Safe Haskell	None
Language	Haskell2010

Coya

Description

Consider some log-semiring R. Then, for any two x,y :: R, the following holds:

x ^ log y == y ^ log x == e ^ (log x * log y)

Coya is a commutative monoid (R, y = x ^ log y.

The following laws hold:

Left Identity: e # x == x
Right Identity: x # e == x
Associativity: (x # y) # z == x # (y # z)
Commutativity: x # y == y # x

If R is a poset where all elements in R are greater than one, then R also forms a group:

x # (exp (1 / log x)) = x

Synopsis

newtype Coya a = Coya {
- getCoya :: a
}
newtype CoyaGroup a = CoyaGroup {
- getCoyaGroup :: Refined (From 1) (Coya a)
}
coyaGroup :: forall a. (Ord a, Num a) => a -> Maybe (CoyaGroup a)

Documentation

newtype Coya a Source #

The Coya monoid. Its semigroup instance is a binary operation that distributes over multiplication, i.e:

Coya x <> (Coya y * Coya z) == (Coya x <> Coya y) * (Coya x <> Coya z)

The Semiring and Num instances simply lift the underlying type's.

Constructors

Coya
Fields getCoya :: a

Instances

Eq a => Eq (Coya a) Source #
Instance details Defined in Coya Methods (==) :: Coya a -> Coya a -> Bool # (/=) :: Coya a -> Coya a -> Bool #
Floating a => Floating (Coya a) Source #
Instance details Defined in Coya Methods pi :: Coya a # exp :: Coya a -> Coya a # log :: Coya a -> Coya a # sqrt :: Coya a -> Coya a # (**) :: Coya a -> Coya a -> Coya a # logBase :: Coya a -> Coya a -> Coya a # sin :: Coya a -> Coya a # cos :: Coya a -> Coya a # tan :: Coya a -> Coya a # asin :: Coya a -> Coya a # acos :: Coya a -> Coya a # atan :: Coya a -> Coya a # sinh :: Coya a -> Coya a # cosh :: Coya a -> Coya a # tanh :: Coya a -> Coya a # asinh :: Coya a -> Coya a # acosh :: Coya a -> Coya a # atanh :: Coya a -> Coya a # log1p :: Coya a -> Coya a # expm1 :: Coya a -> Coya a # log1pexp :: Coya a -> Coya a # log1mexp :: Coya a -> Coya a #
Fractional a => Fractional (Coya a) Source #
Instance details Defined in Coya Methods (/) :: Coya a -> Coya a -> Coya a # recip :: Coya a -> Coya a # fromRational :: Rational -> Coya a #
Num a => Num (Coya a) Source #
Instance details Defined in Coya Methods (+) :: Coya a -> Coya a -> Coya a # (-) :: Coya a -> Coya a -> Coya a # (*) :: Coya a -> Coya a -> Coya a # negate :: Coya a -> Coya a # abs :: Coya a -> Coya a # signum :: Coya a -> Coya a # fromInteger :: Integer -> Coya a #
Ord a => Ord (Coya a) Source #
Instance details Defined in Coya Methods compare :: Coya a -> Coya a -> Ordering # (<) :: Coya a -> Coya a -> Bool # (<=) :: Coya a -> Coya a -> Bool # (>) :: Coya a -> Coya a -> Bool # (>=) :: Coya a -> Coya a -> Bool # max :: Coya a -> Coya a -> Coya a # min :: Coya a -> Coya a -> Coya a #
Read a => Read (Coya a) Source #
Instance details Defined in Coya Methods readsPrec :: Int -> ReadS (Coya a) # readList :: ReadS [Coya a] # readPrec :: ReadPrec (Coya a) # readListPrec :: ReadPrec [Coya a] #
Real a => Real (Coya a) Source #
Instance details Defined in Coya Methods toRational :: Coya a -> Rational #
RealFloat a => RealFloat (Coya a) Source #
Instance details Defined in Coya Methods floatRadix :: Coya a -> Integer # floatDigits :: Coya a -> Int # floatRange :: Coya a -> (Int, Int) # decodeFloat :: Coya a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Coya a # exponent :: Coya a -> Int # significand :: Coya a -> Coya a # scaleFloat :: Int -> Coya a -> Coya a # isNaN :: Coya a -> Bool # isInfinite :: Coya a -> Bool # isDenormalized :: Coya a -> Bool # isNegativeZero :: Coya a -> Bool # isIEEE :: Coya a -> Bool # atan2 :: Coya a -> Coya a -> Coya a #
RealFrac a => RealFrac (Coya a) Source #
Instance details Defined in Coya Methods properFraction :: Integral b => Coya a -> (b, Coya a) # truncate :: Integral b => Coya a -> b # round :: Integral b => Coya a -> b # ceiling :: Integral b => Coya a -> b # floor :: Integral b => Coya a -> b #
Show a => Show (Coya a) Source #
Instance details Defined in Coya Methods showsPrec :: Int -> Coya a -> ShowS # show :: Coya a -> String # showList :: [Coya a] -> ShowS #
Floating a => Semigroup (Coya a) Source #	`Coya` x `<>` `Coya` y `==` Coya (x `**` `log` y)
Instance details Defined in Coya Methods (<>) :: Coya a -> Coya a -> Coya a # sconcat :: NonEmpty (Coya a) -> Coya a # stimes :: Integral b => b -> Coya a -> Coya a #
Floating a => Monoid (Coya a) Source #	`mempty` `==` e
Instance details Defined in Coya Methods mempty :: Coya a # mappend :: Coya a -> Coya a -> Coya a # mconcat :: [Coya a] -> Coya a #
Storable a => Storable (Coya a) Source #
Instance details Defined in Coya Methods sizeOf :: Coya a -> Int # alignment :: Coya a -> Int # peekElemOff :: Ptr (Coya a) -> Int -> IO (Coya a) # pokeElemOff :: Ptr (Coya a) -> Int -> Coya a -> IO () # peekByteOff :: Ptr b -> Int -> IO (Coya a) # pokeByteOff :: Ptr b -> Int -> Coya a -> IO () # peek :: Ptr (Coya a) -> IO (Coya a) # poke :: Ptr (Coya a) -> Coya a -> IO () #
Prim a => Prim (Coya a) Source #
Instance details Defined in Coya Methods sizeOf# :: Coya a -> Int# # alignment# :: Coya a -> Int# # indexByteArray# :: ByteArray# -> Int# -> Coya a # readByteArray# :: MutableByteArray# s -> Int# -> State# s -> (#State# s, Coya a#) # writeByteArray# :: MutableByteArray# s -> Int# -> Coya a -> State# s -> State# s # setByteArray# :: MutableByteArray# s -> Int# -> Int# -> Coya a -> State# s -> State# s # indexOffAddr# :: Addr# -> Int# -> Coya a # readOffAddr# :: Addr# -> Int# -> State# s -> (#State# s, Coya a#) # writeOffAddr# :: Addr# -> Int# -> Coya a -> State# s -> State# s # setOffAddr# :: Addr# -> Int# -> Int# -> Coya a -> State# s -> State# s #
Semiring a => Semiring (Coya a) Source #
Instance details Defined in Coya Methods plus :: Coya a -> Coya a -> Coya a # zero :: Coya a # times :: Coya a -> Coya a -> Coya a # one :: Coya a #
Ring a => Ring (Coya a) Source #
Instance details Defined in Coya Methods negate :: Coya a -> Coya a #

newtype CoyaGroup a Source #

The Coya monoid constrained to numbers which are greater than 1. This ensures that the group property of inversion holds:

x <> (exp (1 / log x)) == x

Constructors

CoyaGroup
Fields getCoyaGroup :: Refined (From 1) (Coya a)

Instances

(Floating a, Ord a) => Semigroup (CoyaGroup a) Source #	Equivalent to the `Semigroup` instance for `Coya`.
Instance details Defined in Coya Methods (<>) :: CoyaGroup a -> CoyaGroup a -> CoyaGroup a # sconcat :: NonEmpty (CoyaGroup a) -> CoyaGroup a # stimes :: Integral b => b -> CoyaGroup a -> CoyaGroup a #
(Floating a, Ord a) => Monoid (CoyaGroup a) Source #	Equivalent to the `Monoid` instance for `Coya`.
Instance details Defined in Coya Methods mempty :: CoyaGroup a # mappend :: CoyaGroup a -> CoyaGroup a -> CoyaGroup a # mconcat :: [CoyaGroup a] -> CoyaGroup a #
(Floating a, Ord a) => Group (CoyaGroup a) Source #	x <> (`exp` (1 `/` `log` x)) `==` x
Instance details Defined in Coya Methods invert :: CoyaGroup a -> CoyaGroup a # pow :: Integral x => CoyaGroup a -> x -> CoyaGroup a #

coyaGroup :: forall a. (Ord a, Num a) => a -> Maybe (CoyaGroup a) Source #

A smart constructor for CoyaGroup.