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.

Documentation

data ExactPi

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.

Instances

Floating ExactPi

Fractional ExactPi

Num ExactPi

Show ExactPi

Monoid ExactPi

The multiplicative monoid over Rationals augmented with multiples of pi.

approximateValue :: Floating a => ExactPi -> a

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.

isZero :: ExactPi -> Bool

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

isExact :: ExactPi -> Bool

Identifies whether an ExactPi is an exact value.

isExactZero :: ExactPi -> Bool

Identifies whether an ExactPi is an exact representation of zero.

isExactOne :: ExactPi -> Bool

Identifies whether an ExactPi is an exact representation of one.

areExactlyEqual :: ExactPi -> ExactPi -> Bool

Identifies whether two ExactPi values are exactly equal.

isExactInteger :: ExactPi -> Bool

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

toExactInteger :: ExactPi -> Maybe Integer

Converts an ExactPi to an exact Integer or Nothing.

isExactRational :: ExactPi -> Bool

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

toExactRational :: ExactPi -> Maybe Rational

Converts an ExactPi to an exact Rational or Nothing.

rationalApproximations :: ExactPi -> [Rational]

Converts an ExactPi to a list of increasingly accurate rational approximations, on alternating sides of the actual value. 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 based on work by Anders Kaseorg shared at http://qr.ae/RbXl8M.