{-|
    Module      :  MixedTypesNumPrelude
    Description :  Bottom-up typed numeric expressions
    Copyright   :  (c) Michal Konecny, Pieter Collins
    License     :  BSD3

    Maintainer  :  mikkonecny@gmail.com
    Stability   :  experimental
    Portability :  portable

    @MixedTypesNumPrelude@ provides a version of @Prelude@ where
    unary and binary operations such as @not@, @+@, @==@
    have their result type derived from the parameter type(s).

    This module facilitates a single-line import for the package
    mixed-types-num.  See the re-exported modules for further details.
-}


module MixedTypesNumPrelude
(
  -- * Feature highlights

  -- ** Basics
  -- $basics

  -- ** Type classes
  -- $classes

  -- ** Testable specifications
  -- $specs

  -- * Limitations
  -- $limitations

  -- * Origin
  -- $origin

  -- * Re-exporting Prelude, hiding the operators we are changing
  module Numeric.MixedTypes.PreludeHiding,
  -- * A part of package ``convertible''
  module Data.Convertible.Base,
  -- * Modules with Prelude alternatives
  module Numeric.MixedTypes.Literals,
  module Numeric.MixedTypes.Bool,
  module Numeric.MixedTypes.Kleenean,
  module Numeric.MixedTypes.Eq,
  module Numeric.MixedTypes.Ord,
  module Numeric.MixedTypes.MinMaxAbs,
  module Numeric.MixedTypes.AddSub,
  module Numeric.MixedTypes.Round,
  module Numeric.MixedTypes.Reduce,
  module Numeric.MixedTypes.Ring,
  module Numeric.MixedTypes.Div,
  module Numeric.MixedTypes.Power,
  module Numeric.MixedTypes.Field,
  module Numeric.MixedTypes.Elementary,
  module Numeric.MixedTypes.Complex,
  -- module Numeric.CollectErrors,
  module Utils.TH.DeclForTypes,
  module Utils.Test.EnforceRange,
  -- * Re-export for convenient Rational literals
  (%)
)
where

import Data.Ratio ((%))
import Data.Convertible.Instances.Num()
import Data.Convertible.Base
import Utils.TH.DeclForTypes
import Utils.Test.EnforceRange
import Numeric.MixedTypes.PreludeHiding
import Numeric.MixedTypes.Literals
import Numeric.MixedTypes.Bool
import Numeric.MixedTypes.Kleenean
import Numeric.MixedTypes.Eq
import Numeric.MixedTypes.Ord
import Numeric.MixedTypes.MinMaxAbs
import Numeric.MixedTypes.AddSub
import Numeric.MixedTypes.Round
import Numeric.MixedTypes.Reduce
import Numeric.MixedTypes.Ring
import Numeric.MixedTypes.Div
import Numeric.MixedTypes.Power
import Numeric.MixedTypes.Field
import Numeric.MixedTypes.Elementary
import Numeric.MixedTypes.Complex

{- $basics

To replicate the below in ghci using stack, start it as follows:

>> stack ghci mixed-types-num:lib
>...> :add MixedTypesNumPrelude

=== Literals have a fixed type

>...> :t 1
>... Integer

>...> :t 1.0
>... Rational

>...> 1 :: Rational
>... Couldn't match type ‘Integer’ with ‘GHC.Real.Ratio Integer’ ...

=== Mixed-type operations

>...> :t 1.5 + 1
>... :: Rational

>...> :t 1.5 * (length [[]])
>... :: Rational

=== Dividing integers, dealing with potential error

>...> :t let n = 1 in n/(n+1)
>... :: Rational

To avoid runtime exceptions, it is recommended to use the CN error-collecting wrapper from package collect-errors:

>...> :t let n = cn 1 in n/(n+1)
>... :: CN Rational

@CN@ is a synonym for @CollectErrors [(ErrorCertaintyLevel, NumError)] Rational@ as defined in module "Numeric.CollectErrors".
The @CN@ wrapper indicates that integer division can fail for some values:

>...> let n = cn 1 in n/(n-1)
>{[(division by 0,ERROR)]}

Note that the error printed above is not an exception, but a special value.

All arithmetic operations have been extended to CN types so that it is possible to
have expressions that operate exclusively on CN types:

>...> f (n :: CN Integer) = 1/(1/(n-1) + 1/n) :: CN Rational
>...> f (cn 0)
>{[(division by 0,POTENTIAL ERROR),(division by 0,ERROR)]}
>...> f (cn 1)
>{[(division by 0,POTENTIAL ERROR),(division by 0,ERROR)]}
>...> f (cn 2)
>2 % 3

The function @hasError@ can be used to check whether any error occurred:

>...> hasError (cn 1/0)
>True

>...> hasError (cn 1/1)
>False

To extract a value from the CN wrapper, one can use function @withErrorOrValue@:

>...> withErrorOrValue (const 0.0) id (cn 1/2)
>1 % 2

The following examples require also package <https://github.com/michalkonecny/aern2 aern2-real>.
To get access to this via stack, you can start ghci eg as follows:

> stack ghci aern2-real:lib
>...> :add AERN2.Real

Also other harmless potential errors can be ignored using @(~!)@:

>...> (~!) $ sqrt (pi-pi) ? (bitsS 10)
> [0.000007629... ± 7.6294e-6 <2^(-17)]

>...> sqrt (pi-pi) ? (bitsS 10)
> [0.000007629... ± 7.6294e-6 <2^(-17)]{[(POTENTIAL ERROR,out of range: sqrt: argument must be >= 0: [0 ± 2.3283e-10 <2^(-32)])]}

When an error is present (which can be checked using hasErrorCN), the function hasCertainErrorCN can be used to further distinguish cases where the error is certain or potential:

>...> hasCertainErrorCN (sqrt (-1) ? (bitsS 10))
>True

>...> hasCertainErrorCN (sqrt (pi-pi) ? (bitsS 10))
>False


=== Natural, integer and fractional powers

>...> :t 2^2
>...CN Integer

>...> :t 2.0^(-2)
>...CN Rational

>...> :t (double 2)^(1/!2)
>...Double

The following examples require package <https://github.com/michalkonecny/aern2 aern2-real>:

>...> :t 2^(1/2)
>...CauchyRealCN

>...> :t pi
>...CauchyReal

>...> :t sqrt 2
>...CauchyRealCN

=== Comparing an integer with an (exact) real number

>...> let abs2 x = if x < 0 then -x else x in (abs2 (pi - pi)) ? (bitsS 100)
>[0 ± <2^(-103)]{[(POTENTIAL ERROR,numeric error: union of enclosures: not enclosing the same value)]}

The potential error means that both branches were executed in parallel because
the condition could not be decided, and it was moreover impossible to guarantee
(in general) that both branches will return the same number.  If we make a mistake,
this error may appear with certainty, eg:

>...> let abs2 x = if x < 0 then 1-x else x in (abs2 (pi - pi)) ? (bitsS 100)
>*** Exception: WithGlobalParam ensureNoCE: [(ERROR,numeric error: union of enclosures: not enclosing the same value)]

If we are certain such errors will never appear, we can silence the potential error warnings:

>...> let abs2 x = (~!) (if x < 0 then -x else x) in (abs2 (pi - pi)) ? (bitsS 100)
>[0 ± <2^(-103)]

In these examples, @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.

-}

{- $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@.

    Notably, there are convenience classes `Ring` and `Field`.

    For types that instantiate Prelude classes such as `Num`, one can
    define instances of the new classes using the default implementation, eg:


    > newtype II = II Integer deriving (Eq, Ord, Num) -- assuming -XGeneralizedNewtypeDeriving
    > instance CanAddAsymmetric II II

    Conversely, if one defines instances for classes such as `CanAddAsymmetric`,
    one can then trivially define also instances of `Num` etc:

    > instance Prelude.Num T where
    >   (+) = (+)
    >   ...

-}

{- $specs
    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.

    * 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: (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.

-}