-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Numbers in the range [0.005, 1] as a sum of 2, 3, 4 or 5 unit fractions of special types.
--   
--   Inspired by the ancient Egyptian mathematics. See the ideas
--   description and formulation in the <a>related draft paper.</a>
@package fractionizer
@version 0.13.0.0

module UnitFractionsDecomposition2

-- | Rounding to thousandth.
threeDigitsK :: Double -> Double

-- | Characterizes the impact of the absolute error sign on the
--   approximation.
type ErrorImpact = Int

-- | Absolute error with sign for the two unit fractions approximations and
--   the first argument (a in the related paper) being taken as the second
--   parameter for the function. The second argument here is expected to be
--   <a>fromIntegral</a> <tt>a0</tt> where <a>elem</a> <tt>a0</tt>
--   <tt>[2..] == </tt> <a>True</a>.
absErr2Frac :: Double -> Double -> Double
absErrUDecomp3 :: [Double] -> Double -> Double
elemSolution2 :: Integral a => Int -> a -> Double -> Bool

-- | Searches for the minimum absolute error solution to two unit fractions
--   decomposition (approximation) for the fraction in the <a>isRangeN</a>
--   <a>True</a> values with taking into account the sign of the absolute
--   error.
--   
--   If the <a>ErrorImpact</a> parameter is equal to 0 , then absolute
--   error can be of any sign, otherwise the sign of it is the same as the
--   sign of the <a>ErrorImpact</a> argument.
setOfSolutionsGmin :: ErrorImpact -> Double -> (Double, Double)

-- | Searches for the minimum absolute error solution to two unit fractions
--   decomposition (approximation) for the fraction in <tt>k</tt> the
--   <a>isRangeN</a> <tt>k = </tt> <a>True</a>.
suitable2 :: Double -> (Double, Double)

-- | Allows to take into account the sign of the absolute error of the
--   aproximation. If the <a>ErrorImpact</a> parameter is equal to 0 , then
--   absolute error can be of any sign, otherwise the sign of it is the
--   same as the sign of the <a>ErrorImpact</a> argument.
suitable21G :: ErrorImpact -> Double -> Maybe ([Double], Double)
suitable21 :: Double -> Maybe ([Double], Double)
isRangeN :: Double -> Bool

-- | The preferable range of the argument for <a>suitable2</a> and
--   <a>suitable21</a> functions. For arguments in this range the functions
--   always have informative results.
isRangeNPref :: Double -> Bool

-- | Allows to take into account the sign of the absolute error of the
--   aproximation. If the <a>ErrorImpact</a> parameter is equal to 0 , then
--   absolute error can be of any sign, otherwise the sign of it is the
--   same as the sign of the <a>ErrorImpact</a> argument.
check1FracDecompG :: ErrorImpact -> Double -> Maybe ([Double], Double)

-- | Allows to take into account the sign of the absolute error of the
--   aproximation. If the <a>ErrorImpact</a> parameter is equal to 0 , then
--   absolute error can be of any sign, otherwise the sign of it is the
--   same as the sign of the <a>ErrorImpact</a> argument.
check3FracDecompPartialG :: ErrorImpact -> Bool -> Double -> Maybe ([Double], Double)

-- | Allows to take into account the sign of the absolute error of the
--   aproximation. If the <a>ErrorImpact</a> parameter is equal to 0 , then
--   absolute error can be of any sign, otherwise the sign of it is the
--   same as the sign of the <a>ErrorImpact</a> argument.
check3FracDecompPartialPG :: ErrorImpact -> Bool -> [Int] -> Double -> Maybe ([Double], Double)

-- | Allows to take into account the sign of the absolute error of the
--   aproximation. If the <a>ErrorImpact</a> parameter is equal to 0 , then
--   absolute error can be of any sign, otherwise the sign of it is the
--   same as the sign of the <a>ErrorImpact</a> argument.
lessErrSimpleDecompPG :: ErrorImpact -> [Int] -> Double -> (Int, Maybe ([Double], Double), Double)

-- | Allows to take into account the sign of the absolute error of the
--   aproximation. If the <a>ErrorImpact</a> parameter is equal to 0 , then
--   absolute error can be of any sign, otherwise the sign of it is the
--   same as the sign of the <a>ErrorImpact</a> argument.
lessErrDenomsPG :: ErrorImpact -> [Int] -> Double -> [Integer]

-- | Tries to approximate the fraction by just one unit fraction. Can be
--   used for the numbers between 0.005 and 0.501.
check1FracDecomp :: Double -> Maybe ([Double], Double)

-- | Function to find the less by absolute value error approximation. One
--   of the denominators is taken from the range [2..10]. The two others
--   are taken from the appropriate <a>suitable21</a> applicattion.
check3FracDecompPartial :: Bool -> Double -> Maybe ([Double], Double)

-- | Extended version of the <a>check3FracDecompPartial</a> with the first
--   denominator being taken not - only from the [2..10], but also from the
--   custom user provided list. -
check3FracDecompPartialP :: Bool -> [Int] -> Double -> Maybe ([Double], Double)
lessErrSimpleDecompP :: [Int] -> Double -> (Int, Maybe ([Double], Double), Double)

-- | Returns a list of denominators for fraction decomposition using also
--   those ones from the the list provided as the first argument. Searches
--   for the least error from the checked ones.
lessErrDenomsP :: [Int] -> Double -> [Integer]

-- | Returns a unit fraction decomposition into 4 unit fractions. For the
--   cases where the fraction can be represented as a sum of three or less
--   unit fractions, the last element in the resulting list is really
--   related to the floating-point arithmetic rounding errors and it
--   depends on the machine architecture and the realization of the
--   floating-point arithmetic operations. Almost any <a>Double</a> value
--   inside the [0.005, 2/3] can be rather well approximated by the sum of
--   4 unit fractions from the function here. There are also some intervals
--   in the (2/3, 1] that have also good approximations, though not every
--   fraction is approximated well here.
lessErrSimpleDecomp4PG :: ErrorImpact -> [Int] -> Double -> ([Integer], Double)

-- | Returns a unit fraction decomposition into 5 unit fractions. For the
--   cases where the fraction can be represented as a sum of three or less
--   unit fractions, the last element(s) in the resulting list is (are)
--   really related to the floating-point arithmetic rounding errors and it
--   depends on the machine architecture and the realization of the
--   floating-point arithmetic operations. Almost any <a>Double</a> value
--   inside the [0.005, 2/3] can be rather well approximated by the sum of
--   5 unit fractions from the function here. There are also some intervals
--   in the (2/3, 1] that have also good approximations, though not every
--   fraction is approximated well here.
lessErrSimpleDecomp5PG :: ErrorImpact -> [Int] -> Double -> ([Integer], Double)

-- | The argument should be greater or equal than 0.005 (1/200) though it
--   is not checked. Returns the representation of the fraction using
--   canonical ancient Egyptian representation and its error as
--   <a>Double</a> value in the resulting tuple.
egyptianFractionDecomposition :: Double -> ([(Integer, Integer)], Double)