Data.Real.Constructible

The constructible reals, Construct, 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.

>>> [((1 + sqrt 5)/2)^n - ((1 - sqrt 5)/2)^n :: Construct | n <- [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]

>>> let f (a, b, t, p) = ((a + b)/2, sqrt (a*b), t - p*((a - b)/2)^2, 2*p)
>>> let (a, b, t, p) = f . f . f . f $ (1, 1/sqrt 2, 1/4, 1 :: Construct)
>>> floor $ ((a + b)^2/(4*t))*10**40
31415926535897932384626433832795028841971

>>> let qf (p, q) = ((p + sqrt (p^2 - 4*q))/2, (p - sqrt (p^2 - 4*q))/2 :: Construct)
>>> let [(v, w), (x, _), (y, _), (z, _)] = map qf [(-1, -4), (v, -1), (w, -1), (x, y)]
>>> 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))

Constructible complex numbers may be built from constructible reals using Complex from the complex-generic library.

>>> (z/2 :+ sqrt (1 - (z/2)^2))^17
1 :+ 0

Documentation

data Construct

The type of constructible real numbers.

Instances

Enum Construct

Eq Construct

Floating Construct

This partial Floating instance only supports sqrt and ** where the exponent is a dyadic rational. Passing a negative number to sqrt will throw the ConstructSqrtNegative exception. All other operations will throw the Unconstructible exception.

Fractional Construct

Num Construct

Ord Construct

Read Construct

Real Construct

This Real instance only supports toRational on numbers that are in fact rational. toRational on an irrational number will throw the ConstructIrrational exception.

RealFrac Construct

Show Construct

Floating (Complex Construct)

Fractional (Complex Construct)

Num (Complex Construct)

ComplexRect (Complex Construct) Construct

ComplexPolar (Complex Construct) Construct

deconstruct :: Construct -> Either Rational (Construct, Construct, Construct)

Deconstruct a rational constructible number as a Rational, or an irrational constructible number as a triple (a, b, r) of simpler constructible numbers representing a + b*sqrt r (with b /= 0 and r > 0). Recursively calling deconstruct on all triples will yield a finite tree that terminates in Rational leaves.

Note that two constructible numbers that compare as equal may deconstruct in different ways.

fromConstruct :: Floating a => Construct -> a

Evaluate a floating-point approximation for a constructible number.

To improve numerical stability, addition of numbers with different signs is avoided using quadratic conjugation.

>>> fromConstruct $ sum (map sqrt [7, 14, 39, 70, 72, 76, 85]) - sum (map sqrt [13, 16, 46, 55, 67, 73, 79])
1.8837969820815017e-19

data ConstructException

The type of exceptions thrown by impossible Construct operations.

Constructors

ConstructIrrational

toRational was given an irrational constructible number.

ConstructSqrtNegative

sqrt was given a negative constructible number.

Unconstructible String

** was given an exponent that is not a dyadic rational, or a transcendental function was called.

Instances

Eq ConstructException

Ord ConstructException

Show ConstructException

Typeable ConstructException

Exception ConstructException