Data.CReal

This module exports everything you need to use exact real numbers

Documentation

data CReal (n :: Nat)

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.

Instances

KnownNat n => Eq (CReal n)

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

Defined in Data.CReal.Internal

Methods

(==) :: CReal n -> CReal n -> Bool

(/=) :: CReal n -> CReal n -> Bool

Floating (CReal n)

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)

Taking the reciprocal of zero will not terminate

Defined in Data.CReal.Internal

Methods

(/) :: CReal n -> CReal n -> CReal n

recip :: CReal n -> CReal n

fromRational :: Rational -> CReal n

Num (CReal n)

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

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)

Like equality values of type CReal p are compared at precision p.

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

KnownNat n => Read (CReal n)

The instance of Read will read an optionally signed number expressed in decimal scientific notation

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)

toRational returns the CReal n evaluated at a precision of 2^-n

Defined in Data.CReal.Internal

Methods

toRational :: CReal n -> Rational

KnownNat n => RealFloat (CReal n)

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

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)

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)

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"

Defined in Data.CReal.Internal

Methods

showsPrec :: Int -> CReal n -> ShowS

show :: CReal n -> String

showList :: [CReal n] -> ShowS

KnownNat n => Random (CReal n)

The Random instance for 'CReal' p will return random number with at least p digits of precision, every digit after that is zero.

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]

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"

Defined in Data.CReal.Converge

Associated Types

type Element [CReal n] :: Type

Methods

converge :: [CReal n] -> Maybe (Element [CReal n])

convergeErr :: (Element [CReal n] -> Element [CReal n]) -> [CReal n] -> Maybe (Element [CReal n])

type Element [CReal n]

Defined in Data.CReal.Converge

type Element [CReal n] = CReal n

atPrecision :: CReal n -> Int -> Integer

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

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

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