Moduleexact-pi-0.5.1.0Haskell2010

Data.ExactPi

This type is sufficient to exactly express the closure of Q ∪ {π} under multiplication and division. As a result it is useful for representing conversion factors between physical units. Approximate values are included both to close the remainder of the arithmetic operations in the Num typeclass and to encode conversion factors defined experimentally.

ParallelListComp
PostfixOperators
RankNTypes
ExplicitForAll

Contents

Utils

Synopsis

data ExactPi
value approximateValue
value isZero
value isExact
value isExactZero
value isExactOne
value areExactlyEqual
value isExactInteger
value toExactInteger
value isExactRational
value toExactRational
value rationalApproximations
value getRationalLimit

Packageexact-pi-0.5.1.0
Exports13
LanguageHaskell2010
LicenceMIT
SourceExactPi.hs

datadata ExactPi

Source#

Represents an exact or approximate real value. The exactly representable values are rational multiples of an integer power of pi.

Constructors

Exact Integer Rational
Exact z q = q * pi^z. Note that this means there are many representations of zero.
Approximate (forall a. Floating a => a)
An approximate value. This representation was chosen because it allows conversion to floating types using their native definition of pi.

Instances6Floating, Fractional, Num, Show, Semigroup, Monoid

Floating ExactPiDefined in exact-pi-0.5.1.0 · Data.ExactPi
Fractional ExactPiDefined in exact-pi-0.5.1.0 · Data.ExactPi
Num ExactPiDefined in exact-pi-0.5.1.0 · Data.ExactPi
Show ExactPiDefined in exact-pi-0.5.1.0 · Data.ExactPi
Semigroup ExactPiDefined in exact-pi-0.5.1.0 · Data.ExactPi
The multiplicative semigroup over Rationals augmented with multiples of pi.
Monoid ExactPiDefined in exact-pi-0.5.1.0 · Data.ExactPi
The multiplicative monoid over Rationals augmented with multiples of pi.

valueapproximateValue :: Floating a => ExactPi -> a

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Approximates an exact or approximate value, converting it to a Floating type. This uses the value of pi supplied by the destination type, to provide the appropriate precision.

valueisZero :: ExactPi -> Bool

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Identifies whether an ExactPi is an exact or approximate representation of zero.

valueisExact :: ExactPi -> Bool

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Identifies whether an ExactPi is an exact value.

valueisExactZero :: ExactPi -> Bool

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Identifies whether an ExactPi is an exact representation of zero.

valueisExactOne :: ExactPi -> Bool

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Identifies whether an ExactPi is an exact representation of one.

valueareExactlyEqual :: ExactPi -> ExactPi -> Bool

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Identifies whether two ExactPi values are exactly equal.

valueisExactInteger :: ExactPi -> Bool

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Identifies whether an ExactPi is an exact representation of an integer.

valuetoExactInteger :: ExactPi -> Maybe Integer

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Converts an ExactPi to an exact Integer or Nothing.

valueisExactRational :: ExactPi -> Bool

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Identifies whether an ExactPi is an exact representation of a rational.

valuetoExactRational :: ExactPi -> Maybe Rational

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Converts an ExactPi to an exact Rational or Nothing.

valuerationalApproximations :: ExactPi -> [Rational]

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Converts an ExactPi to a list of increasingly accurate rational approximations. Note that Approximate values are converted using the Real instance for Double into a singleton list. Note that exact rationals are also converted into a singleton list.

Implementation is based on Chudnovsky's algorithm.

Utils

valuegetRationalLimit :: Fractional a => (a -> a -> Bool) -> [Rational] -> a

Source#

Given an infinite converging sequence of rationals, find their limit. Takes a comparison function to determine when convergence is close enough.

Example1 expression

>>> getRationalLimit (==) (rationalApproximations (Exact 1 1)) :: Double3.141592653589793