Copyright | (c) Michal Konecny Pieter Collins |
---|---|
License | BSD3 |
Maintainer | mikkonecny@gmail.com |
Stability | experimental |
Portability | portable |
Safe Haskell | None |
Language | Haskell98 |
Main purpose
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),
allowing, e.g.:
- Dividing an integer by an integer, giving a rational, wrapped in the CN (ie Collecting NumErrors) monad:
>>>
:t let n = 1 :: Integer in n/(n+1)
...CN Rational
>>>
:t 1/2
...CN Rational
(Integer literals are always of type Integer
, not Num t => t
.)
- Adding an integer and a rational, giving a rational:
>>>
:t (length [])+1/3
...CN Rational
The CN
monad is required because integer division can, in general, fail as it is a partial operation:
>>>
1/0
{[(ERROR,division by 0)]}
Note that when evaluating 1/0
, it evaluates to the error value printed above.
This is not an exception, but a special value.
When one is certain the division is well defined, one can remove CN
in several ways:
>>>
:t (1%2)
...Rational
Above we use (re-exported) Data.Ratio.(%), which means this trick works only for Integers.
>>>
:t (1/!2)
...Rational
This works also for non-integer division.
>>>
:t (~!) (1/2)
...Rational
The (~!) operator removes CN from any type, throwing an exception if there are collected errors.
The operator (/!) stands for division which throws an exception is the denominator is 0. It "propagates" any potential errors from the sub-expressions. For example:
>>>
:t 1/!(1 - 1/n)
...CN Rational
The above expression will throw an error exception when evaluated with n=1
but when n=0
, it will not throw an excetion but return an error value.
- taking natural, integer and fractional power using the same operator:
>>>
:t 2^2
...CN Integer
>>>
:t 2.0^(-2)
...CN Rational
>>>
:t (double 2)^(1/!2)
...Double
The following examples require package aern2-real:
>>>
:t 2^(1/2)
...CauchyRealCN
>>>
:t pi
...CauchyReal
>>>
:t sqrt 2
...CauchyRealCN
- comparing an integer with an (exact) real number, giving a sequence of
Maybe Bool
:
>>>
let abs2 x = if x < 0 then -x else x in (abs2 (pi - pi)) ? (bitsS 100)
[0 ± <2^(-102)]
In the last example, if
is overloaded so that it works for conditions
of other types than Bool
. Here the condition has the type Sequence (Maybe Bool)
.
The whole expression is the sequence of balls in which those balls for which the condition
is inconclusive are the union of the balls computed by both branches.
Type classes
Arithmetic operations are provided via multi-parameter type classes and the result type is given by associated type families. For example:
(+) :: (CanAddAsymmetric t1 t2) => t1 -> t2 -> AddType t1 t2
The type constraint CanAdd t1 t2
implies 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
.
Testable specification
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.
Limitations
- Not all numerical operations are supported yet.
Eg
tan
,atan
are missing at the moment.
- Inferred types can be very large. Eg for
f a b c = sqrt (a + b * c + 1)
the inferred type is:
f: (CanMulAsymmetric t1 t2, CanAddAsymmetric t4 (MulType t1 t2), CanAddAsymmetric (AddType t4 (MulType t1 t2)) Integer, CanSqrt (AddType (AddType t4 (MulType t1 t2)) Integer)) => t4 -> t1 -> t2 -> SqrtType (AddType (AddType t4 (MulType t1 t2)) Integer)
- Due to limitations of some versions of ghc, type inferrence sometimes fails.
Eg
add1 = (+ 1)
fails (eg with ghc 8.0.2) unless we explicitly declare the typeadd1 :: (CanAdd Integer t) => t -> AddType t Integer
or use an explicit parameter, egadd1 x = x + 1
.
Origin
The idea of having numeric expressions in Haskell with types derived bottom-up was initially suggested and implemented by Pieter Collins. This version is a fresh rewrite by Michal Konečný.
More details
This module facilitates a single-line import for the package mixed-types-num. See the re-exported modules for further details.
- module Numeric.MixedTypes.PreludeHiding
- module Numeric.MixedTypes.Literals
- module Numeric.MixedTypes.Bool
- module Numeric.MixedTypes.Eq
- module Numeric.MixedTypes.Ord
- module Numeric.MixedTypes.MinMaxAbs
- module Numeric.MixedTypes.AddSub
- module Numeric.MixedTypes.Round
- module Numeric.MixedTypes.Ring
- module Numeric.MixedTypes.Field
- module Numeric.MixedTypes.Elementary
- module Numeric.MixedTypes.Complex
- module Numeric.CollectErrors
- module Utils.TH.DeclForTypes
- (%) :: Integral a => a -> a -> Ratio a
Re-exporting Prelude, hiding the operators we are changing
Modules with Prelude alternatives
module Numeric.MixedTypes.Literals
module Numeric.MixedTypes.Bool
module Numeric.MixedTypes.Eq
module Numeric.MixedTypes.Ord
module Numeric.MixedTypes.MinMaxAbs
module Numeric.MixedTypes.AddSub
module Numeric.MixedTypes.Round
module Numeric.MixedTypes.Ring
module Numeric.MixedTypes.Field
module Numeric.MixedTypes.Complex
module Numeric.CollectErrors
module Utils.TH.DeclForTypes