exact-pi 0.4.1.4 → 0.5.0.0

raw patch · 5 files changed

+280/−21 lines, 5 filesdep +QuickCheckdep +exact-pidep +tastydep ~basedep ~semigroupsPVP ok

version bump matches the API change (PVP)

Dependencies added: QuickCheck, exact-pi, tasty, tasty-hunit, tasty-quickcheck

Dependency ranges changed: base, semigroups

API changes (from Hackage documentation)

Files

• changelog.md +4/−0
• exact-pi.cabal +18/−1
• src/Data/ExactPi.hs +49/−20
• test-suite/Test.hs +146/−0
• test-suite/TestUtils.hs +63/−0

@@ -1,3 +1,7 @@+0.5.0.0+-------+* Change implementation of 'rationalApproximations' to use Chudnovsky's approximations.+ 0.4.1.4 ------- * Comply with NoStarIsType pragma.

@@ -1,5 +1,5 @@ name:                exact-pi-version:             0.4.1.4+version:             0.5.0.0 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.@@ -30,6 +30,23 @@                        semigroups >=0.8   ghc-options:         -Wall   hs-source-dirs:      src+  default-language:    Haskell2010++test-suite spec+  main-is:             Test.hs+  build-depends:       base >=4.7 && <4.12,+                       exact-pi,+                       numtype-dk >= 0.5,+                       QuickCheck >=2.10 && <2.12,+                       tasty >=0.10 && <1.2,+                       tasty-hunit >=0.9 && <0.11,+                       tasty-quickcheck >= 0.9 && <0.11+  if impl(ghc < 8.0)+    build-depends:     semigroups >=0.9 && < 1.0+  other-modules:       TestUtils+  type:                exitcode-stdio-1.0+  ghc-options:         -Wall+  hs-source-dirs:      test-suite   default-language:    Haskell2010  source-repository head

@@ -1,4 +1,5 @@-{-# LANGUAGE RankNTypes #-}+{-# LANGUAGE RankNTypes          #-}+{-# LANGUAGE ParallelListComp    #-}  {-# OPTIONS_HADDOCK show-extensions #-} @@ -28,7 +29,9 @@   toExactInteger,   isExactRational,   toExactRational,-  rationalApproximations+  rationalApproximations,+  -- * Utils+  getRationalLimit ) where @@ -95,28 +98,54 @@ 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.+-- | 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 based on work by Anders Kaseorg shared at http://qr.ae/RbXl8M.+-- Implementation is based on Chudnovsky's algorithm. rationalApproximations :: ExactPi -> [Rational] rationalApproximations (Approximate x) = [toRational (x :: Double)]-rationalApproximations (Exact 0 q) = [q]-rationalApproximations (Exact z q) = fmap (\pi' -> q * (pi' ^^ z)) piConvergents+rationalApproximations (Exact _ 0)     = [0]+rationalApproximations (Exact 0 q)     = [q]+rationalApproximations (Exact z q)+  | even z    = [q * 10005^^k * c^^z     | c <- chudnovsky]+  | otherwise = [q * 10005^^k * c^^z * r | c <- chudnovsky | r <- rootApproximation]+  where k = z `div` 2++chudnovsky :: [Rational]+chudnovsky = [426880 / s | s <- partials]+  where lk = iterate (+545140134) 13591409+        xk = iterate (*(-262537412640768000)) 1+        kk = iterate (+12) 6+        mk = 1: [m * ((k^(3::Int) - 16*k) % (n+1)^(3::Int)) | m <- mk | k <- kk | n <- [0..]]+        values = [m * l / x | m <- mk | l <- lk | x <- xk]+        partials = scanl1 (+) values++-- | Given an infinite converging sequence of rationals, find their limit.+-- Takes a comparison function to determine when convergence is close enough.+--+-- >>> getRationalLimit (==) (rationalApproximations (Exact 1 1)) :: Double+-- 3.141592653589793+getRationalLimit :: Fractional a => (a -> a -> Bool) -> [Rational] -> a+getRationalLimit cmp = go . map fromRational+  where go (x:y:xs)+          | cmp x y   = y+          | otherwise = go (y:xs)+        go [x] = x+        go _ = error "did not converge"++-- | A sequence of convergents approximating @sqrt 10005@, intended to be zipped+-- with 'chudnovksy' in 'rationalApproximations'. Carefully chosen so that+-- the denominator does not increase too rapidly but approximations are still+-- appropriately precise.+--+-- Chudnovsky's series provides no more than 15 digits+-- per iteration, so the root approximation should not+-- have a more rapid rate of convergence.+rootApproximation :: [Rational]+rootApproximation = map head . iterate (drop 4) $ go 1 0 100 1 40   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))+    go pk' qk' pk qk a = (pk % qk): go pk qk (pk' + a*pk) (qk' + a*qk) (240-a)  instance Show ExactPi where   show (Exact z q) | z == 0 = "Exactly " ++ show q

@@ -0,0 +1,146 @@+{-# LANGUAGE DataKinds #-}+{-# OPTIONS_GHC -fno-warn-type-defaults #-}+import Data.Fixed            (Fixed(..))+import Data.Ratio            ((%))+import Test.Tasty            (TestTree, testGroup, defaultMain)+import Test.Tasty.HUnit      ((@?=), Assertion, testCase)+import Test.Tasty.QuickCheck (testProperty)+import Test.QuickCheck       (Positive(..))++import Data.ExactPi+import TestUtils             (E, getValue, getDigit, getDigitBBP)++-- test pi^2 first since it does not rely on square roots+piSquaredDouble :: Assertion+piSquaredDouble = getValue (Exact 2 1) @?= (pi^2 :: Double)++-- first 57 digits of pi^2+-- http://www.wolframalpha.com/input/?i=pi%5E2+piSquaredWAstart :: Assertion+piSquaredWAstart = getValue (Exact 2 1) @?= piSquared++piSquared :: Fixed (E 57)+piSquared = 9.869604401089358618834490999876151135313699407240790626413++-- last 21 digits of pi^2 on wolfram alpha http://www.wolframalpha.com/input/?i=pi%5E2+-- by asking for more digits as much as possible+piSquaredWAend :: Assertion+piSquaredWAend = x `mod` (10^21) @?= 643271910414561208753+  where+    MkFixed x = getValue (Exact 2 1) :: Fixed (E 3647)++-- test first term matches formula of chudnovsky's algorithm+firstApproximation :: Assertion+firstApproximation = head (rationalApproximations (Exact 2 1)) @?= (426880 % 13591409)^2 * 10005++-- pi tests+piDouble :: Assertion+piDouble = getValue (Exact 1 1) @?= (pi :: Double)++piMatchesOeis :: Assertion+piMatchesOeis = getValue (Exact 1 1) @?= oeisValue++-- https://oeis.org/A000796+oeisValue :: Fixed (E 104)+oeisValue = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214++-- digits 762 to 767 of pi are 999999+feynmanPoint :: Assertion+feynmanPoint = x `mod` 1000000 @?= 999999+  where+    MkFixed x = getValue (Exact 1 1) :: Fixed (E 767)++-- last 21 digits of pi on wolfram alpha (http://www.wolframalpha.com/input/?i=pi)+-- by asking for more digits as much as possible+piWAend :: Assertion+piWAend = x `mod` (10^21) @?= 706420467525907091548+  where+    MkFixed x = getValue (Exact 1 1) :: Fixed (E 3647)++-- pi power tests+-- http://www.wolframalpha.com/input/?i=1000th+digit+of+pi%5E3%2F10+pi3 :: Assertion+pi3 = x `mod` 100 @?= 98+  where+    MkFixed x = getValue (Exact 3 (1 % 10)) :: Fixed (E 1000)++-- http://www.wolframalpha.com/input/?i=1000th+digit+of+pi%5E-1+*+10+piNegOne :: Assertion+piNegOne = x `mod` 100 @?= 87+  where+    MkFixed x = getValue (Exact (-1) 10) :: Fixed (E 1000)++-- http://www.wolframalpha.com/input/?i=1000th+digit+of+pi%5E10+%2F+10%5E4+pi10 :: Assertion+pi10 = x `mod` 100 @?= 58+  where+    MkFixed x = getValue (Exact 10 (1 % 10^4)) :: Fixed (E 1000)++-- http://www.wolframalpha.com/input/?i=1000th+digit+of+pi%5E-10+*+100000+piNeg10 :: Assertion+piNeg10 = x `mod` 100 @?= 01+  where+    MkFixed x = getValue (Exact (-10) (10^5)) :: Fixed (E 1000)++-- http://www.wolframalpha.com/input/?i=400th+digit+of+pi%5E51+*+10%5E-25+pi51 :: Assertion+pi51 = x `mod` 100 @?= 39+  where+    MkFixed x = getValue (Exact 51 (1 % 10^25)) :: Fixed (E 400)++-- http://www.wolframalpha.com/input/?i=400th+digit+of+pi%5E-51+*+10%5E26+piNeg51 :: Assertion+piNeg51 = x `mod` 100 @?= 93+  where+    MkFixed x = getValue (Exact (-51) (10^26)) :: Fixed (E 400)++-- exact value of riemann zeta(50): should be very near 1+zeta50 :: ExactPi+zeta50 = Exact 50 (39604576419286371856998202 % 285258771457546764463363635252374414183254365234375)++zeta200 :: ExactPi+zeta200 = Exact 200 (996768098856666829529857264280799324216991774914413349936111645234527339243047375137731604604421998265202825395226558782117309054290681031680198580956052700765605768743424718675968548245722319600560038220395777111787342302 % 2682678748792657844957504192313280657551803049278355275671666881580642758576467817615493645217977237214155689404787155170845497733836863647685885197919191727452679238952541411298115541287013688972773507748859386210346035176197388875022427877722880764252312145081723341902733317236524547144682628641021437942981719970703125)++-- value of zeta(50) - 1 from wolfram alpha (up to a Double)+-- http://www.wolframalpha.com/input/?i=zeta(50)-1+zeta50MinusOne :: Assertion+zeta50MinusOne = t @?= 8.8817842109308159e-16+  where+    t = getRationalLimit (==) . map (subtract 1) . rationalApproximations $ zeta50 :: Double++-- http://www.wolframalpha.com/input/?i=zeta(200)-1+zeta200MinusOne :: Assertion+zeta200MinusOne = t @?= 6.2230152778611417071e-61+  where+    t = getRationalLimit (==) . map (subtract 1) . rationalApproximations $ zeta200 :: Double++-- test against bbp formula+prop :: Positive Integer -> Bool+prop (Positive n) = getDigit n == getDigitBBP (n - 1)++tests :: TestTree+tests = testGroup "Rational approximation tests"+  [ testGroup "π² tests" [ testCase "matches double precision"       piSquaredDouble+                         , testCase "matches start of wolfram alpha" piSquaredWAstart+                         , testCase "matches end of wolfram alpha"   piSquaredWAend+                         , testCase "first term matches chudnovsky"  firstApproximation+                         ]+  , testGroup "π tests"  [ testCase "matches double precision"       piDouble+                         , testCase "matches oeis digits"            piMatchesOeis+                         , testCase "has feynman point"              feynmanPoint+                         , testCase "matches end of wolfram alpha"   piWAend+                         ]+  , testGroup "πᵏ tests" [ testCase "digits near 1000 of k=3"        pi3+                         , testCase "digits near 1000 of k=-1"       piNegOne+                         , testCase "digits near 1000 of k=10"       pi10+                         , testCase "digits near 1000 of k=-10"      piNeg10+                         , testCase "digits near 400 of k=51"        pi51+                         , testCase "digits near 400 of k=-51"       piNeg51+                         , testCase "ζ(50)-1 double precision"       zeta50MinusOne+                         , testCase "ζ(500)-1 double precision"      zeta200MinusOne+                         ]+  , testProperty "hex digits match BBP formula" prop+  ]++main :: IO ()+main = defaultMain tests

@@ -0,0 +1,63 @@+{-# LANGUAGE KindSignatures #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE DataKinds #-}+module TestUtils+  ( getValue+  , getDigit+  , getDigitBBP+  , E+  ) where++import Data.Proxy   (Proxy)+import Data.List    (foldl')+import Data.Fixed   (mod', HasResolution(..), Fixed)++import GHC.TypeLits (Nat, KnownNat, SomeNat(..), natVal, someNatVal)++import Data.ExactPi++-- E n generalises E2/E3/E6/E12 from Data.Fixed to give more precise+-- fixed-precision arithmetic: Fixed (E 30) has 30 decimal places.+data E (n :: Nat)++instance KnownNat n => HasResolution (E n) where+  resolution _ = 10^natVal (undefined :: E n)++-- this function is not necessarily in general safe but is fine in the cases used here+getValue :: (Eq a, Fractional a) => ExactPi -> a+getValue = getRationalLimit (==) . rationalApproximations++getDigit :: Integer -> Int+getDigit n = case someNatVal d of+               Just (SomeNat (_ :: Proxy m)) -> (floor $ 16^n * (getValue (Exact 1 1) :: Fixed (E m))) `mod` 16+               Nothing -> error "negative digit requested"+             where d = fromInteger $ 4 * n `div` 3 + 1+--------------------------------------------------------------------------------+powModInteger :: Integer -> Integer -> Integer -> Integer+powModInteger a k n = a^k `mod` n++infTerms :: Integer -> Int -> Integer -> Float+infTerms n j k = 16^^(n-k) / (fromIntegral $ 8*k + fromIntegral j)++finiteTerms :: Integer -> Int -> Integer -> Float+finiteTerms n j k = (fromIntegral $ powModInteger 16 (n-k) (8*k + j')) / (fromIntegral $ 8*k + j')+  where j' = fromIntegral j++summation :: Integer -> Int -> Float+summation n j = stabilise $ scanl plus finitePart [infTerms n j k | k <- [n+1..]]+  where finitePart = foldl' plus 0 [finiteTerms n j k | k <- [0..n]]++mod1 :: Float -> Float+mod1 x = mod' x 1++plus :: Float -> Float -> Float+plus x y = mod1 (x + y)++stabilise :: Eq a => [a] -> a+stabilise (x:y:xs)+  | x == y    = x+  | otherwise = stabilise (y:xs)+stabilise _ = error "finite list"++getDigitBBP :: Integer -> Int+getDigitBBP n = floor . (16 *) . mod1 $ 4 * summation n 1 - 2 * summation n 4 - summation n 5 - summation n 6