exact-real-0.5.0.0: src/Data/CReal/Internal.hs
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE MagicHash #-}
{-# LANGUAGE MultiWayIf #-}
{-# LANGUAGE PostfixOperators #-}
-----------------------------------------------------------------------------
-- | This module exports a bunch of utilities for working inside the CReal
-- datatype. One should be careful to maintain the CReal invariant when using
-- these functions
----------------------------------------------------------------------------
module Data.CReal.Internal
( CReal(..)
, atPrecision
, crealPrecision
, expBounded
, logBounded
, atanBounded
, sinBounded
, cosBounded
, shiftL
, shiftR
, powerSeries
, alternateSign
, (/.)
, log2
, log10
, isqrt
, showAtPrecision
, decimalDigitsAtPrecision
, rationalToDecimal
) where
import Data.List (scanl')
import Data.Ratio (numerator,denominator,(%))
import GHC.Base (Int(..))
import GHC.Integer.Logarithms (integerLog2#, integerLogBase#)
import GHC.TypeLits
-- $setup
-- >>> :set -XDataKinds
infixl 7 /.
default ()
-- | The type CReal represents a fast binary Cauchy sequence. This is
-- a Cauchy sequence with the invariant that the pth element will be within
-- 2^-p of the true value. Internally this sequence is represented as
-- a function from Ints to Integers.
newtype CReal (n :: Nat) = CR (Int -> Integer)
-- | crealPrecision x returns the type level parameter representing x's default
-- precision.
--
-- >>> crealPrecision (1 :: CReal 10)
-- 10
crealPrecision :: KnownNat n => CReal n -> Int
crealPrecision = fromInteger . natVal
-- | @x \`atPrecision\` p@ returns the numerator of the pth element in the
-- Cauchy sequence represented by x. The denominator is 2^p.
--
-- >>> 10 `atPrecision` 10
-- 10240
atPrecision :: CReal n -> Int -> Integer
(CR x) `atPrecision` p = x p
-- | A CReal with precision p is shown as a decimal number d such that d is
-- within 2^-p of the true value.
--
-- >>> show (47176870 :: CReal 0)
-- "47176870"
instance KnownNat n => Show (CReal n) where
show x = showAtPrecision (crealPrecision x) x
-- | @signum (x :: CReal p)@ returns the sign of @x@ at precision @p@. It's
-- important to remember that this /may not/ represent the actual sign of @x@ if
-- the distance between @x@ and zero is less than 2^-@p@.
--
-- This is a little bit of a fudge, but it's probably better than failing to
-- terminate when trying to find the sign of zero. The class still respects the
-- abs-signum law though.
--
-- >>> signum (0.1 :: CReal 2)
-- 0.0
--
-- >>> signum (0.1 :: CReal 3)
-- 1.0
instance Num (CReal n) where
fromInteger i = CR (\p -> i * 2 ^ p)
negate (CR x) = CR (negate . x)
abs (CR x) = CR (abs . x)
{-# INLINE (+) #-}
CR x1 + CR x2 = CR (\p -> let n1 = x1 (p + 2)
n2 = x2 (p + 2)
in (n1 + n2) /. 4)
{-# INLINE (*) #-}
CR x1 * CR x2 = CR (\p -> let s1 = log2 (abs (x1 0) + 2) + 3
s2 = log2 (abs (x2 0) + 2) + 3
n1 = x1 (p + s2)
n2 = x2 (p + s1)
in (n1 * n2) /. 2^(p + s1 + s2) )
signum x = CR (\p -> signum (x `atPrecision` p) * 2^p)
-- | Taking the reciprocal of zero will not terminate
instance Fractional (CReal n) where
-- This should be in base
fromRational n = fromInteger (numerator n) / fromInteger (denominator n)
{-# INLINE recip #-}
-- TODO: Make recip 0 throw an error (if, for example, it would take more
-- than 4GB of memory to represent the result)
recip (CR x) = CR (\p -> let s = findFirstMonotonic ((3 <=) . abs . x)
n = x (p + 2 * s + 2)
in 2^(2 * p + 2 * s + 2) /. n)
instance Floating (CReal n) where
-- TODO: Could we use something faster such as Ramanujan's formula
pi = 4 * piBy4
exp x = let CR o = x / ln2
l = o 0
y = x - fromInteger l * ln2
in if l == 0
then expBounded x
else expBounded y `shiftL` fromInteger l
-- | Range reduction on the principle that ln (a * b) = ln a + ln b
log x = let CR o = x
l = log2 (o 2) - 2
a = x `shiftR` l
in if | l < 0 -> - log (recip x)
| l == 0 -> logBounded x
| l > 0 -> logBounded a + fromIntegral l * ln2
sqrt (CR x) = CR (\p -> let n = x (2 * p)
in isqrt n)
-- | This will diverge when the base is not positive
x ** y = exp (log x * y)
logBase x y = log y / log x
sin x = cos (x - pi / 2)
cos x = let CR o = x / piBy4
s = o 1 /. 2
octant = fromInteger $ s `mod` 8
offset = x - (fromIntegral s * piBy4)
fs = [ cosBounded
, negate . sinBounded . subtract piBy4
, negate . sinBounded
, negate . cosBounded . (piBy4-)
, negate . cosBounded
, sinBounded . subtract piBy4
, sinBounded
, cosBounded . (piBy4-)]
in (fs !! octant) offset
-- TODO: use multiplyBounded here
tan x = sin x / cos x
asin x = 2 * atan (x / (1 + sqrt (1 - x*x)))
acos x = pi/2 - asin x
atan x = let -- q is x to the nearest 1/4
q = x `atPrecision` 2
in if | q < -4 -> atanBounded (negate (recip x)) - pi / 2
| q == -4 -> -pi / 4 - atanBounded ((x + 1) / (x - 1))
| q == 4 -> pi / 4 + atanBounded ((x - 1) / (x + 1))
| q > 4 -> pi / 2 - atanBounded (recip x)
| otherwise -> atanBounded x
-- TODO: benchmark replacing these with their series expansion
sinh x = (exp x - exp (-x)) / 2
cosh x = (exp x + exp (-x)) / 2
tanh x = let e2x = exp (2 * x)
in (e2x - 1) / (e2x + 1)
asinh x = log (x + sqrt (x * x + 1))
acosh x = log (x + sqrt (x + 1) * sqrt (x - 1))
atanh x = (log (1 + x) - log (1 - x)) / 2
-- | 'toRational' returns the CReal n evaluated at a precision of 2^-n
instance KnownNat n => Real (CReal n) where
toRational x = let p = crealPrecision x
in x `atPrecision` p % 2^p
-- | Values of type @CReal p@ are compared for equality at precision @p@. This
-- may cause values which differ by less than 2^-p to compare as equal.
--
-- >>> 0 == (0.1 :: CReal 1)
-- True
instance KnownNat n => Eq (CReal n) where
-- TODO, should this try smaller values first?
x == y = let p = crealPrecision x
in (x - y) `atPrecision` p == 0
-- | Like equality values of type @CReal p@ are compared at precision @p@.
instance KnownNat n => Ord (CReal n) where
compare x y = let p = crealPrecision x
in compare ((x - y) `atPrecision` p) 0
max (CR x) (CR y) = CR (\p -> max (x p) (y p))
min (CR x) (CR y) = CR (\p -> min (x p) (y p))
--------------------------------------------------------------------------------
-- Some utility functions
--------------------------------------------------------------------------------
--
-- Constants
--
piBy4 :: CReal n
piBy4 = 4 * atanBounded (1/5) - atanBounded (1 / 239) -- Machin Formula
ln2 :: CReal n
ln2 = logBounded 2
--
-- Bounded exponential functions
--
-- | The input to expBounded must be in the range (-1..1)
expBounded :: CReal n -> CReal n
expBounded x = let q = [1 % (n!) | n <- [0..]]
in powerSeries q (max 5) x
-- | The input must be in [1..2]
logBounded :: CReal n -> CReal n
logBounded x = let q = [1 % n | n <- [1..]]
y = (x - 1) / x
in y * powerSeries q (*2) y
--
-- Bounded trigonometric functions
--
-- | The input to sinBounded must be in (-1..1)
sinBounded :: CReal n -> CReal n
sinBounded x = let q = alternateSign (scanl' (*) 1 [ 1 % (n*(n+1)) | n <- [2,4..]])
in x * powerSeries q (max 1) (x*x)
-- | The input to cosBounded must be in (-1..1)
cosBounded :: CReal n -> CReal n
cosBounded x = let q = alternateSign (scanl' (*) 1 [1 % (n*(n+1)) | n <- [1,3..]])
in powerSeries q (max 1) (x*x)
-- | The input to atanBounded must be in [-1..1]
atanBounded :: CReal n -> CReal n
atanBounded x = let q = scanl' (*) 1 [n % (n + 1) | n <- [2,4..]]
d = 1 + x * x
in CR (\p -> ((x/d) * powerSeries q (+1) (x*x/d)) `atPrecision` p)
--
-- Multiplication with powers of two
--
-- | @x \`shiftR\` n@ is equal to @x@ divided by 2^@n@
--
-- @n@ can be negative or zero
--
-- This can be faster than doing the division
shiftR :: CReal n -> Int -> CReal n
shiftR (CR x) n = CR (\p -> let p' = p - n
in if p' >= 0
then x p'
else x 0 /. 2^(-p'))
-- | @x \`shiftL\` n@ is equal to @x@ multiplied by 2^@n@
--
-- @n@ can be negative or zero
--
-- This can be faster than doing the multiplication
shiftL :: CReal n -> Int -> CReal n
shiftL x = shiftR x . negate
--
-- Showing CReals
--
-- | Return a string representing a decimal number within 2^-p of the value
-- represented by the given @CReal p@.
showAtPrecision :: Int -> CReal n -> String
showAtPrecision p (CR x) = let places = decimalDigitsAtPrecision p
r = x p % 2^p
in rationalToDecimal places r
-- | How many decimal digits are required to represent a number to within 2^-p
decimalDigitsAtPrecision :: Int -> Int
decimalDigitsAtPrecision 0 = 0
decimalDigitsAtPrecision p = log10 (2^p) + 1
-- | @rationalToDecimal p x@ returns a string representing @x@ at @p@ decimal
-- places.
rationalToDecimal :: Int -> Rational -> String
rationalToDecimal places r = p ++ is ++ if places > 0 then "." ++ fs else ""
where r' = abs r
p = case signum r of
-1 -> "-"
_ -> ""
ds = show ((numerator r' * 10^places) /. denominator r')
l = length ds
(is, fs) = if | l <= places -> ("0", replicate (places - l) '0' ++ ds)
| otherwise -> splitAt (length ds - places) ds
--
-- Integer operations
--
-- | Division rounding to the nearest integer and rounding half integers to the
-- nearest even integer.
(/.) :: Integer -> Integer -> Integer
n /. d = round (n % d)
-- | @log2 x@ returns the base 2 logarithm of @x@ rounded towards zero.
log2 :: Integer -> Int
log2 x = I# (integerLog2# x)
-- | @log10 x@ returns the base 10 logarithm of @x@ rounded towards zero.
log10 :: Integer -> Int
log10 x = I# (integerLogBase# 10 x)
-- | @isqrt x@ returns the square root of @x@ rounded towards zero.
isqrt :: Integer -> Integer
isqrt x | x < 0 = error "Sqrt applied to negative Integer"
| x == 0 = 0
| otherwise = until satisfied improve initialGuess
where improve r = (r + (x `div` r)) `div` 2
satisfied r = sq r <= x && sq (r + 1) > x
initialGuess = 2 ^ (log2 x `div` 2)
sq r = r * r
-- | Factorial function
(!) :: Integer -> Integer
(!) x = product [2..x]
--
-- Searching
--
-- | Given a monotonic function
findFirstMonotonic :: (Int -> Bool) -> Int
findFirstMonotonic p = binarySearch l' u'
where (l', u') = findBounds 0 1
findBounds l u = if p u then (l, u)
else findBounds u (u*2)
binarySearch l u = let m = l + ((u - l) `div` 2)
in if | l+1 == u -> l
| p m -> binarySearch l m
| otherwise -> binarySearch m u
--
-- Power series
--
-- | Apply 'negate' to every other element, starting with the second
--
-- >>> alternateSign [1..5]
-- [1,-2,3,-4,5]
alternateSign :: Num a => [a] -> [a]
alternateSign = zipWith ($) (cycle [id, negate])
-- | @powerSeries q f x `atPrecision` p@ will evaluate the power series with
-- coefficients @q@ at precision @f p@ at @x@
--
-- @f@ should be a function such that the CReal invariant is maintained
--
-- See any of the trig functions for an example
powerSeries :: [Rational] -> (Int -> Int) -> CReal n -> CReal n
powerSeries q termsAtPrecision (CR x) =
CR (\p -> let t = termsAtPrecision p
d = log2 (toInteger t) + 2
p' = p + d
p'' = p' + d
m = x p''
xs = (%1) <$> iterate (\e -> m * e /. 2^p'') (2^p')
r = sum . take (t + 1) . fmap (round . (* (2^d))) $ zipWith (*) q xs
in r /. 4^d)