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


-- | Exact computation with constructible real numbers
--   
--   The constructible reals are the subset of the real numbers that can be
--   represented exactly using field operations (addition, subtraction,
--   multiplication, division) and positive square roots. They support
--   exact computations, equality comparisons, and ordering.
@package constructible
@version 0.1.1


-- | The constructible reals, <a>Construct</a>, are the subset of the real
--   numbers that can be represented exactly using field operations
--   (addition, subtraction, multiplication, division) and positive square
--   roots. They support exact computations, equality comparisons, and
--   ordering.
--   
--   <pre>
--   &gt;&gt;&gt; [((1 + sqrt 5)/2)^n - ((1 - sqrt 5)/2)^n :: Construct | n &lt;- [1..10]]
--   [sqrt 5,sqrt 5,2*sqrt 5,3*sqrt 5,5*sqrt 5,8*sqrt 5,13*sqrt 5,21*sqrt 5,34*sqrt 5,55*sqrt 5]
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; let f (a, b, t, p) = ((a + b)/2, sqrt (a*b), t - p*((a - b)/2)^2, 2*p)
--   
--   &gt;&gt;&gt; let (a, b, t, p) = f . f . f . f $ (1, 1/sqrt 2, 1/4, 1 :: Construct)
--   
--   &gt;&gt;&gt; floor $ ((a + b)^2/(4*t))*10**40
--   31415926535897932384626433832795028841971
--   </pre>
--   
--   <pre>
--   &gt;&gt;&gt; let qf (p, q) = ((p + sqrt (p^2 - 4*q))/2, (p - sqrt (p^2 - 4*q))/2 :: Construct)
--   
--   &gt;&gt;&gt; let [(v, w), (x, _), (y, _), (z, _)] = map qf [(-1, -4), (v, -1), (w, -1), (x, y)]
--   
--   &gt;&gt;&gt; z/2
--   -1/16 + 1/16*sqrt 17 + 1/8*sqrt (17/2 - 1/2*sqrt 17) + 1/4*sqrt (17/4 + 3/4*sqrt 17 - (3/4 + 1/4*sqrt 17)*sqrt (17/2 - 1/2*sqrt 17))
--   </pre>
--   
--   Constructible complex numbers may be built from constructible reals
--   using <a>Complex</a> from the complex-generic library.
--   
--   <pre>
--   &gt;&gt;&gt; (z/2 :+ sqrt (1 - (z/2)^2))^17
--   1 :+ 0
--   </pre>
module Data.Real.Constructible

-- | The type of constructible real numbers.
data Construct

-- | Deconstruct a rational constructible number as a <a>Rational</a>, or
--   an irrational constructible number as a triple <tt>(a, b, r)</tt> of
--   simpler constructible numbers representing <tt>a + b*sqrt r</tt> (with
--   <tt>b /= 0</tt> and <tt>r &gt; 0</tt>). Recursively calling
--   <a>deconstruct</a> on all triples will yield a finite tree that
--   terminates in <a>Rational</a> leaves.
--   
--   Note that two constructible numbers that compare as equal may
--   deconstruct in different ways.
deconstruct :: Construct -> Either Rational (Construct, Construct, Construct)

-- | Evaluate a floating-point approximation for a constructible number.
--   
--   To improve numerical stability, addition of numbers with different
--   signs is avoided using quadratic conjugation.
--   
--   <pre>
--   &gt;&gt;&gt; fromConstruct $ sum (map sqrt [7, 14, 39, 70, 72, 76, 85]) - sum (map sqrt [13, 16, 46, 55, 67, 73, 79])
--   1.8837969820815017e-19
--   </pre>
fromConstruct :: Floating a => Construct -> a

-- | The type of exceptions thrown by impossible <a>Construct</a>
--   operations.
data ConstructException

-- | <a>toRational</a> was given an irrational constructible number.
ConstructIrrational :: ConstructException

-- | <a>sqrt</a> was given a negative constructible number.
ConstructSqrtNegative :: ConstructException

-- | <a>**</a> was given an exponent that is not a dyadic rational, or a
--   transcendental function was called.
Unconstructible :: String -> ConstructException
instance Data.Complex.Generic.Class.ComplexRect (Data.Complex.Generic.Complex Data.Real.Constructible.Construct) Data.Real.Constructible.Construct
instance Data.Complex.Generic.Class.ComplexPolar (Data.Complex.Generic.Complex Data.Real.Constructible.Construct) Data.Real.Constructible.Construct
instance GHC.Num.Num (Data.Complex.Generic.Complex Data.Real.Constructible.Construct)
instance GHC.Real.Fractional (Data.Complex.Generic.Complex Data.Real.Constructible.Construct)
instance GHC.Float.Floating (Data.Complex.Generic.Complex Data.Real.Constructible.Construct)
instance GHC.Classes.Ord Data.Real.Constructible.ConstructException
instance GHC.Classes.Eq Data.Real.Constructible.ConstructException
instance GHC.Show.Show Data.Real.Constructible.FieldShape
instance GHC.Show.Show Data.Real.Constructible.ConstructException
instance GHC.Exception.Type.Exception Data.Real.Constructible.ConstructException
instance GHC.Float.Floating Data.Real.Constructible.Construct
instance GHC.Real.Real Data.Real.Constructible.Construct
instance GHC.Classes.Eq Data.Real.Constructible.Construct
instance GHC.Classes.Ord Data.Real.Constructible.Construct
instance GHC.Num.Num Data.Real.Constructible.Construct
instance GHC.Real.Fractional Data.Real.Constructible.Construct
instance GHC.Show.Show Data.Real.Constructible.Construct
instance GHC.Read.Read Data.Real.Constructible.Construct
instance GHC.Real.RealFrac Data.Real.Constructible.Construct
instance GHC.Enum.Enum Data.Real.Constructible.Construct
instance GHC.Show.Show (Data.Real.Constructible.Field k)