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

Methods

pi :: ExactPi

exp :: ExactPi -> ExactPi

log :: ExactPi -> ExactPi

sqrt :: ExactPi -> ExactPi

(**) :: ExactPi -> ExactPi -> ExactPi

logBase :: ExactPi -> ExactPi -> ExactPi

sin :: ExactPi -> ExactPi

cos :: ExactPi -> ExactPi

tan :: ExactPi -> ExactPi

asin :: ExactPi -> ExactPi

acos :: ExactPi -> ExactPi

atan :: ExactPi -> ExactPi

sinh :: ExactPi -> ExactPi

cosh :: ExactPi -> ExactPi

tanh :: ExactPi -> ExactPi

asinh :: ExactPi -> ExactPi

acosh :: ExactPi -> ExactPi

atanh :: ExactPi -> ExactPi

log1p :: ExactPi -> ExactPi

expm1 :: ExactPi -> ExactPi

log1pexp :: ExactPi -> ExactPi

log1mexp :: ExactPi -> ExactPi

Fractional ExactPi

Methods

(/) :: ExactPi -> ExactPi -> ExactPi

recip :: ExactPi -> ExactPi

fromRational :: Rational -> ExactPi

Num ExactPi

Methods

(+) :: ExactPi -> ExactPi -> ExactPi

(-) :: ExactPi -> ExactPi -> ExactPi

(*) :: ExactPi -> ExactPi -> ExactPi

negate :: ExactPi -> ExactPi

abs :: ExactPi -> ExactPi

signum :: ExactPi -> ExactPi

fromInteger :: Integer -> ExactPi

Show ExactPi

Methods

showsPrec :: Int -> ExactPi -> ShowS

show :: ExactPi -> String

showList :: [ExactPi] -> ShowS

Semigroup ExactPi

The multiplicative semigroup over Rationals augmented with multiples of pi.

Methods

(<>) :: ExactPi -> ExactPi -> ExactPi

sconcat :: NonEmpty ExactPi -> ExactPi

stimes :: Integral b => b -> ExactPi -> ExactPi

Monoid ExactPi

The multiplicative monoid over Rationals augmented with multiples of pi.

Methods

mempty :: ExactPi

mappend :: ExactPi -> ExactPi -> ExactPi

mconcat :: [ExactPi] -> ExactPi

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.