{-# OPTIONS_GHC -Wall #-}
{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveLift #-}
{-# LANGUAGE DeriveTraversable #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Data.ExtendedReal
-- Copyright   :  (c) Masahiro Sakai 2014
-- License     :  BSD-style
-- Maintainer  :  masahiro.sakai@gmail.com
--
-- Extension of real numbers with positive/negative infinities (±∞).
-- It is useful for describing various limiting behaviors in mathematics.
--
-- Remarks:
--
-- * @∞ - ∞@ is left undefined as usual,
--   but we define @0 × ∞ = 0 × -∞ = 0@ by following the convention of
--   probability or measure theory.
--
-- References:
--
-- * Wikipedia contributors, "Extended real number line," Wikipedia,
--   The Free Encyclopedia, https://en.wikipedia.org/wiki/Extended_real_number_line
--   (accessed September 1, 2014).
--
-----------------------------------------------------------------------------
module Data.ExtendedReal
  ( Extended (..)
  , inf
  , isFinite
  , isInfinite
  , fromRealFloat
  ) where

import Prelude hiding (isInfinite)
import qualified Prelude as P
import Control.DeepSeq
import Data.Data (Data)
import Data.Hashable
import GHC.Generics (Generic)
import Language.Haskell.TH.Syntax (Lift)

-- | @Extended r@ is an extension of /r/ with positive/negative infinity (±∞).
data Extended r
  = NegInf    -- ^ negative infinity (-∞)
  | Finite !r -- ^ finite value
  | PosInf    -- ^ positive infinity (+∞)
  deriving
  ( Eq (Extended r)
Eq (Extended r) =>
(Extended r -> Extended r -> Ordering)
-> (Extended r -> Extended r -> Bool)
-> (Extended r -> Extended r -> Bool)
-> (Extended r -> Extended r -> Bool)
-> (Extended r -> Extended r -> Bool)
-> (Extended r -> Extended r -> Extended r)
-> (Extended r -> Extended r -> Extended r)
-> Ord (Extended r)
Extended r -> Extended r -> Bool
Extended r -> Extended r -> Ordering
Extended r -> Extended r -> Extended r
forall a.
Eq a =>
(a -> a -> Ordering)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> a)
-> (a -> a -> a)
-> Ord a
forall r. Ord r => Eq (Extended r)
forall r. Ord r => Extended r -> Extended r -> Bool
forall r. Ord r => Extended r -> Extended r -> Ordering
forall r. Ord r => Extended r -> Extended r -> Extended r
$ccompare :: forall r. Ord r => Extended r -> Extended r -> Ordering
compare :: Extended r -> Extended r -> Ordering
$c< :: forall r. Ord r => Extended r -> Extended r -> Bool
< :: Extended r -> Extended r -> Bool
$c<= :: forall r. Ord r => Extended r -> Extended r -> Bool
<= :: Extended r -> Extended r -> Bool
$c> :: forall r. Ord r => Extended r -> Extended r -> Bool
> :: Extended r -> Extended r -> Bool
$c>= :: forall r. Ord r => Extended r -> Extended r -> Bool
>= :: Extended r -> Extended r -> Bool
$cmax :: forall r. Ord r => Extended r -> Extended r -> Extended r
max :: Extended r -> Extended r -> Extended r
$cmin :: forall r. Ord r => Extended r -> Extended r -> Extended r
min :: Extended r -> Extended r -> Extended r
Ord
  , Extended r -> Extended r -> Bool
(Extended r -> Extended r -> Bool)
-> (Extended r -> Extended r -> Bool) -> Eq (Extended r)
forall r. Eq r => Extended r -> Extended r -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
$c== :: forall r. Eq r => Extended r -> Extended r -> Bool
== :: Extended r -> Extended r -> Bool
$c/= :: forall r. Eq r => Extended r -> Extended r -> Bool
/= :: Extended r -> Extended r -> Bool
Eq
  , Int -> Extended r -> ShowS
[Extended r] -> ShowS
Extended r -> String
(Int -> Extended r -> ShowS)
-> (Extended r -> String)
-> ([Extended r] -> ShowS)
-> Show (Extended r)
forall r. Show r => Int -> Extended r -> ShowS
forall r. Show r => [Extended r] -> ShowS
forall r. Show r => Extended r -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
$cshowsPrec :: forall r. Show r => Int -> Extended r -> ShowS
showsPrec :: Int -> Extended r -> ShowS
$cshow :: forall r. Show r => Extended r -> String
show :: Extended r -> String
$cshowList :: forall r. Show r => [Extended r] -> ShowS
showList :: [Extended r] -> ShowS
Show
  , ReadPrec [Extended r]
ReadPrec (Extended r)
Int -> ReadS (Extended r)
ReadS [Extended r]
(Int -> ReadS (Extended r))
-> ReadS [Extended r]
-> ReadPrec (Extended r)
-> ReadPrec [Extended r]
-> Read (Extended r)
forall r. Read r => ReadPrec [Extended r]
forall r. Read r => ReadPrec (Extended r)
forall r. Read r => Int -> ReadS (Extended r)
forall r. Read r => ReadS [Extended r]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
$creadsPrec :: forall r. Read r => Int -> ReadS (Extended r)
readsPrec :: Int -> ReadS (Extended r)
$creadList :: forall r. Read r => ReadS [Extended r]
readList :: ReadS [Extended r]
$creadPrec :: forall r. Read r => ReadPrec (Extended r)
readPrec :: ReadPrec (Extended r)
$creadListPrec :: forall r. Read r => ReadPrec [Extended r]
readListPrec :: ReadPrec [Extended r]
Read
  , Typeable (Extended r)
Typeable (Extended r) =>
(forall (c :: * -> *).
 (forall d b. Data d => c (d -> b) -> d -> c b)
 -> (forall g. g -> c g) -> Extended r -> c (Extended r))
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c (Extended r))
-> (Extended r -> Constr)
-> (Extended r -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c (Extended r)))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e))
    -> Maybe (c (Extended r)))
-> ((forall b. Data b => b -> b) -> Extended r -> Extended r)
-> (forall r r'.
    (r -> r' -> r)
    -> r -> (forall d. Data d => d -> r') -> Extended r -> r)
-> (forall r r'.
    (r' -> r -> r)
    -> r -> (forall d. Data d => d -> r') -> Extended r -> r)
-> (forall u. (forall d. Data d => d -> u) -> Extended r -> [u])
-> (forall u.
    Int -> (forall d. Data d => d -> u) -> Extended r -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> Extended r -> m (Extended r))
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> Extended r -> m (Extended r))
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> Extended r -> m (Extended r))
-> Data (Extended r)
Extended r -> Constr
Extended r -> DataType
(forall b. Data b => b -> b) -> Extended r -> Extended r
forall r. Data r => Typeable (Extended r)
forall r. Data r => Extended r -> Constr
forall r. Data r => Extended r -> DataType
forall r.
Data r =>
(forall b. Data b => b -> b) -> Extended r -> Extended r
forall r u.
Data r =>
Int -> (forall d. Data d => d -> u) -> Extended r -> u
forall r u.
Data r =>
(forall d. Data d => d -> u) -> Extended r -> [u]
forall r r r'.
Data r =>
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
forall r r r'.
Data r =>
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
forall r (m :: * -> *).
(Data r, Monad m) =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
forall r (m :: * -> *).
(Data r, MonadPlus m) =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
forall r (c :: * -> *).
Data r =>
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Extended r)
forall r (c :: * -> *).
Data r =>
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Extended r -> c (Extended r)
forall r (t :: * -> *) (c :: * -> *).
(Data r, Typeable t) =>
(forall d. Data d => c (t d)) -> Maybe (c (Extended r))
forall r (t :: * -> * -> *) (c :: * -> *).
(Data r, Typeable t) =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (Extended r))
forall a.
Typeable a =>
(forall (c :: * -> *).
 (forall d b. Data d => c (d -> b) -> d -> c b)
 -> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
    (forall b r. Data b => c (b -> r) -> c r)
    -> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
    Typeable t =>
    (forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
    Typeable t =>
    (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
    (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
    (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
    Monad m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
    MonadPlus m =>
    (forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall u. Int -> (forall d. Data d => d -> u) -> Extended r -> u
forall u. (forall d. Data d => d -> u) -> Extended r -> [u]
forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Extended r)
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Extended r -> c (Extended r)
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c (Extended r))
forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (Extended r))
$cgfoldl :: forall r (c :: * -> *).
Data r =>
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Extended r -> c (Extended r)
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Extended r -> c (Extended r)
$cgunfold :: forall r (c :: * -> *).
Data r =>
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Extended r)
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Extended r)
$ctoConstr :: forall r. Data r => Extended r -> Constr
toConstr :: Extended r -> Constr
$cdataTypeOf :: forall r. Data r => Extended r -> DataType
dataTypeOf :: Extended r -> DataType
$cdataCast1 :: forall r (t :: * -> *) (c :: * -> *).
(Data r, Typeable t) =>
(forall d. Data d => c (t d)) -> Maybe (c (Extended r))
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c (Extended r))
$cdataCast2 :: forall r (t :: * -> * -> *) (c :: * -> *).
(Data r, Typeable t) =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (Extended r))
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (Extended r))
$cgmapT :: forall r.
Data r =>
(forall b. Data b => b -> b) -> Extended r -> Extended r
gmapT :: (forall b. Data b => b -> b) -> Extended r -> Extended r
$cgmapQl :: forall r r r'.
Data r =>
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
$cgmapQr :: forall r r r'.
Data r =>
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
gmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Extended r -> r
$cgmapQ :: forall r u.
Data r =>
(forall d. Data d => d -> u) -> Extended r -> [u]
gmapQ :: forall u. (forall d. Data d => d -> u) -> Extended r -> [u]
$cgmapQi :: forall r u.
Data r =>
Int -> (forall d. Data d => d -> u) -> Extended r -> u
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> Extended r -> u
$cgmapM :: forall r (m :: * -> *).
(Data r, Monad m) =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
$cgmapMp :: forall r (m :: * -> *).
(Data r, MonadPlus m) =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
$cgmapMo :: forall r (m :: * -> *).
(Data r, MonadPlus m) =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Extended r -> m (Extended r)
Data
  , (forall a b. (a -> b) -> Extended a -> Extended b)
-> (forall a b. a -> Extended b -> Extended a) -> Functor Extended
forall a b. a -> Extended b -> Extended a
forall a b. (a -> b) -> Extended a -> Extended b
forall (f :: * -> *).
(forall a b. (a -> b) -> f a -> f b)
-> (forall a b. a -> f b -> f a) -> Functor f
$cfmap :: forall a b. (a -> b) -> Extended a -> Extended b
fmap :: forall a b. (a -> b) -> Extended a -> Extended b
$c<$ :: forall a b. a -> Extended b -> Extended a
<$ :: forall a b. a -> Extended b -> Extended a
Functor
  , (forall m. Monoid m => Extended m -> m)
-> (forall m a. Monoid m => (a -> m) -> Extended a -> m)
-> (forall m a. Monoid m => (a -> m) -> Extended a -> m)
-> (forall a b. (a -> b -> b) -> b -> Extended a -> b)
-> (forall a b. (a -> b -> b) -> b -> Extended a -> b)
-> (forall b a. (b -> a -> b) -> b -> Extended a -> b)
-> (forall b a. (b -> a -> b) -> b -> Extended a -> b)
-> (forall a. (a -> a -> a) -> Extended a -> a)
-> (forall a. (a -> a -> a) -> Extended a -> a)
-> (forall a. Extended a -> [a])
-> (forall a. Extended a -> Bool)
-> (forall a. Extended a -> Int)
-> (forall a. Eq a => a -> Extended a -> Bool)
-> (forall a. Ord a => Extended a -> a)
-> (forall a. Ord a => Extended a -> a)
-> (forall a. Num a => Extended a -> a)
-> (forall a. Num a => Extended a -> a)
-> Foldable Extended
forall a. Eq a => a -> Extended a -> Bool
forall a. Num a => Extended a -> a
forall a. Ord a => Extended a -> a
forall m. Monoid m => Extended m -> m
forall a. Extended a -> Bool
forall a. Extended a -> Int
forall a. Extended a -> [a]
forall a. (a -> a -> a) -> Extended a -> a
forall m a. Monoid m => (a -> m) -> Extended a -> m
forall b a. (b -> a -> b) -> b -> Extended a -> b
forall a b. (a -> b -> b) -> b -> Extended a -> b
forall (t :: * -> *).
(forall m. Monoid m => t m -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. t a -> [a])
-> (forall a. t a -> Bool)
-> (forall a. t a -> Int)
-> (forall a. Eq a => a -> t a -> Bool)
-> (forall a. Ord a => t a -> a)
-> (forall a. Ord a => t a -> a)
-> (forall a. Num a => t a -> a)
-> (forall a. Num a => t a -> a)
-> Foldable t
$cfold :: forall m. Monoid m => Extended m -> m
fold :: forall m. Monoid m => Extended m -> m
$cfoldMap :: forall m a. Monoid m => (a -> m) -> Extended a -> m
foldMap :: forall m a. Monoid m => (a -> m) -> Extended a -> m
$cfoldMap' :: forall m a. Monoid m => (a -> m) -> Extended a -> m
foldMap' :: forall m a. Monoid m => (a -> m) -> Extended a -> m
$cfoldr :: forall a b. (a -> b -> b) -> b -> Extended a -> b
foldr :: forall a b. (a -> b -> b) -> b -> Extended a -> b
$cfoldr' :: forall a b. (a -> b -> b) -> b -> Extended a -> b
foldr' :: forall a b. (a -> b -> b) -> b -> Extended a -> b
$cfoldl :: forall b a. (b -> a -> b) -> b -> Extended a -> b
foldl :: forall b a. (b -> a -> b) -> b -> Extended a -> b
$cfoldl' :: forall b a. (b -> a -> b) -> b -> Extended a -> b
foldl' :: forall b a. (b -> a -> b) -> b -> Extended a -> b
$cfoldr1 :: forall a. (a -> a -> a) -> Extended a -> a
foldr1 :: forall a. (a -> a -> a) -> Extended a -> a
$cfoldl1 :: forall a. (a -> a -> a) -> Extended a -> a
foldl1 :: forall a. (a -> a -> a) -> Extended a -> a
$ctoList :: forall a. Extended a -> [a]
toList :: forall a. Extended a -> [a]
$cnull :: forall a. Extended a -> Bool
null :: forall a. Extended a -> Bool
$clength :: forall a. Extended a -> Int
length :: forall a. Extended a -> Int
$celem :: forall a. Eq a => a -> Extended a -> Bool
elem :: forall a. Eq a => a -> Extended a -> Bool
$cmaximum :: forall a. Ord a => Extended a -> a
maximum :: forall a. Ord a => Extended a -> a
$cminimum :: forall a. Ord a => Extended a -> a
minimum :: forall a. Ord a => Extended a -> a
$csum :: forall a. Num a => Extended a -> a
sum :: forall a. Num a => Extended a -> a
$cproduct :: forall a. Num a => Extended a -> a
product :: forall a. Num a => Extended a -> a
Foldable    -- ^ @since 0.2.6.0
  , Functor Extended
Foldable Extended
(Functor Extended, Foldable Extended) =>
(forall (f :: * -> *) a b.
 Applicative f =>
 (a -> f b) -> Extended a -> f (Extended b))
-> (forall (f :: * -> *) a.
    Applicative f =>
    Extended (f a) -> f (Extended a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> Extended a -> m (Extended b))
-> (forall (m :: * -> *) a.
    Monad m =>
    Extended (m a) -> m (Extended a))
-> Traversable Extended
forall (t :: * -> *).
(Functor t, Foldable t) =>
(forall (f :: * -> *) a b.
 Applicative f =>
 (a -> f b) -> t a -> f (t b))
-> (forall (f :: * -> *) a. Applicative f => t (f a) -> f (t a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> t a -> m (t b))
-> (forall (m :: * -> *) a. Monad m => t (m a) -> m (t a))
-> Traversable t
forall (m :: * -> *) a. Monad m => Extended (m a) -> m (Extended a)
forall (f :: * -> *) a.
Applicative f =>
Extended (f a) -> f (Extended a)
forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Extended a -> m (Extended b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Extended a -> f (Extended b)
$ctraverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Extended a -> f (Extended b)
traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Extended a -> f (Extended b)
$csequenceA :: forall (f :: * -> *) a.
Applicative f =>
Extended (f a) -> f (Extended a)
sequenceA :: forall (f :: * -> *) a.
Applicative f =>
Extended (f a) -> f (Extended a)
$cmapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Extended a -> m (Extended b)
mapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Extended a -> m (Extended b)
$csequence :: forall (m :: * -> *) a. Monad m => Extended (m a) -> m (Extended a)
sequence :: forall (m :: * -> *) a. Monad m => Extended (m a) -> m (Extended a)
Traversable -- ^ @since 0.2.6.0
  , (forall x. Extended r -> Rep (Extended r) x)
-> (forall x. Rep (Extended r) x -> Extended r)
-> Generic (Extended r)
forall x. Rep (Extended r) x -> Extended r
forall x. Extended r -> Rep (Extended r) x
forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
forall r x. Rep (Extended r) x -> Extended r
forall r x. Extended r -> Rep (Extended r) x
$cfrom :: forall r x. Extended r -> Rep (Extended r) x
from :: forall x. Extended r -> Rep (Extended r) x
$cto :: forall r x. Rep (Extended r) x -> Extended r
to :: forall x. Rep (Extended r) x -> Extended r
Generic     -- ^ @since 0.2.6.0
  , (forall (m :: * -> *). Quote m => Extended r -> m Exp)
-> (forall (m :: * -> *).
    Quote m =>
    Extended r -> Code m (Extended r))
-> Lift (Extended r)
forall r (m :: * -> *). (Lift r, Quote m) => Extended r -> m Exp
forall r (m :: * -> *).
(Lift r, Quote m) =>
Extended r -> Code m (Extended r)
forall t.
(forall (m :: * -> *). Quote m => t -> m Exp)
-> (forall (m :: * -> *). Quote m => t -> Code m t) -> Lift t
forall (m :: * -> *). Quote m => Extended r -> m Exp
forall (m :: * -> *). Quote m => Extended r -> Code m (Extended r)
$clift :: forall r (m :: * -> *). (Lift r, Quote m) => Extended r -> m Exp
lift :: forall (m :: * -> *). Quote m => Extended r -> m Exp
$cliftTyped :: forall r (m :: * -> *).
(Lift r, Quote m) =>
Extended r -> Code m (Extended r)
liftTyped :: forall (m :: * -> *). Quote m => Extended r -> Code m (Extended r)
Lift        -- ^ @since 0.2.6.0
  )

instance Bounded (Extended r) where
  minBound :: Extended r
minBound = Extended r
forall r. Extended r
NegInf
  maxBound :: Extended r
maxBound = Extended r
forall r. Extended r
PosInf

instance NFData r => NFData (Extended r)

instance Hashable r => Hashable (Extended r)

-- | Infinity (∞)
inf :: Extended r
inf :: forall r. Extended r
inf = Extended r
forall r. Extended r
PosInf

-- | @isFinite x = not (isInfinite x)@.
isFinite :: Extended r -> Bool
isFinite :: forall a. Extended a -> Bool
isFinite (Finite r
_) = Bool
True
isFinite Extended r
_ = Bool
False

-- | @isInfinite x@ returns @True@ iff @x@ is @PosInf@ or @NegInf@.
isInfinite :: Extended r -> Bool
isInfinite :: forall a. Extended a -> Bool
isInfinite = Bool -> Bool
not (Bool -> Bool) -> (Extended r -> Bool) -> Extended r -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Extended r -> Bool
forall a. Extended a -> Bool
isFinite

-- | Note that @Extended r@ is /not/ a field, nor a ring.
-- 
-- @PosInf + NegInf@ is left undefined as usual,
-- but we define @0 * PosInf = 0 * NegInf = 0@ by following the convention of probability or measure theory.
instance (Num r, Ord r) => Num (Extended r) where
  Finite r
a + :: Extended r -> Extended r -> Extended r
+ Finite r
b = r -> Extended r
forall r. r -> Extended r
Finite (r
ar -> r -> r
forall a. Num a => a -> a -> a
+r
b)
  Extended r
PosInf + Extended r
NegInf = String -> Extended r
forall a. HasCallStack => String -> a
error String
"PosInf + NegInf is undefined"
  Extended r
NegInf + Extended r
PosInf = String -> Extended r
forall a. HasCallStack => String -> a
error String
"NegInf + PosInf is undefined"
  Extended r
PosInf + Extended r
_ = Extended r
forall r. Extended r
PosInf
  Extended r
_ + Extended r
PosInf = Extended r
forall r. Extended r
PosInf
  Extended r
NegInf + Extended r
_ = Extended r
forall r. Extended r
NegInf
  Extended r
_ + Extended r
NegInf = Extended r
forall r. Extended r
NegInf

  Finite r
x1 * :: Extended r -> Extended r -> Extended r
* Extended r
e = r -> Extended r -> Extended r
forall r. (Num r, Ord r) => r -> Extended r -> Extended r
scale r
x1 Extended r
e
  Extended r
e * Finite r
x2 = r -> Extended r -> Extended r
forall r. (Num r, Ord r) => r -> Extended r -> Extended r
scale r
x2 Extended r
e
  Extended r
PosInf * Extended r
PosInf = Extended r
forall r. Extended r
PosInf
  Extended r
PosInf * Extended r
NegInf = Extended r
forall r. Extended r
NegInf
  Extended r
NegInf * Extended r
PosInf = Extended r
forall r. Extended r
NegInf
  Extended r
NegInf * Extended r
NegInf = Extended r
forall r. Extended r
PosInf

  negate :: Extended r -> Extended r
negate Extended r
NegInf = Extended r
forall r. Extended r
PosInf
  negate (Finite r
x) = r -> Extended r
forall r. r -> Extended r
Finite (r -> r
forall a. Num a => a -> a
negate r
x)
  negate Extended r
PosInf = Extended r
forall r. Extended r
NegInf

  abs :: Extended r -> Extended r
abs Extended r
NegInf = Extended r
forall r. Extended r
PosInf
  abs (Finite r
x) = r -> Extended r
forall r. r -> Extended r
Finite (r -> r
forall a. Num a => a -> a
abs r
x)
  abs Extended r
PosInf = Extended r
forall r. Extended r
PosInf

  signum :: Extended r -> Extended r
signum Extended r
NegInf = r -> Extended r
forall r. r -> Extended r
Finite (-r
1)
  signum (Finite r
x) = r -> Extended r
forall r. r -> Extended r
Finite (r -> r
forall a. Num a => a -> a
signum r
x)
  signum Extended r
PosInf = r -> Extended r
forall r. r -> Extended r
Finite r
1

  fromInteger :: Integer -> Extended r
fromInteger = r -> Extended r
forall r. r -> Extended r
Finite (r -> Extended r) -> (Integer -> r) -> Integer -> Extended r
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Integer -> r
forall a. Num a => Integer -> a
fromInteger  

-- | Note that @Extended r@ is /not/ a field, nor a ring.
instance (Fractional r, Ord r) => Fractional (Extended r) where
  recip :: Extended r -> Extended r
recip (Finite r
x) = r -> Extended r
forall r. r -> Extended r
Finite (r
1r -> r -> r
forall a. Fractional a => a -> a -> a
/r
x)
  recip Extended r
_ = r -> Extended r
forall r. r -> Extended r
Finite r
0

  fromRational :: Rational -> Extended r
fromRational = r -> Extended r
forall r. r -> Extended r
Finite (r -> Extended r) -> (Rational -> r) -> Rational -> Extended r
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Rational -> r
forall a. Fractional a => Rational -> a
fromRational

-- Note that we define @0 * PosInf = 0 * NegInf = 0@ by the convention of probability or measure theory.
scale :: (Num r, Ord r) => r -> Extended r -> Extended r
scale :: forall r. (Num r, Ord r) => r -> Extended r -> Extended r
scale r
a Extended r
e = Extended r -> Extended r -> Extended r
forall a b. a -> b -> b
seq Extended r
e (Extended r -> Extended r) -> Extended r -> Extended r
forall a b. (a -> b) -> a -> b
$
  case r
a r -> r -> Ordering
forall a. Ord a => a -> a -> Ordering
`compare` r
0 of
    Ordering
EQ -> r -> Extended r
forall r. r -> Extended r
Finite r
0
    Ordering
GT ->
      case Extended r
e of
        Extended r
NegInf   -> Extended r
forall r. Extended r
NegInf
        Finite r
b -> r -> Extended r
forall r. r -> Extended r
Finite (r
ar -> r -> r
forall a. Num a => a -> a -> a
*r
b)
        Extended r
PosInf   -> Extended r
forall r. Extended r
PosInf
    Ordering
LT ->
      case Extended r
e of
        Extended r
NegInf   -> Extended r
forall r. Extended r
PosInf
        Finite r
b -> r -> Extended r
forall r. r -> Extended r
Finite (r
ar -> r -> r
forall a. Num a => a -> a -> a
*r
b)
        Extended r
PosInf   -> Extended r
forall r. Extended r
NegInf

-- | Helper to convert 'Double' and 'Float' to 'Extended',
-- taking care of infinite values automatically.
--
-- >>> fromRealFloat (1 / 0)
-- PosInf
-- >>> fromRealFloat (-1 / 0)
-- NegInf
-- >>> fromRealFloat (0 / 0)
-- *** Exception: fromRealFloat: argument should not be NaN
--
-- @since 0.2.5.0
fromRealFloat :: RealFloat r => r -> Extended r
fromRealFloat :: forall r. RealFloat r => r -> Extended r
fromRealFloat r
x
  | r -> Bool
forall a. RealFloat a => a -> Bool
isNaN r
x = String -> Extended r
forall a. HasCallStack => String -> a
error String
"fromRealFloat: argument should not be NaN"
  | r -> Bool
forall a. RealFloat a => a -> Bool
P.isInfinite r
x = if r
x r -> r -> Bool
forall a. Ord a => a -> a -> Bool
> r
0 then Extended r
forall r. Extended r
PosInf else Extended r
forall r. Extended r
NegInf
  | Bool
otherwise = r -> Extended r
forall r. r -> Extended r
Finite r
x