Safe Haskell	None
Language	Haskell98

NumericPrelude.Numeric

Synopsis

(+), (-) :: C a => a -> a -> a
(+), (-) :: C a => a -> a -> a
negate :: C a => a -> a
zero :: C a => a
subtract :: C a => a -> a -> a
sum :: C a => [a] -> a
sum1 :: C a => [a] -> a
isZero :: C a => a -> Bool
(*) :: C a => a -> a -> a
one :: C a => a
fromInteger :: C a => Integer -> a
(^) :: C a => a -> Integer -> a
ringPower :: (C a, C b) => b -> a -> a
sqr :: C a => a -> a
product :: C a => [a] -> a
product1 :: C a => [a] -> a
div, mod :: C a => a -> a -> a
div, mod :: C a => a -> a -> a
divMod :: C a => a -> a -> (a, a)
divides :: (C a, C a) => a -> a -> Bool
even :: (C a, C a) => a -> Bool
odd :: (C a, C a) => a -> Bool
(/) :: C a => a -> a -> a
recip :: C a => a -> a
fromRational' :: C a => Rational -> a
(^-) :: C a => a -> Integer -> a
fieldPower :: (C a, C b) => b -> a -> a
fromRational :: C a => Rational -> a
(^/) :: C a => a -> Rational -> a
sqrt :: C a => a -> a
pi :: C a => a
exp, log :: C a => a -> a
exp, log :: C a => a -> a
logBase, (**) :: C a => a -> a -> a
logBase, (**) :: C a => a -> a -> a
(^?) :: C a => a -> a -> a
sin, cos, tan :: C a => a -> a
sin, cos, tan :: C a => a -> a
sin, cos, tan :: C a => a -> a
asin, acos, atan :: C a => a -> a
asin, acos, atan :: C a => a -> a
asin, acos, atan :: C a => a -> a
sinh, cosh, tanh :: C a => a -> a
sinh, cosh, tanh :: C a => a -> a
sinh, cosh, tanh :: C a => a -> a
asinh, acosh, atanh :: C a => a -> a
asinh, acosh, atanh :: C a => a -> a
asinh, acosh, atanh :: C a => a -> a
abs :: C a => a -> a
signum :: C a => a -> a
quot, rem :: C a => a -> a -> a
quot, rem :: C a => a -> a -> a
quotRem :: C a => a -> a -> (a, a)
splitFraction :: (C a, C b) => a -> (b, a)
fraction :: C a => a -> a
truncate :: (C a, C b) => a -> b
round :: (C a, C b) => a -> b
ceiling, floor :: (C a, C b) => a -> b
ceiling, floor :: (C a, C b) => a -> b
approxRational :: (C a, C a) => a -> a -> Rational
atan2 :: C a => a -> a -> a
toRational :: C a => a -> Rational
toInteger :: C a => a -> Integer
fromIntegral :: (C a, C b) => a -> b
isUnit :: C a => a -> Bool
stdAssociate, stdUnit, stdUnitInv :: C a => a -> a
stdAssociate, stdUnit, stdUnitInv :: C a => a -> a
stdAssociate, stdUnit, stdUnitInv :: C a => a -> a
extendedGCD :: C a => a -> a -> (a, (a, a))
gcd :: C a => a -> a -> a
lcm :: C a => a -> a -> a
euclid :: (C a, C a) => (a -> a -> a) -> a -> a -> a
extendedEuclid :: (C a, C a) => (a -> a -> (a, a)) -> a -> a -> (a, (a, a))
type Rational = T Integer
(%) :: C a => a -> a -> T a
numerator :: T a -> a
denominator :: T a -> a
data Integer
data Int
data Float
data Double
(*>) :: C a v => a -> v -> v

Documentation

(+), (-) :: C a => a -> a -> a infixl 6 +, - #

add and subtract elements

(+), (-) :: C a => a -> a -> a infixl 6 +, - #

add and subtract elements

negate :: C a => a -> a #

inverse with respect to +

zero :: C a => a #

zero element of the vector space

subtract :: C a => a -> a -> a #

subtract is (-) with swapped operand order. This is the operand order which will be needed in most cases of partial application.

sum :: C a => [a] -> a #

Sum up all elements of a list. An empty list yields zero.

This function is inappropriate for number types like Peano. Maybe we should make sum a method of Additive. This would also make lengthLeft and lengthRight superfluous.

sum1 :: C a => [a] -> a #

Sum up all elements of a non-empty list. This avoids including a zero which is useful for types where no universal zero is available. ToDo: Should have NonEmpty type.

isZero :: C a => a -> Bool #

(*) :: C a => a -> a -> a infixl 7 #

one :: C a => a #

fromInteger :: C a => Integer -> a #

(^) :: C a => a -> Integer -> a infixr 8 #

The exponent has fixed type Integer in order to avoid an arbitrarily limitted range of exponents, but to reduce the need for the compiler to guess the type (default type). In practice the exponent is most oftenly fixed, and is most oftenly 2. Fixed exponents can be optimized away and thus the expensive computation of Integers doesn't matter. The previous solution used a C constrained type and the exponent was converted to Integer before computation. So the current solution is not less efficient.

A variant of ^ with more flexibility is provided by ringPower.

ringPower :: (C a, C b) => b -> a -> a #

A prefix function of '(Algebra.Ring.^)' with a parameter order that fits the needs of partial application and function composition. It has generalised exponent.

See: Argument order of expNat on http://www.haskell.org/pipermail/haskell-cafe/2006-September/018022.html

sqr :: C a => a -> a #

product :: C a => [a] -> a #

product1 :: C a => [a] -> a #

div, mod :: C a => a -> a -> a infixl 7 `div`, `mod` #

divMod :: C a => a -> a -> (a, a) #

divides :: (C a, C a) => a -> a -> Bool #

even :: (C a, C a) => a -> Bool #

odd :: (C a, C a) => a -> Bool #

(/) :: C a => a -> a -> a infixl 7 #

recip :: C a => a -> a #

fromRational' :: C a => Rational -> a #

(^-) :: C a => a -> Integer -> a infixr 8 #

fieldPower :: (C a, C b) => b -> a -> a #

A prefix function of '(Algebra.Field.^-)'. It has a generalised exponent.

fromRational :: C a => Rational -> a #

Needed to work around shortcomings in GHC.

(^/) :: C a => a -> Rational -> a infixr 8 #

sqrt :: C a => a -> a #

pi :: C a => a #

exp, log :: C a => a -> a #

logBase, (**) :: C a => a -> a -> a infixr 8 #

(^?) :: C a => a -> a -> a infixr 8 #

sin, cos, tan :: C a => a -> a #

asin, acos, atan :: C a => a -> a #

sinh, cosh, tanh :: C a => a -> a #

asinh, acosh, atanh :: C a => a -> a #

abs :: C a => a -> a #

signum :: C a => a -> a #

quot, rem :: C a => a -> a -> a infixl 7 `quot`, `rem` #

quotRem :: C a => a -> a -> (a, a) #

splitFraction :: (C a, C b) => a -> (b, a) #

fraction :: C a => a -> a #

truncate :: (C a, C b) => a -> b #

round :: (C a, C b) => a -> b #

ceiling, floor :: (C a, C b) => a -> b #

approxRational :: (C a, C a) => a -> a -> Rational #

TODO: Should be moved to a continued fraction module.

atan2 :: C a => a -> a -> a #

toRational :: C a => a -> Rational #

Lossless conversion from any representation of a rational to Rational

toInteger :: C a => a -> Integer #

fromIntegral :: (C a, C b) => a -> b #

isUnit :: C a => a -> Bool #

stdAssociate, stdUnit, stdUnitInv :: C a => a -> a #

extendedGCD :: C a => a -> a -> (a, (a, a)) #

Compute the greatest common divisor and solve a respective Diophantine equation.

  (g,(a,b)) = extendedGCD x y ==>
       g==a*x+b*y   &&  g == gcd x y

TODO: This method is not appropriate for the PID class, because there are rings like the one of the multivariate polynomials, where for all x and y greatest common divisors of x and y exist, but they cannot be represented as a linear combination of x and y. TODO: The definition of extendedGCD does not return the canonical associate.

gcd :: C a => a -> a -> a #

The Greatest Common Divisor is defined by:

  gcd x y == gcd y x
  divides z x && divides z y ==> divides z (gcd x y)   (specification)
  divides (gcd x y) x

lcm :: C a => a -> a -> a #

Least common multiple

euclid :: (C a, C a) => (a -> a -> a) -> a -> a -> a #

extendedEuclid :: (C a, C a) => (a -> a -> (a, a)) -> a -> a -> (a, (a, a)) #

type Rational = T Integer #

(%) :: C a => a -> a -> T a infixl 7 #

numerator :: T a -> a #

denominator :: T a -> a #

data Integer #

Invariant: Jn# and Jp# are used iff value doesn't fit in S#

Useful properties resulting from the invariants:

```
abs (S# _) <= abs (Jp# _)
```
```
abs (S# _) <  abs (Jn# _)
```

Instances

Enum Integer	Since: base-2.1
Instance details Defined in GHC.Enum Methods succ :: Integer -> Integer # pred :: Integer -> Integer # toEnum :: Int -> Integer # fromEnum :: Integer -> Int # enumFrom :: Integer -> [Integer] # enumFromThen :: Integer -> Integer -> [Integer] # enumFromTo :: Integer -> Integer -> [Integer] # enumFromThenTo :: Integer -> Integer -> Integer -> [Integer] #
Eq Integer
Instance details Defined in GHC.Integer.Type Methods (==) :: Integer -> Integer -> Bool # (/=) :: Integer -> Integer -> Bool #
Integral Integer	Since: base-2.0.1
Instance details Defined in GHC.Real Methods quot :: Integer -> Integer -> Integer # rem :: Integer -> Integer -> Integer # div :: Integer -> Integer -> Integer # mod :: Integer -> Integer -> Integer # quotRem :: Integer -> Integer -> (Integer, Integer) # divMod :: Integer -> Integer -> (Integer, Integer) # toInteger :: Integer -> Integer #
Num Integer	Since: base-2.1
Instance details Defined in GHC.Num Methods (+) :: Integer -> Integer -> Integer # (-) :: Integer -> Integer -> Integer # (*) :: Integer -> Integer -> Integer # negate :: Integer -> Integer # abs :: Integer -> Integer # signum :: Integer -> Integer # fromInteger :: Integer -> Integer #
Ord Integer
Instance details Defined in GHC.Integer.Type Methods compare :: Integer -> Integer -> Ordering # (<) :: Integer -> Integer -> Bool # (<=) :: Integer -> Integer -> Bool # (>) :: Integer -> Integer -> Bool # (>=) :: Integer -> Integer -> Bool # max :: Integer -> Integer -> Integer # min :: Integer -> Integer -> Integer #
Read Integer	Since: base-2.1
Instance details Defined in GHC.Read Methods readsPrec :: Int -> ReadS Integer # readList :: ReadS [Integer] # readPrec :: ReadPrec Integer # readListPrec :: ReadPrec [Integer] #
Real Integer	Since: base-2.0.1
Instance details Defined in GHC.Real Methods toRational :: Integer -> Rational #
Show Integer	Since: base-2.1
Instance details Defined in GHC.Show Methods showsPrec :: Int -> Integer -> ShowS # show :: Integer -> String # showList :: [Integer] -> ShowS #
Ix Integer	Since: base-2.1
Instance details Defined in GHC.Arr Methods range :: (Integer, Integer) -> [Integer] # index :: (Integer, Integer) -> Integer -> Int # unsafeIndex :: (Integer, Integer) -> Integer -> Int inRange :: (Integer, Integer) -> Integer -> Bool # rangeSize :: (Integer, Integer) -> Int # unsafeRangeSize :: (Integer, Integer) -> Int
Lift Integer
Instance details Defined in Language.Haskell.TH.Syntax Methods lift :: Integer -> Q Exp #
Arbitrary Integer
Instance details Defined in Test.QuickCheck.Arbitrary Methods arbitrary :: Gen Integer # shrink :: Integer -> [Integer] #
CoArbitrary Integer
Instance details Defined in Test.QuickCheck.Arbitrary Methods coarbitrary :: Integer -> Gen b -> Gen b #
NFData Integer
Instance details Defined in Control.DeepSeq Methods rnf :: Integer -> () #
Random Integer
Instance details Defined in System.Random Methods randomR :: RandomGen g => (Integer, Integer) -> g -> (Integer, g) # random :: RandomGen g => g -> (Integer, g) # randomRs :: RandomGen g => (Integer, Integer) -> g -> [Integer] # randoms :: RandomGen g => g -> [Integer] # randomRIO :: (Integer, Integer) -> IO Integer # randomIO :: IO Integer #
C Integer #
Instance details Defined in Algebra.Indexable Methods compare :: Integer -> Integer -> Ordering #
C Integer #
Instance details Defined in Algebra.Additive Methods zero :: Integer # (+) :: Integer -> Integer -> Integer # (-) :: Integer -> Integer -> Integer # negate :: Integer -> Integer #
C Integer #
Instance details Defined in Algebra.ZeroTestable Methods isZero :: Integer -> Bool #
C Integer #
Instance details Defined in Algebra.Ring Methods (*) :: Integer -> Integer -> Integer # one :: Integer # fromInteger :: Integer -> Integer # (^) :: Integer -> Integer -> Integer #
C Integer #
Instance details Defined in Algebra.IntegralDomain Methods div :: Integer -> Integer -> Integer # mod :: Integer -> Integer -> Integer # divMod :: Integer -> Integer -> (Integer, Integer) #
C Integer #
Instance details Defined in Algebra.Units Methods isUnit :: Integer -> Bool # stdAssociate :: Integer -> Integer # stdUnit :: Integer -> Integer # stdUnitInv :: Integer -> Integer #
C Integer #
Instance details Defined in Algebra.PrincipalIdealDomain Methods extendedGCD :: Integer -> Integer -> (Integer, (Integer, Integer)) # gcd :: Integer -> Integer -> Integer # lcm :: Integer -> Integer -> Integer #
C Integer #
Instance details Defined in Algebra.Absolute Methods abs :: Integer -> Integer # signum :: Integer -> Integer #
C Integer #
Instance details Defined in Algebra.ToRational Methods toRational :: Integer -> Rational #
C Integer #
Instance details Defined in Algebra.RealIntegral Methods quot :: Integer -> Integer -> Integer # rem :: Integer -> Integer -> Integer # quotRem :: Integer -> Integer -> (Integer, Integer) #
C Integer #
Instance details Defined in Algebra.ToInteger Methods toInteger :: Integer -> Integer #
C Integer #
Instance details Defined in Algebra.RealRing Methods splitFraction :: C b => Integer -> (b, Integer) # fraction :: Integer -> Integer # ceiling :: C b => Integer -> b # floor :: C b => Integer -> b # truncate :: C b => Integer -> b # round :: C b => Integer -> b #
C Integer #
Instance details Defined in Algebra.Lattice Methods up :: Integer -> Integer -> Integer # dn :: Integer -> Integer -> Integer #
C Integer Integer #
Instance details Defined in Algebra.Module Methods (*>) :: Integer -> Integer -> Integer #
C Integer Integer #
Instance details Defined in Algebra.ModuleBasis Methods basis :: Integer -> [Integer] # flatten :: Integer -> [Integer] # dimension :: Integer -> Integer -> Int #
C Integer Integer #
Instance details Defined in Algebra.NormedSpace.Sum Methods norm :: Integer -> Integer #
C Integer Integer #
Instance details Defined in Algebra.NormedSpace.Maximum Methods norm :: Integer -> Integer #
C Integer Integer #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods norm :: Integer -> Integer #
Sqr Integer Integer #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods normSqr :: Integer -> Integer #
C a => C Integer (T a) #
Instance details Defined in Algebra.Module Methods (*>) :: Integer -> T a -> T a #

data Int #

A fixed-precision integer type with at least the range [-2^29 .. 2^29-1]. The exact range for a given implementation can be determined by using minBound and maxBound from the Bounded class.

Instances

Bounded Int	Since: base-2.1
Instance details Defined in GHC.Enum Methods minBound :: Int # maxBound :: Int #
Enum Int	Since: base-2.1
Instance details Defined in GHC.Enum Methods succ :: Int -> Int # pred :: Int -> Int # toEnum :: Int -> Int # fromEnum :: Int -> Int # enumFrom :: Int -> [Int] # enumFromThen :: Int -> Int -> [Int] # enumFromTo :: Int -> Int -> [Int] # enumFromThenTo :: Int -> Int -> Int -> [Int] #
Eq Int
Instance details Defined in GHC.Classes Methods (==) :: Int -> Int -> Bool # (/=) :: Int -> Int -> Bool #
Integral Int	Since: base-2.0.1
Instance details Defined in GHC.Real Methods quot :: Int -> Int -> Int # rem :: Int -> Int -> Int # div :: Int -> Int -> Int # mod :: Int -> Int -> Int # quotRem :: Int -> Int -> (Int, Int) # divMod :: Int -> Int -> (Int, Int) # toInteger :: Int -> Integer #
Num Int	Since: base-2.1
Instance details Defined in GHC.Num Methods (+) :: Int -> Int -> Int # (-) :: Int -> Int -> Int # (*) :: Int -> Int -> Int # negate :: Int -> Int # abs :: Int -> Int # signum :: Int -> Int # fromInteger :: Integer -> Int #
Ord Int
Instance details Defined in GHC.Classes Methods compare :: Int -> Int -> Ordering # (<) :: Int -> Int -> Bool # (<=) :: Int -> Int -> Bool # (>) :: Int -> Int -> Bool # (>=) :: Int -> Int -> Bool # max :: Int -> Int -> Int # min :: Int -> Int -> Int #
Read Int	Since: base-2.1
Instance details Defined in GHC.Read Methods readsPrec :: Int -> ReadS Int # readList :: ReadS [Int] # readPrec :: ReadPrec Int # readListPrec :: ReadPrec [Int] #
Real Int	Since: base-2.0.1
Instance details Defined in GHC.Real Methods toRational :: Int -> Rational #
Show Int	Since: base-2.1
Instance details Defined in GHC.Show Methods showsPrec :: Int -> Int -> ShowS # show :: Int -> String # showList :: [Int] -> ShowS #
Ix Int	Since: base-2.1
Instance details Defined in GHC.Arr Methods range :: (Int, Int) -> [Int] # index :: (Int, Int) -> Int -> Int # unsafeIndex :: (Int, Int) -> Int -> Int inRange :: (Int, Int) -> Int -> Bool # rangeSize :: (Int, Int) -> Int # unsafeRangeSize :: (Int, Int) -> Int
Lift Int
Instance details Defined in Language.Haskell.TH.Syntax Methods lift :: Int -> Q Exp #
Arbitrary Int
Instance details Defined in Test.QuickCheck.Arbitrary Methods arbitrary :: Gen Int # shrink :: Int -> [Int] #
CoArbitrary Int
Instance details Defined in Test.QuickCheck.Arbitrary Methods coarbitrary :: Int -> Gen b -> Gen b #
Storable Int	Since: base-2.1
Instance details Defined in Foreign.Storable Methods sizeOf :: Int -> Int # alignment :: Int -> Int # peekElemOff :: Ptr Int -> Int -> IO Int # pokeElemOff :: Ptr Int -> Int -> Int -> IO () # peekByteOff :: Ptr b -> Int -> IO Int # pokeByteOff :: Ptr b -> Int -> Int -> IO () # peek :: Ptr Int -> IO Int # poke :: Ptr Int -> Int -> IO () #
NFData Int
Instance details Defined in Control.DeepSeq Methods rnf :: Int -> () #
Random Int
Instance details Defined in System.Random Methods randomR :: RandomGen g => (Int, Int) -> g -> (Int, g) # random :: RandomGen g => g -> (Int, g) # randomRs :: RandomGen g => (Int, Int) -> g -> [Int] # randoms :: RandomGen g => g -> [Int] # randomRIO :: (Int, Int) -> IO Int # randomIO :: IO Int #
C Int #
Instance details Defined in Algebra.Additive Methods zero :: Int # (+) :: Int -> Int -> Int # (-) :: Int -> Int -> Int # negate :: Int -> Int #
C Int #
Instance details Defined in Algebra.ZeroTestable Methods isZero :: Int -> Bool #
C Int #
Instance details Defined in Algebra.Ring Methods (*) :: Int -> Int -> Int # one :: Int # fromInteger :: Integer -> Int # (^) :: Int -> Integer -> Int #
C Int #
Instance details Defined in Algebra.IntegralDomain Methods div :: Int -> Int -> Int # mod :: Int -> Int -> Int # divMod :: Int -> Int -> (Int, Int) #
C Int #
Instance details Defined in Algebra.Units Methods isUnit :: Int -> Bool # stdAssociate :: Int -> Int # stdUnit :: Int -> Int # stdUnitInv :: Int -> Int #
C Int #
Instance details Defined in Algebra.PrincipalIdealDomain Methods extendedGCD :: Int -> Int -> (Int, (Int, Int)) # gcd :: Int -> Int -> Int # lcm :: Int -> Int -> Int #
C Int #
Instance details Defined in Algebra.Absolute Methods abs :: Int -> Int # signum :: Int -> Int #
C Int #
Instance details Defined in Algebra.ToRational Methods toRational :: Int -> Rational #
C Int #
Instance details Defined in Algebra.RealIntegral Methods quot :: Int -> Int -> Int # rem :: Int -> Int -> Int # quotRem :: Int -> Int -> (Int, Int) #
C Int #
Instance details Defined in Algebra.ToInteger Methods toInteger :: Int -> Integer #
C Int #
Instance details Defined in Algebra.RealRing Methods splitFraction :: C b => Int -> (b, Int) # fraction :: Int -> Int # ceiling :: C b => Int -> b # floor :: C b => Int -> b # truncate :: C b => Int -> b # round :: C b => Int -> b #
C Int Int #
Instance details Defined in Algebra.Module Methods (*>) :: Int -> Int -> Int #
C Int Int #
Instance details Defined in Algebra.ModuleBasis Methods basis :: Int -> [Int] # flatten :: Int -> [Int] # dimension :: Int -> Int -> Int #
C Int Int #
Instance details Defined in Algebra.NormedSpace.Sum Methods norm :: Int -> Int #
C Int Int #
Instance details Defined in Algebra.NormedSpace.Maximum Methods norm :: Int -> Int #
C Int Int #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods norm :: Int -> Int #
Sqr Int Int #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods normSqr :: Int -> Int #
Generic1 (URec Int :: k -> Type)
Instance details Defined in GHC.Generics Associated Types type Rep1 (URec Int) :: k -> Type # Methods from1 :: URec Int a -> Rep1 (URec Int) a # to1 :: Rep1 (URec Int) a -> URec Int a #
Functor (URec Int :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods fmap :: (a -> b) -> URec Int a -> URec Int b # (<$) :: a -> URec Int b -> URec Int a #
Foldable (URec Int :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Int m -> m # foldMap :: Monoid m => (a -> m) -> URec Int a -> m # foldr :: (a -> b -> b) -> b -> URec Int a -> b # foldr' :: (a -> b -> b) -> b -> URec Int a -> b # foldl :: (b -> a -> b) -> b -> URec Int a -> b # foldl' :: (b -> a -> b) -> b -> URec Int a -> b # foldr1 :: (a -> a -> a) -> URec Int a -> a # foldl1 :: (a -> a -> a) -> URec Int a -> a # toList :: URec Int a -> [a] # null :: URec Int a -> Bool # length :: URec Int a -> Int # elem :: Eq a => a -> URec Int a -> Bool # maximum :: Ord a => URec Int a -> a # minimum :: Ord a => URec Int a -> a # sum :: Num a => URec Int a -> a # product :: Num a => URec Int a -> a #
Traversable (URec Int :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Int a -> f (URec Int b) # sequenceA :: Applicative f => URec Int (f a) -> f (URec Int a) # mapM :: Monad m => (a -> m b) -> URec Int a -> m (URec Int b) # sequence :: Monad m => URec Int (m a) -> m (URec Int a) #
Eq (URec Int p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods (==) :: URec Int p -> URec Int p -> Bool # (/=) :: URec Int p -> URec Int p -> Bool #
Ord (URec Int p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods compare :: URec Int p -> URec Int p -> Ordering # (<) :: URec Int p -> URec Int p -> Bool # (<=) :: URec Int p -> URec Int p -> Bool # (>) :: URec Int p -> URec Int p -> Bool # (>=) :: URec Int p -> URec Int p -> Bool # max :: URec Int p -> URec Int p -> URec Int p # min :: URec Int p -> URec Int p -> URec Int p #
Show (URec Int p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods showsPrec :: Int -> URec Int p -> ShowS # show :: URec Int p -> String # showList :: [URec Int p] -> ShowS #
Generic (URec Int p)
Instance details Defined in GHC.Generics Associated Types type Rep (URec Int p) :: Type -> Type # Methods from :: URec Int p -> Rep (URec Int p) x # to :: Rep (URec Int p) x -> URec Int p #
data URec Int (p :: k)	Used for marking occurrences of `Int#` Since: base-4.9.0.0
Instance details Defined in GHC.Generics data URec Int (p :: k) = UInt { uInt# :: Int# }
type Rep1 (URec Int :: k -> Type)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics type Rep1 (URec Int :: k -> Type) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UInt" PrefixI True) (S1 (MetaSel (Just "uInt#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (UInt :: k -> Type)))
type Rep (URec Int p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics type Rep (URec Int p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UInt" PrefixI True) (S1 (MetaSel (Just "uInt#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (UInt :: Type -> Type)))

data Float #

Single-precision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE single-precision type.

Instances

Eq Float	Note that due to the presence of `NaN`, `Float`'s `Eq` instance does not satisfy reflexivity. `>>>` `0/0 == (0/0 :: Float)`False Also note that `Float`'s `Eq` instance does not satisfy substitutivity: `>>>` `0 == (-0 :: Float)`True `>>>` `recip 0 == recip (-0 :: Float)`False
Instance details Defined in GHC.Classes Methods (==) :: Float -> Float -> Bool # (/=) :: Float -> Float -> Bool #
Floating Float	Since: base-2.1
Instance details Defined in GHC.Float Methods pi :: Float # exp :: Float -> Float # log :: Float -> Float # sqrt :: Float -> Float # (**) :: Float -> Float -> Float # logBase :: Float -> Float -> Float # sin :: Float -> Float # cos :: Float -> Float # tan :: Float -> Float # asin :: Float -> Float # acos :: Float -> Float # atan :: Float -> Float # sinh :: Float -> Float # cosh :: Float -> Float # tanh :: Float -> Float # asinh :: Float -> Float # acosh :: Float -> Float # atanh :: Float -> Float # log1p :: Float -> Float # expm1 :: Float -> Float # log1pexp :: Float -> Float # log1mexp :: Float -> Float #
Ord Float	Note that due to the presence of `NaN`, `Float`'s `Ord` instance does not satisfy reflexivity. `>>>` `0/0 <= (0/0 :: Float)`False Also note that, due to the same, `Ord`'s operator interactions are not respected by `Float`'s instance: `>>>` `(0/0 :: Float) > 1`False `>>>` `compare (0/0 :: Float) 1`GT
Instance details Defined in GHC.Classes Methods compare :: Float -> Float -> Ordering # (<) :: Float -> Float -> Bool # (<=) :: Float -> Float -> Bool # (>) :: Float -> Float -> Bool # (>=) :: Float -> Float -> Bool # max :: Float -> Float -> Float # min :: Float -> Float -> Float #
Read Float	Since: base-2.1
Instance details Defined in GHC.Read Methods readsPrec :: Int -> ReadS Float # readList :: ReadS [Float] # readPrec :: ReadPrec Float # readListPrec :: ReadPrec [Float] #
RealFloat Float	Since: base-2.1
Instance details Defined in GHC.Float Methods floatRadix :: Float -> Integer # floatDigits :: Float -> Int # floatRange :: Float -> (Int, Int) # decodeFloat :: Float -> (Integer, Int) # encodeFloat :: Integer -> Int -> Float # exponent :: Float -> Int # significand :: Float -> Float # scaleFloat :: Int -> Float -> Float # isNaN :: Float -> Bool # isInfinite :: Float -> Bool # isDenormalized :: Float -> Bool # isNegativeZero :: Float -> Bool # isIEEE :: Float -> Bool # atan2 :: Float -> Float -> Float #
Lift Float
Instance details Defined in Language.Haskell.TH.Syntax Methods lift :: Float -> Q Exp #
Arbitrary Float
Instance details Defined in Test.QuickCheck.Arbitrary Methods arbitrary :: Gen Float # shrink :: Float -> [Float] #
CoArbitrary Float
Instance details Defined in Test.QuickCheck.Arbitrary Methods coarbitrary :: Float -> Gen b -> Gen b #
Storable Float	Since: base-2.1
Instance details Defined in Foreign.Storable Methods sizeOf :: Float -> Int # alignment :: Float -> Int # peekElemOff :: Ptr Float -> Int -> IO Float # pokeElemOff :: Ptr Float -> Int -> Float -> IO () # peekByteOff :: Ptr b -> Int -> IO Float # pokeByteOff :: Ptr b -> Int -> Float -> IO () # peek :: Ptr Float -> IO Float # poke :: Ptr Float -> Float -> IO () #
NFData Float
Instance details Defined in Control.DeepSeq Methods rnf :: Float -> () #
Random Float
Instance details Defined in System.Random Methods randomR :: RandomGen g => (Float, Float) -> g -> (Float, g) # random :: RandomGen g => g -> (Float, g) # randomRs :: RandomGen g => (Float, Float) -> g -> [Float] # randoms :: RandomGen g => g -> [Float] # randomRIO :: (Float, Float) -> IO Float # randomIO :: IO Float #
C Float #
Instance details Defined in Algebra.Additive Methods zero :: Float # (+) :: Float -> Float -> Float # (-) :: Float -> Float -> Float # negate :: Float -> Float #
C Float #
Instance details Defined in Algebra.ZeroTestable Methods isZero :: Float -> Bool #
C Float #
Instance details Defined in Algebra.Ring Methods (*) :: Float -> Float -> Float # one :: Float # fromInteger :: Integer -> Float # (^) :: Float -> Integer -> Float #
C Float #
Instance details Defined in Algebra.Absolute Methods abs :: Float -> Float # signum :: Float -> Float #
C Float #
Instance details Defined in Algebra.Field Methods (/) :: Float -> Float -> Float # recip :: Float -> Float # fromRational' :: Rational -> Float # (^-) :: Float -> Integer -> Float #
C Float #
Instance details Defined in Algebra.ToRational Methods toRational :: Float -> Rational #
C Float #
Instance details Defined in Algebra.Algebraic Methods sqrt :: Float -> Float # root :: Integer -> Float -> Float # (^/) :: Float -> Rational -> Float #
C Float #
Instance details Defined in Algebra.Transcendental Methods pi :: Float # exp :: Float -> Float # log :: Float -> Float # logBase :: Float -> Float -> Float # (**) :: Float -> Float -> Float # sin :: Float -> Float # cos :: Float -> Float # tan :: Float -> Float # asin :: Float -> Float # acos :: Float -> Float # atan :: Float -> Float # sinh :: Float -> Float # cosh :: Float -> Float # tanh :: Float -> Float # asinh :: Float -> Float # acosh :: Float -> Float # atanh :: Float -> Float #
C Float #
Instance details Defined in Algebra.RealRing Methods splitFraction :: C b => Float -> (b, Float) # fraction :: Float -> Float # ceiling :: C b => Float -> b # floor :: C b => Float -> b # truncate :: C b => Float -> b # round :: C b => Float -> b #
C Float #
Instance details Defined in Algebra.RealField
C Float #
Instance details Defined in Algebra.RealTranscendental Methods atan2 :: Float -> Float -> Float #
C Float #
Instance details Defined in Algebra.FloatingPoint Methods radix :: Float -> Integer # digits :: Float -> Int # range :: Float -> (Int, Int) # decode :: Float -> (Integer, Int) # encode :: Integer -> Int -> Float # exponent :: Float -> Int # significand :: Float -> Float # scale :: Int -> Float -> Float # isNaN :: Float -> Bool # isInfinite :: Float -> Bool # isDenormalized :: Float -> Bool # isNegativeZero :: Float -> Bool # isIEEE :: Float -> Bool #
Power Float #
Instance details Defined in Number.Complex Methods power :: Rational -> T Float -> T Float #
C Float Float #
Instance details Defined in Algebra.Module Methods (*>) :: Float -> Float -> Float #
C Float Float #
Instance details Defined in Algebra.VectorSpace
C Float Float #
Instance details Defined in Algebra.ModuleBasis Methods basis :: Float -> [Float] # flatten :: Float -> [Float] # dimension :: Float -> Float -> Int #
C Float Float #
Instance details Defined in Algebra.OccasionallyScalar Methods toScalar :: Float -> Float # toMaybeScalar :: Float -> Maybe Float # fromScalar :: Float -> Float #
C Float Float #
Instance details Defined in Algebra.NormedSpace.Sum Methods norm :: Float -> Float #
C Float Float #
Instance details Defined in Algebra.NormedSpace.Maximum Methods norm :: Float -> Float #
C Float Float #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods norm :: Float -> Float #
Sqr Float Float #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods normSqr :: Float -> Float #
Generic1 (URec Float :: k -> Type)
Instance details Defined in GHC.Generics Associated Types type Rep1 (URec Float) :: k -> Type # Methods from1 :: URec Float a -> Rep1 (URec Float) a # to1 :: Rep1 (URec Float) a -> URec Float a #
Functor (URec Float :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods fmap :: (a -> b) -> URec Float a -> URec Float b # (<$) :: a -> URec Float b -> URec Float a #
Foldable (URec Float :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Float m -> m # foldMap :: Monoid m => (a -> m) -> URec Float a -> m # foldr :: (a -> b -> b) -> b -> URec Float a -> b # foldr' :: (a -> b -> b) -> b -> URec Float a -> b # foldl :: (b -> a -> b) -> b -> URec Float a -> b # foldl' :: (b -> a -> b) -> b -> URec Float a -> b # foldr1 :: (a -> a -> a) -> URec Float a -> a # foldl1 :: (a -> a -> a) -> URec Float a -> a # toList :: URec Float a -> [a] # null :: URec Float a -> Bool # length :: URec Float a -> Int # elem :: Eq a => a -> URec Float a -> Bool # maximum :: Ord a => URec Float a -> a # minimum :: Ord a => URec Float a -> a # sum :: Num a => URec Float a -> a # product :: Num a => URec Float a -> a #
Traversable (URec Float :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Float a -> f (URec Float b) # sequenceA :: Applicative f => URec Float (f a) -> f (URec Float a) # mapM :: Monad m => (a -> m b) -> URec Float a -> m (URec Float b) # sequence :: Monad m => URec Float (m a) -> m (URec Float a) #
Eq (URec Float p)
Instance details Defined in GHC.Generics Methods (==) :: URec Float p -> URec Float p -> Bool # (/=) :: URec Float p -> URec Float p -> Bool #
Ord (URec Float p)
Instance details Defined in GHC.Generics Methods compare :: URec Float p -> URec Float p -> Ordering # (<) :: URec Float p -> URec Float p -> Bool # (<=) :: URec Float p -> URec Float p -> Bool # (>) :: URec Float p -> URec Float p -> Bool # (>=) :: URec Float p -> URec Float p -> Bool # max :: URec Float p -> URec Float p -> URec Float p # min :: URec Float p -> URec Float p -> URec Float p #
Show (URec Float p)
Instance details Defined in GHC.Generics Methods showsPrec :: Int -> URec Float p -> ShowS # show :: URec Float p -> String # showList :: [URec Float p] -> ShowS #
Generic (URec Float p)
Instance details Defined in GHC.Generics Associated Types type Rep (URec Float p) :: Type -> Type # Methods from :: URec Float p -> Rep (URec Float p) x # to :: Rep (URec Float p) x -> URec Float p #
data URec Float (p :: k)	Used for marking occurrences of `Float#` Since: base-4.9.0.0
Instance details Defined in GHC.Generics data URec Float (p :: k) = UFloat { uFloat# :: Float# }
type Rep1 (URec Float :: k -> Type)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics type Rep1 (URec Float :: k -> Type) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UFloat" PrefixI True) (S1 (MetaSel (Just "uFloat#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (UFloat :: k -> Type)))
type Rep (URec Float p)
Instance details Defined in GHC.Generics type Rep (URec Float p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UFloat" PrefixI True) (S1 (MetaSel (Just "uFloat#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (UFloat :: Type -> Type)))

data Double #

Double-precision floating point numbers. It is desirable that this type be at least equal in range and precision to the IEEE double-precision type.

Instances

Eq Double	Note that due to the presence of `NaN`, `Double`'s `Eq` instance does not satisfy reflexivity. `>>>` `0/0 == (0/0 :: Double)`False Also note that `Double`'s `Eq` instance does not satisfy substitutivity: `>>>` `0 == (-0 :: Double)`True `>>>` `recip 0 == recip (-0 :: Double)`False
Instance details Defined in GHC.Classes Methods (==) :: Double -> Double -> Bool # (/=) :: Double -> Double -> Bool #
Floating Double	Since: base-2.1
Instance details Defined in GHC.Float Methods pi :: Double # exp :: Double -> Double # log :: Double -> Double # sqrt :: Double -> Double # (**) :: Double -> Double -> Double # logBase :: Double -> Double -> Double # sin :: Double -> Double # cos :: Double -> Double # tan :: Double -> Double # asin :: Double -> Double # acos :: Double -> Double # atan :: Double -> Double # sinh :: Double -> Double # cosh :: Double -> Double # tanh :: Double -> Double # asinh :: Double -> Double # acosh :: Double -> Double # atanh :: Double -> Double # log1p :: Double -> Double # expm1 :: Double -> Double # log1pexp :: Double -> Double # log1mexp :: Double -> Double #
Ord Double	Note that due to the presence of `NaN`, `Double`'s `Ord` instance does not satisfy reflexivity. `>>>` `0/0 <= (0/0 :: Double)`False Also note that, due to the same, `Ord`'s operator interactions are not respected by `Double`'s instance: `>>>` `(0/0 :: Double) > 1`False `>>>` `compare (0/0 :: Double) 1`GT
Instance details Defined in GHC.Classes Methods compare :: Double -> Double -> Ordering # (<) :: Double -> Double -> Bool # (<=) :: Double -> Double -> Bool # (>) :: Double -> Double -> Bool # (>=) :: Double -> Double -> Bool # max :: Double -> Double -> Double # min :: Double -> Double -> Double #
Read Double	Since: base-2.1
Instance details Defined in GHC.Read Methods readsPrec :: Int -> ReadS Double # readList :: ReadS [Double] # readPrec :: ReadPrec Double # readListPrec :: ReadPrec [Double] #
RealFloat Double	Since: base-2.1
Instance details Defined in GHC.Float Methods floatRadix :: Double -> Integer # floatDigits :: Double -> Int # floatRange :: Double -> (Int, Int) # decodeFloat :: Double -> (Integer, Int) # encodeFloat :: Integer -> Int -> Double # exponent :: Double -> Int # significand :: Double -> Double # scaleFloat :: Int -> Double -> Double # isNaN :: Double -> Bool # isInfinite :: Double -> Bool # isDenormalized :: Double -> Bool # isNegativeZero :: Double -> Bool # isIEEE :: Double -> Bool # atan2 :: Double -> Double -> Double #
Lift Double
Instance details Defined in Language.Haskell.TH.Syntax Methods lift :: Double -> Q Exp #
Arbitrary Double
Instance details Defined in Test.QuickCheck.Arbitrary Methods arbitrary :: Gen Double # shrink :: Double -> [Double] #
CoArbitrary Double
Instance details Defined in Test.QuickCheck.Arbitrary Methods coarbitrary :: Double -> Gen b -> Gen b #
Storable Double	Since: base-2.1
Instance details Defined in Foreign.Storable Methods sizeOf :: Double -> Int # alignment :: Double -> Int # peekElemOff :: Ptr Double -> Int -> IO Double # pokeElemOff :: Ptr Double -> Int -> Double -> IO () # peekByteOff :: Ptr b -> Int -> IO Double # pokeByteOff :: Ptr b -> Int -> Double -> IO () # peek :: Ptr Double -> IO Double # poke :: Ptr Double -> Double -> IO () #
NFData Double
Instance details Defined in Control.DeepSeq Methods rnf :: Double -> () #
Random Double
Instance details Defined in System.Random Methods randomR :: RandomGen g => (Double, Double) -> g -> (Double, g) # random :: RandomGen g => g -> (Double, g) # randomRs :: RandomGen g => (Double, Double) -> g -> [Double] # randoms :: RandomGen g => g -> [Double] # randomRIO :: (Double, Double) -> IO Double # randomIO :: IO Double #
C Double #
Instance details Defined in Algebra.Additive Methods zero :: Double # (+) :: Double -> Double -> Double # (-) :: Double -> Double -> Double # negate :: Double -> Double #
C Double #
Instance details Defined in Algebra.ZeroTestable Methods isZero :: Double -> Bool #
C Double #
Instance details Defined in Algebra.Ring Methods (*) :: Double -> Double -> Double # one :: Double # fromInteger :: Integer -> Double # (^) :: Double -> Integer -> Double #
C Double #
Instance details Defined in Algebra.Absolute Methods abs :: Double -> Double # signum :: Double -> Double #
C Double #
Instance details Defined in Algebra.Field Methods (/) :: Double -> Double -> Double # recip :: Double -> Double # fromRational' :: Rational -> Double # (^-) :: Double -> Integer -> Double #
C Double #
Instance details Defined in Algebra.ToRational Methods toRational :: Double -> Rational #
C Double #
Instance details Defined in Algebra.Algebraic Methods sqrt :: Double -> Double # root :: Integer -> Double -> Double # (^/) :: Double -> Rational -> Double #
C Double #
Instance details Defined in Algebra.Transcendental Methods pi :: Double # exp :: Double -> Double # log :: Double -> Double # logBase :: Double -> Double -> Double # (**) :: Double -> Double -> Double # sin :: Double -> Double # cos :: Double -> Double # tan :: Double -> Double # asin :: Double -> Double # acos :: Double -> Double # atan :: Double -> Double # sinh :: Double -> Double # cosh :: Double -> Double # tanh :: Double -> Double # asinh :: Double -> Double # acosh :: Double -> Double # atanh :: Double -> Double #
C Double #
Instance details Defined in Algebra.RealRing Methods splitFraction :: C b => Double -> (b, Double) # fraction :: Double -> Double # ceiling :: C b => Double -> b # floor :: C b => Double -> b # truncate :: C b => Double -> b # round :: C b => Double -> b #
C Double #
Instance details Defined in Algebra.RealField
C Double #
Instance details Defined in Algebra.RealTranscendental Methods atan2 :: Double -> Double -> Double #
C Double #
Instance details Defined in Algebra.FloatingPoint Methods radix :: Double -> Integer # digits :: Double -> Int # range :: Double -> (Int, Int) # decode :: Double -> (Integer, Int) # encode :: Integer -> Int -> Double # exponent :: Double -> Int # significand :: Double -> Double # scale :: Int -> Double -> Double # isNaN :: Double -> Bool # isInfinite :: Double -> Bool # isDenormalized :: Double -> Bool # isNegativeZero :: Double -> Bool # isIEEE :: Double -> Bool #
Power Double #
Instance details Defined in Number.Complex Methods power :: Rational -> T Double -> T Double #
C Double Double #
Instance details Defined in Algebra.Module Methods (*>) :: Double -> Double -> Double #
C Double Double #
Instance details Defined in Algebra.VectorSpace
C Double Double #
Instance details Defined in Algebra.ModuleBasis Methods basis :: Double -> [Double] # flatten :: Double -> [Double] # dimension :: Double -> Double -> Int #
C Double Double #
Instance details Defined in Algebra.OccasionallyScalar Methods toScalar :: Double -> Double # toMaybeScalar :: Double -> Maybe Double # fromScalar :: Double -> Double #
C Double Double #
Instance details Defined in Algebra.NormedSpace.Sum Methods norm :: Double -> Double #
C Double Double #
Instance details Defined in Algebra.NormedSpace.Maximum Methods norm :: Double -> Double #
C Double Double #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods norm :: Double -> Double #
Sqr Double Double #
Instance details Defined in Algebra.NormedSpace.Euclidean Methods normSqr :: Double -> Double #
Generic1 (URec Double :: k -> Type)
Instance details Defined in GHC.Generics Associated Types type Rep1 (URec Double) :: k -> Type # Methods from1 :: URec Double a -> Rep1 (URec Double) a # to1 :: Rep1 (URec Double) a -> URec Double a #
Functor (URec Double :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods fmap :: (a -> b) -> URec Double a -> URec Double b # (<$) :: a -> URec Double b -> URec Double a #
Foldable (URec Double :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in Data.Foldable Methods fold :: Monoid m => URec Double m -> m # foldMap :: Monoid m => (a -> m) -> URec Double a -> m # foldr :: (a -> b -> b) -> b -> URec Double a -> b # foldr' :: (a -> b -> b) -> b -> URec Double a -> b # foldl :: (b -> a -> b) -> b -> URec Double a -> b # foldl' :: (b -> a -> b) -> b -> URec Double a -> b # foldr1 :: (a -> a -> a) -> URec Double a -> a # foldl1 :: (a -> a -> a) -> URec Double a -> a # toList :: URec Double a -> [a] # null :: URec Double a -> Bool # length :: URec Double a -> Int # elem :: Eq a => a -> URec Double a -> Bool # maximum :: Ord a => URec Double a -> a # minimum :: Ord a => URec Double a -> a # sum :: Num a => URec Double a -> a # product :: Num a => URec Double a -> a #
Traversable (URec Double :: Type -> Type)	Since: base-4.9.0.0
Instance details Defined in Data.Traversable Methods traverse :: Applicative f => (a -> f b) -> URec Double a -> f (URec Double b) # sequenceA :: Applicative f => URec Double (f a) -> f (URec Double a) # mapM :: Monad m => (a -> m b) -> URec Double a -> m (URec Double b) # sequence :: Monad m => URec Double (m a) -> m (URec Double a) #
Eq (URec Double p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods (==) :: URec Double p -> URec Double p -> Bool # (/=) :: URec Double p -> URec Double p -> Bool #
Ord (URec Double p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods compare :: URec Double p -> URec Double p -> Ordering # (<) :: URec Double p -> URec Double p -> Bool # (<=) :: URec Double p -> URec Double p -> Bool # (>) :: URec Double p -> URec Double p -> Bool # (>=) :: URec Double p -> URec Double p -> Bool # max :: URec Double p -> URec Double p -> URec Double p # min :: URec Double p -> URec Double p -> URec Double p #
Show (URec Double p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics Methods showsPrec :: Int -> URec Double p -> ShowS # show :: URec Double p -> String # showList :: [URec Double p] -> ShowS #
Generic (URec Double p)
Instance details Defined in GHC.Generics Associated Types type Rep (URec Double p) :: Type -> Type # Methods from :: URec Double p -> Rep (URec Double p) x # to :: Rep (URec Double p) x -> URec Double p #
data URec Double (p :: k)	Used for marking occurrences of `Double#` Since: base-4.9.0.0
Instance details Defined in GHC.Generics data URec Double (p :: k) = UDouble { uDouble# :: Double# }
type Rep1 (URec Double :: k -> Type)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics type Rep1 (URec Double :: k -> Type) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UDouble" PrefixI True) (S1 (MetaSel (Just "uDouble#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (UDouble :: k -> Type)))
type Rep (URec Double p)	Since: base-4.9.0.0
Instance details Defined in GHC.Generics type Rep (URec Double p) = D1 (MetaData "URec" "GHC.Generics" "base" False) (C1 (MetaCons "UDouble" PrefixI True) (S1 (MetaSel (Just "uDouble#") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (UDouble :: Type -> Type)))

(*>) :: C a v => a -> v -> v infixr 7 #

scale a vector by a scalar