Numeric.MixedTypes

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:

let n = 1 :: Integer in n/(n+1) :: Rational

1/2 :: Rational

(The type Rational would be derived automatically because integer literals are always of type Integer, not Num t => t.)

• adding an integer and a rational, giving a rational:

(length [x])+1/3 :: Rational

• taking natural, integer and fractional power using the same operator:

2^2 :: Integer

2.0^(-2) :: Rational

(double 2)^(1/2) :: Double

The following examples require package aern2-real:

2^(1/2) :: CauchyReal

pi :: CauchyReal

sqrt 2 :: CauchyReal

• comparing an integer with an (exact) real number, giving a Maybe Bool:

... x :: CauchyReal ... if (isCertainlyTrue (x > 1)) then ...

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 type add1 :: (CanAdd Integer t) => t -> AddType t Integer or use an explicit parameter, eg add1 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.

Re-exporting Prelude, hiding the operators we are changing

module Numeric.MixedTypes.PreludeHiding

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.Elementary