Field.hs

{-# LANGUAGE RebindableSyntax #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# OPTIONS_GHC -Wall #-}

-- | Field classes
module NumHask.Algebra.Field
  ( Semifield
  , Field
  , ExpField(..)
  , QuotientField(..)
  , UpperBoundedField(..)
  , LowerBoundedField(..)
  , BoundedField
  , TrigField(..)
  ) where

import Data.Complex (Complex(..))
import NumHask.Algebra.Additive
import NumHask.Algebra.Multiplicative
import NumHask.Algebra.Ring
import NumHask.Algebra.Integral
import Data.Bool (bool)
import Prelude (Double, Float, Integer, (||))
import qualified Prelude as P

-- | A Semifield is chosen here to be a Field without an Additive Inverse
class (MultiplicativeInvertible a, MultiplicativeGroup a, Semiring a) =>
      Semifield a

instance Semifield Double

instance Semifield Float

instance (Semifield a, AdditiveGroup a) => Semifield (Complex a)

-- | A Field is a Ring plus additive invertible and multiplicative invertible operations.
--
-- A summary of the rules inherited from super-classes of Field
--
-- > zero + a == a
-- > a + zero == a
-- > (a + b) + c == a + (b + c)
-- > a + b == b + a
-- > a - a = zero
-- > negate a = zero - a
-- > negate a + a = zero
-- > a + negate a = zero
-- > one * a == a
-- > a * one == a
-- > (a * b) * c == a * (b * c)
-- > a * (b + c) == a * b + a * c
-- > (a + b) * c == a * c + b * c
-- > a * zero == zero
-- > zero * a == zero
-- > a * b == b * a
-- > a / a = one
-- > recip a = one / a
-- > recip a * a = one
-- > a * recip a = one
class (AdditiveGroup a, MultiplicativeGroup a, Ring a) =>
      Field a

instance Field Double

instance Field Float

instance (Field a) => Field (Complex a)

-- | A hyperbolic field class
--
-- > sqrt . (**2) == identity
-- > log . exp == identity
-- > for +ive b, a != 0,1: a ** logBase a b ≈ b
class (Field a) =>
      ExpField a where
  exp :: a -> a
  log :: a -> a
  logBase :: a -> a -> a
  logBase a b = log b / log a
  (**) :: a -> a -> a
  (**) a b = exp (log a * b)
  sqrt :: a -> a
  sqrt a = a ** (one / (one + one))

instance ExpField Double where
  exp = P.exp
  log = P.log
  (**) = (P.**)

instance ExpField Float where
  exp = P.exp
  log = P.log
  (**) = (P.**)

-- | todo: bottom is here somewhere???
instance (P.Ord a, TrigField a, ExpField a) => ExpField (Complex a) where
  exp (rx :+ ix) = exp rx * cos ix :+ exp rx * sin ix
  log (rx :+ ix) = log (sqrt (rx * rx + ix * ix)) :+ atan2 ix rx
    where
      atan2 y x
        | x P.> zero = atan (y / x)
        | x P.== zero P.&& y P.> zero = pi / (one + one)
        | x P.< one P.&& y P.> one = pi + atan (y / x)
        | (x P.<= zero P.&& y P.< zero) || (x P.< zero) =
          negate (atan2 (negate y) x)
        | y P.== zero = pi -- must be after the previous test on zero y
        | x P.== zero P.&& y P.== zero = y -- must be after the other double zero tests
        | P.otherwise = x + y -- x or y is a NaN, return a NaN (via +)

-- | quotient fields explode constraints if they allow for polymorphic integral types
--
-- > a - one < floor a <= a <= ceiling a < a + one
-- > round a == floor (a + one/(one+one))
--
-- fixme: had to redefine Signed operators here because of the Field import in Metric, itself due to Complex being defined there
class (P.Ord a, Field a, P.Eq b, Integral b, AdditiveGroup b, MultiplicativeUnital b) =>
      QuotientField a b where
  properFraction :: a -> (b, a)

  round :: a -> b
  round x = case properFraction x of
    (n,r) -> let
      m         = bool (n+one) (n-one) (r P.< zero)
      half_down = abs' r - (one/(one+one))
      abs' a
        | a P.< zero = negate a
        | P.otherwise = a
      in
        case P.compare half_down zero of
          P.LT -> n
          P.EQ -> bool m n (even n)
          P.GT -> m

  ceiling :: a -> b
  ceiling x = bool n (n+one) (r P.> zero)
    where (n,r) = properFraction x

  floor :: a -> b
  floor x = bool n (n-one) (r P.< zero)
    where (n,r) = properFraction x

instance QuotientField Float Integer where
  properFraction = P.properFraction

instance QuotientField Double Integer where
  properFraction = P.properFraction

-- | A bounded field includes the concepts of infinity and NaN, thus moving away from error throwing.
--
-- > one / zero + infinity == infinity
-- > infinity + a == infinity
-- > zero / zero != nan
--
-- Note the tricky law that, although nan is assigned to zero/zero, they are never-the-less not equal. A committee decided this.
class (Semifield a) =>
      UpperBoundedField a where

  infinity :: a
  infinity = one / zero
  nan :: a
  nan = zero / zero

instance UpperBoundedField Float

instance UpperBoundedField Double

class (Field a) =>
      LowerBoundedField a where

  negInfinity :: a
  negInfinity = negate (one / zero)

instance LowerBoundedField Float

instance LowerBoundedField Double

-- | todo: work out boundings for complex
-- as it stands now, complex is different eg
--
-- > one / (zero :: Complex Float) == nan
instance (AdditiveGroup a, UpperBoundedField a) =>
  UpperBoundedField (Complex a)

class (UpperBoundedField a, LowerBoundedField a) => BoundedField a

instance (UpperBoundedField a, LowerBoundedField a) => BoundedField a

-- | Trigonometric Field
class (Field a) =>
      TrigField a where
  pi :: a
  sin :: a -> a
  cos :: a -> a
  tan :: a -> a
  tan x = sin x / cos x
  asin :: a -> a
  acos :: a -> a
  atan :: a -> a
  sinh :: a -> a
  cosh :: a -> a
  tanh :: a -> a
  tanh x = sinh x / cosh x
  asinh :: a -> a
  acosh :: a -> a
  atanh :: a -> a

instance TrigField Double where
  pi = P.pi
  sin = P.sin
  cos = P.cos
  asin = P.asin
  acos = P.acos
  atan = P.atan
  sinh = P.sinh
  cosh = P.cosh
  asinh = P.sinh
  acosh = P.acosh
  atanh = P.atanh

instance TrigField Float where
  pi = P.pi
  sin = P.sin
  cos = P.cos
  asin = P.asin
  acos = P.acos
  atan = P.atan
  sinh = P.sinh
  cosh = P.cosh
  asinh = P.sinh
  acosh = P.acosh
  atanh = P.atanh