exact-pi-0.4.1.2/src/0000755000000000000000000000000012562121754012350 5ustar0000000000000000exact-pi-0.4.1.2/src/Data/0000755000000000000000000000000012716345540013223 5ustar0000000000000000exact-pi-0.4.1.2/src/Data/ExactPi/0000755000000000000000000000000012623661351014556 5ustar0000000000000000exact-pi-0.4.1.2/src/Data/ExactPi.hs0000644000000000000000000001516312716345540015122 0ustar0000000000000000{-# LANGUAGE RankNTypes #-} {-# OPTIONS_HADDOCK show-extensions #-} {-| Module : Data.ExactPi Description : Exact rational multiples of powers of pi License : MIT Maintainer : douglas.mcclean@gmail.com Stability : experimental 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. -} module Data.ExactPi ( ExactPi(..), approximateValue, isZero, isExact, isExactZero, isExactOne, areExactlyEqual, isExactInteger, toExactInteger, isExactRational, toExactRational, rationalApproximations ) where import Data.Monoid import Data.Ratio ((%), numerator, denominator) import Prelude -- | Represents an exact or approximate real value. -- The exactly representable values are rational multiples of an integer power of pi. data ExactPi = 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'. -- | 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. approximateValue :: Floating a => ExactPi -> a approximateValue (Exact z q) = (pi ^^ z) * (fromRational q) approximateValue (Approximate x) = x -- | Identifies whether an 'ExactPi' is an exact or approximate representation of zero. isZero :: ExactPi -> Bool isZero (Exact _ 0) = True isZero (Approximate x) = x == (0 :: Double) isZero _ = False -- | Identifies whether an 'ExactPi' is an exact value. isExact :: ExactPi -> Bool isExact (Exact _ _) = True isExact _ = False -- | Identifies whether an 'ExactPi' is an exact representation of zero. isExactZero :: ExactPi -> Bool isExactZero (Exact _ 0) = True isExactZero _ = False -- | Identifies whether an 'ExactPi' is an exact representation of one. isExactOne :: ExactPi -> Bool isExactOne (Exact 0 1) = True isExactOne _ = False -- | Identifies whether two 'ExactPi' values are exactly equal. areExactlyEqual :: ExactPi -> ExactPi -> Bool areExactlyEqual (Exact z1 q1) (Exact z2 q2) = (z1 == z2 && q1 == q2) || (q1 == 0 && q2 == 0) areExactlyEqual _ _ = False -- | Identifies whether an 'ExactPi' is an exact representation of an integer. isExactInteger :: ExactPi -> Bool isExactInteger (Exact 0 q) | denominator q == 1 = True isExactInteger _ = False -- | Converts an 'ExactPi' to an exact 'Integer' or 'Nothing'. toExactInteger :: ExactPi -> Maybe Integer toExactInteger (Exact 0 q) | denominator q == 1 = Just $ numerator q toExactInteger _ = Nothing -- | Identifies whether an 'ExactPi' is an exact representation of a rational. isExactRational :: ExactPi -> Bool isExactRational (Exact 0 _) = True isExactRational _ = False -- | Converts an 'ExactPi' to an exact 'Rational' or 'Nothing'. toExactRational :: ExactPi -> Maybe Rational toExactRational (Exact 0 q) = Just q toExactRational _ = Nothing -- | 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. rationalApproximations :: ExactPi -> [Rational] rationalApproximations (Approximate x) = [toRational (x :: Double)] rationalApproximations (Exact 0 q) = [q] rationalApproximations (Exact z q) = fmap (\pi' -> q * (pi' ^^ z)) piConvergents where piConvergents :: [Rational] piConvergents = go True 2 4 where go s p' q' | ltPi m = [q' | not s] ++ go True m q' | otherwise = [p' | s] ++ go False p' m where m = (numerator p' + numerator q')%(denominator p' + denominator q') ltPi :: Rational -> Bool ltPi x = ok x 1 where ok y i = y <= (27*i - 12)%5 || (y < (675*i - 216)%125 && ok ((y - fromInteger (5*i - 2))*(3*(3*i + 1)*(3*i + 2)%(i*(2*i - 1)))) (i + 1)) instance Show ExactPi where show (Exact z q) | z == 0 = "Exactly " ++ show q | z == 1 = "Exactly pi * " ++ show q | otherwise = "Exactly pi^" ++ show z ++ " * " ++ show q show (Approximate x) = "Approximately " ++ show (x :: Double) instance Num ExactPi where fromInteger n = Exact 0 (fromInteger n) (Exact z1 q1) * (Exact z2 q2) = Exact (z1 + z2) (q1 * q2) (Exact _ 0) * _ = 0 _ * (Exact _ 0) = 0 x * y = Approximate $ approximateValue x * approximateValue y (Exact z1 q1) + (Exact z2 q2) | z1 == z2 = Exact z1 (q1 + q2) -- by distributive property x + y = Approximate $ approximateValue x + approximateValue y abs (Exact z q) = Exact z (abs q) abs (Approximate x) = Approximate $ abs x signum (Exact _ q) = Exact 0 (signum q) signum (Approximate x) = Approximate $ signum x -- we leave this tagged as approximate because we don't know "how" approximate the input was. a case could be made for exact answers here. negate x = (Exact 0 (-1)) * x instance Fractional ExactPi where fromRational = Exact 0 recip (Exact z q) = Exact (negate z) (recip q) recip (Approximate x) = Approximate (recip x) instance Floating ExactPi where pi = Exact 1 1 exp x | isExactZero x = 1 | otherwise = approx1 exp x log (Exact 0 1) = 0 log x = approx1 log x -- It would be possible to give tighter bounds to the trig functions, preserving exactness for arguments that have an exactly representable result. sin = approx1 sin cos = approx1 cos tan = approx1 tan asin = approx1 asin atan = approx1 atan acos = approx1 acos sinh = approx1 sinh cosh = approx1 cosh tanh = approx1 tanh asinh = approx1 asinh acosh = approx1 acosh atanh = approx1 atanh approx1 :: (forall a.Floating a => a -> a) -> ExactPi -> ExactPi approx1 f x = Approximate (f (approximateValue x)) -- | The multiplicative monoid over 'Rational's augmented with multiples of 'pi'. instance Monoid ExactPi where mempty = 1 mappend = (*) exact-pi-0.4.1.2/src/Data/ExactPi/TypeLevel.hs0000644000000000000000000001257412623661351017034 0ustar0000000000000000{-# OPTIONS_HADDOCK show-extensions #-} {-# LANGUAGE ConstraintKinds #-} {-# LANGUAGE DataKinds #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE KindSignatures #-} {-# LANGUAGE ScopedTypeVariables #-} {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE TypeOperators #-} {-| Module : Data.ExactPi.TypeLevel Description : Exact non-negative rational multiples of powers of pi at the type level License : MIT Maintainer : douglas.mcclean@gmail.com Stability : experimental This kind 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. -} module Data.ExactPi.TypeLevel ( -- * Type Level ExactPi Values type ExactPi'(..), KnownExactPi(..), -- * Arithmetic type (*), type (/), type Recip, type ExactNatural, type One, type Pi, -- * Conversion to Term Level type MinCtxt, type MinCtxt', injMin ) where import Data.ExactPi import Data.Maybe (fromJust) import Data.Proxy import Data.Ratio import GHC.Exts (Constraint) import GHC.TypeLits hiding (type (*), type (^)) import qualified GHC.TypeLits as N import Numeric.NumType.DK.Integers hiding (type (*), type (/)) import qualified Numeric.NumType.DK.Integers as Z -- | A type-level representation of a non-negative rational multiple of an integer power of pi. -- -- Each type in this kind can be exactly represented at the term level by a value of type 'ExactPi', -- provided that its denominator is non-zero. -- -- Note that there are many representations of zero, and many representations of dividing by zero. -- These are not excluded because doing so introduces a lot of extra machinery. Play nice! Future -- versions may not include a representation for zero. -- -- Of course there are also many representations of every value, because the numerator need not be -- comprime to the denominator. For many purposes it is not necessary to maintain the types in reduced -- form, they will be appropriately reduced when converted to terms. data ExactPi' = ExactPi' TypeInt -- Exponent of pi Nat -- Numerator Nat -- Denominator -- | A KnownDimension is one for which we can construct a term-level representation. -- -- Each validly constructed type of kind 'ExactPi'' has a 'KnownExactPi' instance, provided that -- its denominator is non-zero. class KnownExactPi (v :: ExactPi') where -- | Converts an 'ExactPi'' type to an 'ExactPi' value. exactPiVal :: Proxy v -> ExactPi -- | Determines the minimum context required for a numeric type to hold the value -- associated with a specific 'ExactPi'' type. type family MinCtxt' (v :: ExactPi') :: * -> Constraint where MinCtxt' ('ExactPi' 'Zero p 1) = Num MinCtxt' ('ExactPi' 'Zero p q) = Fractional MinCtxt' ('ExactPi' z p q) = Floating type MinCtxt v a = (KnownExactPi v, MinCtxt' v a, KnownMinCtxt (MinCtxt' v)) -- | A KnownMinCtxt is a contraint on values sufficient to allow us to inject certain -- 'ExactPi' values into types that satisfy the constraint. class KnownMinCtxt (c :: * -> Constraint) where -- | Injects an 'ExactPi' value into a specified type satisfying this constraint. -- -- The injection is permitted to fail if type constraint does not entail the 'MinCtxt' -- required by the 'ExactPi'' representation of the supplied 'ExactPi' value. inj :: c a => Proxy c -- ^ A proxy for identifying the required constraint. -> ExactPi -- ^ The value to inject. -> a -- ^ A value of the constrained type corresponding to the supplied 'ExactPi' value. instance KnownMinCtxt Num where inj _ = fromInteger . fromJust . toExactInteger instance KnownMinCtxt Fractional where inj _ = fromRational . fromJust . toExactRational instance KnownMinCtxt Floating where inj _ = approximateValue -- | Converts an 'ExactPi'' type to a numeric value with the minimum required context. -- -- When the value is known to be an integer, it can be returned as any instance of 'Num'. Similarly, -- rationals require 'Fractional', and values that involve 'pi' require 'Floating'. injMin :: forall v a.(MinCtxt v a) => Proxy v -> a injMin = inj (Proxy :: Proxy (MinCtxt' v)) . exactPiVal instance (KnownTypeInt z, KnownNat p, KnownNat q, 1 <= q) => KnownExactPi ('ExactPi' z p q) where exactPiVal _ = Exact z' (p' % q') where z' = toNum (Proxy :: Proxy z) p' = natVal (Proxy :: Proxy p) q' = natVal (Proxy :: Proxy q) -- | Forms the product of 'ExactPi'' types (in the arithmetic sense). type family (a :: ExactPi') * (b :: ExactPi') :: ExactPi' where ('ExactPi' z p q) * ('ExactPi' z' p' q') = 'ExactPi' (z Z.+ z') (p N.* p') (q N.* q') -- | Forms the quotient of 'ExactPi'' types (in the arithmetic sense). type family (a :: ExactPi') / (b :: ExactPi') :: ExactPi' where ('ExactPi' z p q) / ('ExactPi' z' p' q') = 'ExactPi' (z Z.- z') (p N.* q') (q N.* p') -- | Forms the reciprocal of an 'ExactPi'' type. type family Recip (a :: ExactPi') :: ExactPi' where Recip ('ExactPi' z p q) = 'ExactPi' (Negate z) q p -- | Converts a type-level natural to an 'ExactPi'' type. type ExactNatural n = 'ExactPi' 'Zero n 1 -- | The 'ExactPi'' type representing the number 1. type One = ExactNatural 1 -- | The 'ExactPi'' type representing the number 'pi'. type Pi = 'ExactPi' 'Pos1 1 1 exact-pi-0.4.1.2/LICENSE0000644000000000000000000000206712562121754012573 0ustar0000000000000000Copyright (c) 2015 Douglas McClean Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. exact-pi-0.4.1.2/Setup.hs0000644000000000000000000000006012562121754013211 0ustar0000000000000000import Distribution.Simple main = defaultMain exact-pi-0.4.1.2/exact-pi.cabal0000644000000000000000000000260012716345764014267 0ustar0000000000000000-- Initial exact-pi.cabal generated by cabal init. For further -- documentation, see http://haskell.org/cabal/users-guide/ name: exact-pi version: 0.4.1.2 synopsis: Exact rational multiples of pi (and integer powers of pi) description: Provides an exact representation for rational multiples of pi alongside an approximate representation of all reals. Useful for storing and computing with conversion factors between physical units. homepage: https://github.com/dmcclean/exact-pi/ bug-reports: https://github.com/dmcclean/exact-pi/issues/ license: MIT license-file: LICENSE author: Douglas McClean maintainer: douglas.mcclean@gmail.com -- copyright: category: Data build-type: Simple extra-source-files: README.md, changelog.md cabal-version: >=1.10 library exposed-modules: Data.ExactPi, Data.ExactPi.TypeLevel -- other-modules: -- other-extensions: build-depends: base >=4.7 && <5, numtype-dk >= 0.5 ghc-options: -Wall hs-source-dirs: src default-language: Haskell2010 source-repository head type: git location: https://github.com/dmcclean/exact-pi.git exact-pi-0.4.1.2/README.md0000644000000000000000000000047512562121754013046 0ustar0000000000000000# exact-pi Exact rational multiples of pi (and integer powers of pi) in Haskell [![Build Status](https://travis-ci.org/dmcclean/exact-pi.svg?branch=master)](https://travis-ci.org/dmcclean/exact-pi) [![Hackage Version](https://img.shields.io/hackage/v/exact-pi.svg)](http://hackage.haskell.org/package/exact-pi) exact-pi-0.4.1.2/changelog.md0000644000000000000000000000163112716346012014030 0ustar00000000000000000.4.1.2 ------- * Bump base dependency. 0.4.1.1 ------- * Fixed infinite loop in definition of negate. 0.4.1.0 ------- * Added function for computing rational approximations of ExactPi values. 0.4.0.0 ------- * Added simpler constraints for converting ExactPi types to terms with the minimal context. 0.3.1.0 ------- * Added support for exactly comparing values. 0.3.0.0 ------- * Added a type-level representation of ExactPi values. 0.2.1.2 ------- * Fixed a bug in recip. * Fixed approximation of exact values with a negative exponent for pi. 0.2.1.1 ------- * Fixed a missing case in isZero. 0.2.1.0 ------- * Added support for converting to exact integers or exact rationals. 0.2.0.0 ------- * Removed dependency on groups package, since it appears not to be widely used. * Fixed a missing case alternative in recip. 0.1.2.0 ------- * Added support for GHC 7.8.