary operations such as not
, +
, ==
have their result type derived from the parameter type(s)
and thus supports mixedtype arithmetic and comparisons such as:
> a = [1..10]; b = [1..11]
> length a > 2^((length b)/3)
{?(prec 36): CertainFalse}
Partial operations such as division, sqrt and power
do not throw exceptions even when errors such as division by zero
occur. Instead, these errors are propagated bottomup in
a bespoke erroraccumulating functor from package collecterrors.
This library (as well as collecterrors) arose while developing the
AERN2 library for interval and exact real computation.
Certain aspects are specifically tailored for interval or exact real arithmetics,
including threevalued numerical comparisons and distinguishing potential and certain errors.
API documentation available on the Hackage page.
Table of contents
1. Examples
To replicate the examples included below, start ghci as follows:
$ stack ghci mixedtypesnum:lib noload ghcioptions MixedTypesNumPrelude
*MixedTypesNumPrelude>
1.1. Main idea
Literals have a fixed type:
...> :t 1
... Integer
...> :t 1.0
... Rational
...> 1 :: Rational
... Couldn't match type ‘Integer’ with ‘GHC.Real.Ratio Integer’ ...
Operations permit operands of mixed types, types inferred bottomup:
...> :t 1/2
... :: Rational
...> :t 1.5 * (length [[]])
... :: Rational
1.2. Dealing with numerical errors
To avoid runtime exceptions, it is recommended to use the CN errorcollecting wrapper from package collecterrors.
All arithmetic operations have been extended so that it is possible to have expressions that operate exclusively on CNwrapped types:
...> f (n :: CN Integer) = 1/(1/(n1) + 1/n) :: CN Rational
...> f (cn 0)
{{ERROR: division by 0}}
...> f (cn 1)
{{ERROR: division by 0}}
...> f (cn 2)
2 % 3
Note that the errors printed above are not exceptions, but special values. See the collecterrors documentation for more details.
1.3. The generalised power operator
...> :t 2^(2)
... :: Rational
...> :t 2^2
... :: Rational
...> :t round (2^2)
... :: Integer
...> :t (double 2)^(1/2)
... :: Double
The following examples require also package aern2real.
To get access to this via stack, you can start ghci eg as follows:
$ stack ghci aern2real:lib noload ghcioptions AERN2.Real
AERN2.Real> import MixedTypesNumPrelude
...> :t pi
... :: CReal
...> :t sqrt 2
... :: CReal
...> :t 2^(1/2)
... :: CReal
1.4. Undecided comparisons
Comparisons involving intervals are undecided when the intervals overlap:
> pi10 = pi ? (bits 10)
> pi10
[3.1416015625 ± ~9.7656e4 ~2^(10)]
> pi10 > 0
CertainTrue
> pi10 == pi10
TrueOrFalse
The above equality cannot be decided since pi10
is not a single number but a set of numbers spanning the interval and the comparison operator cannot tell if the two operands sets represent the same number or a different number.
Comparison involving real numbers are semidecidable. The result of such a comparison is a lazy Kleenean, ie an infinite sequence of Kleeneans.
Please see package aern2real for further details.
1.5. Fuzzy ifthenelse
This package generalises the Haskell ifthenelse statement so that it admits Kleenean and lazy Kleenean conditions:
...> abs1 x = max 0 (if x < 0 then x else x)
...> abs1 (pi10  pi10)
[0.0009765625 ± ~9.7656e4 ~2^(10)]
Although the condition x < 0
cannot be decided for the interval
pi10pi10 = [0 ± ~1.9531e3 ~2^(9)]
, the ifthenelse statement is resolved by computing both branches and unifying the resulting intervals. This makes sense only if both branches compute the same number whenever the condition cannot be decided, ie when x = 0
in this case, making the function continuous.
If we try to define a discontinuous function this way, we get an error as soon as it is detected:
...> bad1 x = if x < 0 then 1x else x
...> bad1 (pi10  pi10)
[0.5 ± ~0.5020 ~2^(1)]{{ERROR: numeric error: union of enclosures: not enclosing the same value}}
The generalised ifthenelse works also for real numbers with lazy Kleenean comparisons:
...> abs1 (pi  pi)
{?(prec 36): [0.00000000001455191522836685... ± ~1.4552e11 ~2^(36)]}
2. Type classes
Mixedtype arithmetic operations are provided via multiparameter type classes
and the result type is given by associated
type families. For example:
(+) :: (CanAddAsymmetric t1 t2) => t1 > t2 > AddType t1 t2
The constraint CanAdd t1 t2
is a shortcut for both
CanAddAsymmetric t1 t2
and CanAddAsymmetric t2 t1
.
For convenience there are other aggregate type constraints such as
CanAddThis t1 t2
, which implies that the result is of type t1
,
and CanAddSameType t
, which is a shortcut for CanAddThis t t
.
Notably, there are convenience classes Ring
and Field
as well as OrderedRing
and OrderedField
.
For types that instantiate Prelude classes such as Num
, one can
define instances of the new classes using the default implementation, eg:
{# LANGUAGE GeneralizedNewtypeDeriving #}
import MixedTypesPrelude
import qualified Prelude as P
newtype II = II Integer deriving (P.Eq, P.Ord, P.Num)
instance CanAddAsymmetric II II
Conversely, if one defines instances such as CanAddAsymmetric T T
,
one can then trivially define also instances Num T
etc:
instance P.Num T where
(+) = (+)
...
3. Testable specifications
The arithmetic type classes are accompanied by generic hspec test suites,
which are specialised to concrete instance types for their testing.
These test suites include the expected algebraic properties of operations,
such as commutativity and associativity of addition.
4. Limitations

Not all numerical operations are supported yet.
Eg tan
, atan
are missing at the moment.

Not all Prelude numerical types are supported yet.
Eg Natural
and Float
are not supported at present,
but Double
is supported.

Many common operations such as fromEnum
, threadDelay
give or require
an Int
value, which means we sometimes need to convert:
threadDelay (int 1000000)
integer (fromEnum True)
Prelude functions such as take
, !!
and length
that use Int
in Prelude
are shadowed in MixedTypesNumPrelude with more compatible/flexible versions.
Beware that Data.List.length
clashes with length
in MixedTypesNumPrelude.

Inferred types can be very large. Eg for f a b c = sqrt (a + b * c + 1)
the inferred type is:
f :: (CanSqrt (AddType (AddType t2 (MulType t3 t4)) Integer),
CanAddAsymmetric (AddType t2 (MulType t3 t4)) Integer,
CanAddAsymmetric t2 (MulType t3 t4), CanMulAsymmetric t3 t4) =>
t2
> t3
> t4
> SqrtType (AddType (AddType t2 (MulType t3 t4)) Integer)
5. Credits
The idea of having numeric expressions in Haskell with types
derived bottomup was initially suggested and implemented by Pieter Collins.
This version is a fresh rewrite by Michal Konečný.