Safe Haskell	None
Language	Haskell2010

Data.CReal

Description

This module exports everything you need to use exact real numbers

Synopsis

data CReal (n :: Nat)
atPrecision :: CReal n -> Int -> Integer
crealPrecision :: KnownNat n => CReal n -> Int

Documentation

data CReal (n :: Nat) Source #

The type CReal represents a fast binary Cauchy sequence. This is a Cauchy sequence with the invariant that the pth element divided by 2^p will be within 2^-p of the true value. Internally this sequence is represented as a function from Ints to Integers, as well as an MVar to hold the highest precision cached value.

Instances

Instances details

KnownNat n => Eq (CReal n) Source #	Values of type `CReal p` are compared for equality at precision `p`. This may cause values which differ by less than 2^-p to compare as equal. `>>>` `0 == (0.1 :: CReal 1)`True
Instance details Defined in Data.CReal.Internal Methods (==) :: CReal n -> CReal n -> Bool # (/=) :: CReal n -> CReal n -> Bool #
Floating (CReal n) Source #
Instance details Defined in Data.CReal.Internal Methods pi :: CReal n # exp :: CReal n -> CReal n # log :: CReal n -> CReal n # sqrt :: CReal n -> CReal n # (**) :: CReal n -> CReal n -> CReal n # logBase :: CReal n -> CReal n -> CReal n # sin :: CReal n -> CReal n # cos :: CReal n -> CReal n # tan :: CReal n -> CReal n # asin :: CReal n -> CReal n # acos :: CReal n -> CReal n # atan :: CReal n -> CReal n # sinh :: CReal n -> CReal n # cosh :: CReal n -> CReal n # tanh :: CReal n -> CReal n # asinh :: CReal n -> CReal n # acosh :: CReal n -> CReal n # atanh :: CReal n -> CReal n # log1p :: CReal n -> CReal n # expm1 :: CReal n -> CReal n # log1pexp :: CReal n -> CReal n # log1mexp :: CReal n -> CReal n #
Fractional (CReal n) Source #	Taking the reciprocal of zero will not terminate
Instance details Defined in Data.CReal.Internal Methods (/) :: CReal n -> CReal n -> CReal n # recip :: CReal n -> CReal n # fromRational :: Rational -> CReal n #
Num (CReal n) Source #	`signum (x :: CReal p)` returns the sign of `x` at precision `p`. It's important to remember that this may not represent the actual sign of `x` if the distance between `x` and zero is less than 2^-`p`. This is a little bit of a fudge, but it's probably better than failing to terminate when trying to find the sign of zero. The class still respects the abs-signum law though. `>>>` `signum (0.1 :: CReal 2)`0.0 `>>>` `signum (0.1 :: CReal 3)`1.0
Instance details Defined in Data.CReal.Internal Methods (+) :: CReal n -> CReal n -> CReal n # (-) :: CReal n -> CReal n -> CReal n # (*) :: CReal n -> CReal n -> CReal n # negate :: CReal n -> CReal n # abs :: CReal n -> CReal n # signum :: CReal n -> CReal n # fromInteger :: Integer -> CReal n #
KnownNat n => Ord (CReal n) Source #	Like equality values of type `CReal p` are compared at precision `p`.
Instance details Defined in Data.CReal.Internal Methods compare :: CReal n -> CReal n -> Ordering # (<) :: CReal n -> CReal n -> Bool # (<=) :: CReal n -> CReal n -> Bool # (>) :: CReal n -> CReal n -> Bool # (>=) :: CReal n -> CReal n -> Bool # max :: CReal n -> CReal n -> CReal n # min :: CReal n -> CReal n -> CReal n #
Read (CReal n) Source #	The instance of Read will read an optionally signed number expressed in decimal scientific notation
Instance details Defined in Data.CReal.Internal Methods readsPrec :: Int -> ReadS (CReal n) # readList :: ReadS [CReal n] # readPrec :: ReadPrec (CReal n) # readListPrec :: ReadPrec [CReal n] #
KnownNat n => Real (CReal n) Source #	`toRational` returns the CReal n evaluated at a precision of 2^-n
Instance details Defined in Data.CReal.Internal Methods toRational :: CReal n -> Rational #
KnownNat n => RealFloat (CReal n) Source #	Several of the functions in this class (`floatDigits`, `floatRange`, `exponent`, `significand`) only make sense for floats represented by a mantissa and exponent. These are bound to error. `atan2 y x atPrecision p` performs the comparison to determine the quadrant at precision p. This can cause atan2 to be slightly slower than atan
Instance details Defined in Data.CReal.Internal Methods floatRadix :: CReal n -> Integer # floatDigits :: CReal n -> Int # floatRange :: CReal n -> (Int, Int) # decodeFloat :: CReal n -> (Integer, Int) # encodeFloat :: Integer -> Int -> CReal n # exponent :: CReal n -> Int # significand :: CReal n -> CReal n # scaleFloat :: Int -> CReal n -> CReal n # isNaN :: CReal n -> Bool # isInfinite :: CReal n -> Bool # isDenormalized :: CReal n -> Bool # isNegativeZero :: CReal n -> Bool # isIEEE :: CReal n -> Bool # atan2 :: CReal n -> CReal n -> CReal n #
KnownNat n => RealFrac (CReal n) Source #
Instance details Defined in Data.CReal.Internal Methods properFraction :: Integral b => CReal n -> (b, CReal n) # truncate :: Integral b => CReal n -> b # round :: Integral b => CReal n -> b # ceiling :: Integral b => CReal n -> b # floor :: Integral b => CReal n -> b #
KnownNat n => Show (CReal n) Source #	A CReal with precision p is shown as a decimal number d such that d is within 2^-p of the true value. `>>>` `show (47176870 :: CReal 0)`"47176870" `>>>` `show (pi :: CReal 230)`"3.1415926535897932384626433832795028841971693993751058209749445923078164"
Instance details Defined in Data.CReal.Internal Methods showsPrec :: Int -> CReal n -> ShowS # show :: CReal n -> String # showList :: [CReal n] -> ShowS #
KnownNat n => Random (CReal n) Source #	The `Random` instance for `'CReal' p` will return random number with at least `p` digits of precision, every digit after that is zero.
Instance details Defined in Data.CReal.Internal Methods randomR :: RandomGen g => (CReal n, CReal n) -> g -> (CReal n, g) # random :: RandomGen g => g -> (CReal n, g) # randomRs :: RandomGen g => (CReal n, CReal n) -> g -> [CReal n] # randoms :: RandomGen g => g -> [CReal n] # randomRIO :: (CReal n, CReal n) -> IO (CReal n) # randomIO :: IO (CReal n) #
Converge [CReal n] Source #	The overlapping instance for `CReal n` has a slightly different behavior. The instance for `Eq` will cause `converge` to return a value when the list converges to within 2^-n (due to the `Eq` instance for `CReal n`) despite the precision the value is requested at by the surrounding computation. This instance will return a value approximated to the correct precision. It's important to note when the error function reaches zero this function behaves like `converge` as it's not possible to determine the precision at which the error function should be evaluated at. Find where log x = π using Newton's method `>>>` `let initialGuess = 1``>>>` *`let improve x = x - x (log x - pi)``>>>` `let Just y = converge (iterate improve initialGuess)``>>>` `showAtPrecision 10 y`"23.1406" `>>>` `showAtPrecision 50 y`**"23.1406926327792686"
Instance details Defined in Data.CReal.Converge Associated Types type Element [CReal n] Source # Methods converge :: [CReal n] -> Maybe (Element [CReal n]) Source # convergeErr :: (Element [CReal n] -> Element [CReal n]) -> [CReal n] -> Maybe (Element [CReal n]) Source #
type Element [CReal n] Source #
Instance details Defined in Data.CReal.Converge type Element [CReal n] = CReal n

atPrecision :: CReal n -> Int -> Integer Source #

x `atPrecision` p returns the numerator of the pth element in the Cauchy sequence represented by x. The denominator is 2^p.

>>> 10 `atPrecision` 10
10240

crealPrecision :: KnownNat n => CReal n -> Int Source #

crealPrecision x returns the type level parameter representing x's default precision.

>>> crealPrecision (1 :: CReal 10)
10