| Safe Haskell | Safe |
|---|---|
| Language | GHC2021 |
Antelude.Numeric
Description
Synopsis
- data Double
- data Float
- class Fractional a => Floating a where
- class Num a => Fractional a where
- (/) :: a -> a -> a
- recip :: a -> a
- fromRational :: Rational -> a
- data Int
- data Integer
- class (Real a, Enum a) => Integral a where
- class Num a where
- type Rational = Ratio Integer
- fromIntegral :: (Integral a, Num b) => a -> b
- gcd :: Integral a => a -> a -> a
- lcm :: Integral a => a -> a -> a
- realToFrac :: (Real a, Fractional b) => a -> b
- subtract :: Num a => a -> a -> a
- greatestCommonDenominator :: Integral a => a -> a -> a
- isEven :: Integral a => a -> Bool
- isOdd :: Integral a => a -> Bool
- leastCommonMultiple :: Integral a => a -> a -> a
Documentation
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
| Floating Double | Since: base-2.1 |
| RealFloat Double | Since: base-2.1 |
Defined in GHC.Float Methods floatRadix :: Double -> Integer # floatDigits :: Double -> Int # floatRange :: Double -> (Int, Int) # decodeFloat :: Double -> (Integer, Int) # encodeFloat :: Integer -> Int -> Double # significand :: Double -> Double # scaleFloat :: Int -> Double -> Double # isInfinite :: Double -> Bool # isDenormalized :: Double -> Bool # isNegativeZero :: Double -> Bool # | |
| Read Double | Since: base-2.1 |
| Eq Double | Note that due to the presence of
Also note that
|
| Ord Double | Note that due to the presence of
Also note that, due to the same,
|
| Foldable (UDouble :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UDouble m -> m # foldMap :: Monoid m => (a -> m) -> UDouble a -> m # foldMap' :: Monoid m => (a -> m) -> UDouble a -> m # foldr :: (a -> b -> b) -> b -> UDouble a -> b # foldr' :: (a -> b -> b) -> b -> UDouble a -> b # foldl :: (b -> a -> b) -> b -> UDouble a -> b # foldl' :: (b -> a -> b) -> b -> UDouble a -> b # foldr1 :: (a -> a -> a) -> UDouble a -> a # foldl1 :: (a -> a -> a) -> UDouble a -> a # elem :: Eq a => a -> UDouble a -> Bool # maximum :: Ord a => UDouble a -> a # minimum :: Ord a => UDouble a -> a # | |
| Traversable (UDouble :: Type -> Type) | Since: base-4.9.0.0 |
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
| Floating Float | Since: base-2.1 |
| RealFloat Float | Since: base-2.1 |
Defined in GHC.Float Methods floatRadix :: Float -> Integer # floatDigits :: Float -> Int # floatRange :: Float -> (Int, Int) # decodeFloat :: Float -> (Integer, Int) # encodeFloat :: Integer -> Int -> Float # significand :: Float -> Float # scaleFloat :: Int -> Float -> Float # isInfinite :: Float -> Bool # isDenormalized :: Float -> Bool # isNegativeZero :: Float -> Bool # | |
| Read Float | Since: base-2.1 |
| Eq Float | Note that due to the presence of
Also note that
|
| Ord Float | Note that due to the presence of
Also note that, due to the same,
|
| Foldable (UFloat :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UFloat m -> m # foldMap :: Monoid m => (a -> m) -> UFloat a -> m # foldMap' :: Monoid m => (a -> m) -> UFloat a -> m # foldr :: (a -> b -> b) -> b -> UFloat a -> b # foldr' :: (a -> b -> b) -> b -> UFloat a -> b # foldl :: (b -> a -> b) -> b -> UFloat a -> b # foldl' :: (b -> a -> b) -> b -> UFloat a -> b # foldr1 :: (a -> a -> a) -> UFloat a -> a # foldl1 :: (a -> a -> a) -> UFloat a -> a # elem :: Eq a => a -> UFloat a -> Bool # maximum :: Ord a => UFloat a -> a # minimum :: Ord a => UFloat a -> a # | |
| Traversable (UFloat :: Type -> Type) | Since: base-4.9.0.0 |
class Fractional a => Floating a where #
Trigonometric and hyperbolic functions and related functions.
The Haskell Report defines no laws for Floating. However, (, +)(
and *)exp are customarily expected to define an exponential field and have
the following properties:
exp (a + b)=exp a * exp bexp (fromInteger 0)=fromInteger 1
Minimal complete definition
pi, exp, log, sin, cos, asin, acos, atan, sinh, cosh, asinh, acosh, atanh
Instances
class Num a => Fractional a where #
Fractional numbers, supporting real division.
The Haskell Report defines no laws for Fractional. However, ( and
+)( are customarily expected to define a division ring and have the
following properties:*)
recipgives the multiplicative inversex * recip x=recip x * x=fromInteger 1- Totality of
toRational toRationalis total- Coherence with
toRational - if the type also implements
Real, thenfromRationalis a left inverse fortoRational, i.e.fromRational (toRational i) = i
Note that it isn't customarily expected that a type instance of
Fractional implement a field. However, all instances in base do.
Minimal complete definition
fromRational, (recip | (/))
Methods
Fractional division.
Reciprocal fraction.
fromRational :: Rational -> a #
Conversion from a Rational (that is Ratio IntegerfromRational
to a value of type Rational, so such literals have type
(.Fractional a) => a
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 |
| Enum Int | Since: base-2.1 |
| Ix Int | Since: base-2.1 |
| Num Int | Since: base-2.1 |
| Read Int | Since: base-2.1 |
| Integral Int | Since: base-2.0.1 |
| Real Int | Since: base-2.0.1 |
Defined in GHC.Real Methods toRational :: Int -> Rational # | |
| Show Int | Since: base-2.1 |
| Eq Int | |
| Ord Int | |
| Foldable (UInt :: Type -> Type) | Since: base-4.9.0.0 |
Defined in Data.Foldable Methods fold :: Monoid m => UInt m -> m # foldMap :: Monoid m => (a -> m) -> UInt a -> m # foldMap' :: Monoid m => (a -> m) -> UInt a -> m # foldr :: (a -> b -> b) -> b -> UInt a -> b # foldr' :: (a -> b -> b) -> b -> UInt a -> b # foldl :: (b -> a -> b) -> b -> UInt a -> b # foldl' :: (b -> a -> b) -> b -> UInt a -> b # foldr1 :: (a -> a -> a) -> UInt a -> a # foldl1 :: (a -> a -> a) -> UInt a -> a # elem :: Eq a => a -> UInt a -> Bool # maximum :: Ord a => UInt a -> a # | |
| Traversable (UInt :: Type -> Type) | Since: base-4.9.0.0 |
Arbitrary precision integers. In contrast with fixed-size integral types
such as Int, the Integer type represents the entire infinite range of
integers.
Integers are stored in a kind of sign-magnitude form, hence do not expect two's complement form when using bit operations.
If the value is small (fit into an Int), IS constructor is used.
Otherwise Integer and IN constructors are used to store a BigNat
representing respectively the positive or the negative value magnitude.
Invariant: Integer and IN are used iff value doesn't fit in IS
Instances
| Enum Integer | Since: base-2.1 |
| Ix Integer | Since: base-2.1 |
Defined in GHC.Ix | |
| Num Integer | Since: base-2.1 |
| Read Integer | Since: base-2.1 |
| Integral Integer | Since: base-2.0.1 |
Defined in GHC.Real | |
| Real Integer | Since: base-2.0.1 |
Defined in GHC.Real Methods toRational :: Integer -> Rational # | |
| Show Integer | Since: base-2.1 |
| Eq Integer | |
| Ord Integer | |
class (Real a, Enum a) => Integral a where #
Integral numbers, supporting integer division.
The Haskell Report defines no laws for Integral. However, Integral
instances are customarily expected to define a Euclidean domain and have the
following properties for the div/mod and quot/rem pairs, given
suitable Euclidean functions f and g:
x=y * quot x y + rem x ywithrem x y=fromInteger 0org (rem x y)<g yx=y * div x y + mod x ywithmod x y=fromInteger 0orf (mod x y)<f y
An example of a suitable Euclidean function, for Integer's instance, is
abs.
In addition, toInteger should be total, and fromInteger should be a left
inverse for it, i.e. fromInteger (toInteger i) = i.
Methods
quot :: a -> a -> a infixl 7 #
integer division truncated toward zero
WARNING: This function is partial (because it throws when 0 is passed as
the divisor) for all the integer types in base.
integer remainder, satisfying
(x `quot` y)*y + (x `rem` y) == x
WARNING: This function is partial (because it throws when 0 is passed as
the divisor) for all the integer types in base.
integer division truncated toward negative infinity
WARNING: This function is partial (because it throws when 0 is passed as
the divisor) for all the integer types in base.
integer modulus, satisfying
(x `div` y)*y + (x `mod` y) == x
WARNING: This function is partial (because it throws when 0 is passed as
the divisor) for all the integer types in base.
WARNING: This function is partial (because it throws when 0 is passed as
the divisor) for all the integer types in base.
WARNING: This function is partial (because it throws when 0 is passed as
the divisor) for all the integer types in base.
conversion to Integer
Basic numeric class.
The Haskell Report defines no laws for Num. However, ( and +)( are
customarily expected to define a ring and have the following properties:*)
- Associativity of
(+) (x + y) + z=x + (y + z)- Commutativity of
(+) x + y=y + xfromInteger0x + fromInteger 0=xnegategives the additive inversex + negate x=fromInteger 0- Associativity of
(*) (x * y) * z=x * (y * z)fromInteger1x * fromInteger 1=xandfromInteger 1 * x=x- Distributivity of
(with respect to*)(+) a * (b + c)=(a * b) + (a * c)and(b + c) * a=(b * a) + (c * a)- Coherence with
toInteger - if the type also implements
Integral, thenfromIntegeris a left inverse fortoInteger, i.e.fromInteger (toInteger i) == i
Note that it isn't customarily expected that a type instance of both Num
and Ord implement an ordered ring. Indeed, in base only Integer and
Rational do.
Methods
Unary negation.
Absolute value.
Sign of a number.
The functions abs and signum should satisfy the law:
abs x * signum x == x
For real numbers, the signum is either -1 (negative), 0 (zero)
or 1 (positive).
fromInteger :: Integer -> a #
Conversion from an Integer.
An integer literal represents the application of the function
fromInteger to the appropriate value of type Integer,
so such literals have type (.Num a) => a
Instances
| Num Word16 | Since: base-2.1 |
| Num Word32 | Since: base-2.1 |
| Num Word64 | Since: base-2.1 |
| Num Word8 | Since: base-2.1 |
| Num Integer | Since: base-2.1 |
| Num Natural | Note that Since: base-4.8.0.0 |
| Num Int | Since: base-2.1 |
| Num Word | Since: base-2.1 |
| Integral a => Num (Ratio a) | Since: base-2.0.1 |
Reexport from Prelude
fromIntegral :: (Integral a, Num b) => a -> b #
General coercion from Integral types.
WARNING: This function performs silent truncation if the result type is not at least as big as the argument's type.
Reexport from Prelude
gcd :: Integral a => a -> a -> a #
gcd x yx and y of which
every common factor of x and y is also a factor; for example
gcd 4 2 = 2gcd (-4) 6 = 2gcd 0 44. gcd 0 00.
(That is, the common divisor that is "greatest" in the divisibility
preordering.)
Note: Since for signed fixed-width integer types, abs minBound < 0minBound0 or minBound
Reexport from Prelude
lcm :: Integral a => a -> a -> a #
lcm x yx and y divide.
Reexport from Prelude
realToFrac :: (Real a, Fractional b) => a -> b #
General coercion to Fractional types.
WARNING: This function goes through the Rational type, which does not have values for NaN for example.
This means it does not round-trip.
For Double it also behaves differently with or without -O0:
Prelude> realToFrac nan -- With -O0 -Infinity Prelude> realToFrac nan NaN
Reexport from Prelude
greatestCommonDenominator :: Integral a => a -> a -> a Source #
Defined as gcd, get the greatest common denominator of two Integral numbers
leastCommonMultiple :: Integral a => a -> a -> a Source #
Defined as lcm, get the least common multiple of two Integral numbers