Documentation

Create a non-empty interval, turning it around if necessary

Try to create a non-empty interval.

The whole real number line

>>> whole
-Infinity ... Infinity

(x :: Double) `elem` whole

A singleton point

>>> singleton 1
1 ... 1

x `elem` (singleton x)

x /= y ==> y `notElem` (singleton x)

Determine if a point is in the interval.

>>> member 3.2 (1.0 ... 5.0)
True

>>> member 5 (1.0 ... 5.0)
True

>>> member 1 (1.0 ... 5.0)
True

>>> member 8 (1.0 ... 5.0)
False

Determine if a point is not included in the interval

>>> notMember 8 (1.0 ... 5.0)
True

>>> notMember 1.4 (1.0 ... 5.0)
False

Determine if a point is in the interval.

>>> elem 3.2 (1.0 ... 5.0)
True

>>> elem 5 (1.0 ... 5.0)
True

>>> elem 1 (1.0 ... 5.0)
True

>>> elem 8 (1.0 ... 5.0)
False

Determine if a point is not included in the interval

>>> notElem 8 (1.0 ... 5.0)
True

>>> notElem 1.4 (1.0 ... 5.0)
False

The infinumum (lower bound) of an interval

>>> inf (1 ... 20)
1

min x y == inf (x ... y)

inf x <= sup x

The supremum (upper bound) of an interval

>>> sup (1 ... 20)
20

sup x `elem` x

max x y == sup (x ... y)

inf x <= sup x

Is the interval a singleton point? N.B. This is fairly fragile and likely will not hold after even a few operations that only involve singletons

>>> singular (singleton 1)
True

>>> singular (1.0 ... 20.0)
False

Calculate the width of an interval.

>>> width (1 ... 20)
19

>>> width (singleton 1)
0

0 <= width x

Nearest point to the midpoint of the interval.

>>> midpoint (10.0 ... 20.0)
15.0

>>> midpoint (singleton 5.0)
5.0

midpoint x `elem` (x :: Interval Double)

Hausdorff distance between intervals.

>>> distance (1 ... 7) (6 ... 10)
0

>>> distance (1 ... 7) (15 ... 24)
8

>>> distance (1 ... 7) (-10 ... -2)
3

commutative (distance :: Interval Double -> Interval Double -> Double)

0 <= distance x y

Calculate the intersection of two intervals.

>>> intersection (1 ... 10 :: Interval Double) (5 ... 15 :: Interval Double)
Just (5.0 ... 10.0)

Calculate the convex hull of two intervals

>>> hull (0 ... 10 :: Interval Double) (5 ... 15 :: Interval Double)
0.0 ... 15.0

>>> hull (15 ... 85 :: Interval Double) (0 ... 10 :: Interval Double)
0.0 ... 85.0

conservative2 const hull

conservative2 (flip const) hull

Bisect an interval at its midpoint.

>>> bisect (10.0 ... 20.0)
(10.0 ... 15.0,15.0 ... 20.0)

>>> bisect (singleton 5.0)
(5.0 ... 5.0,5.0 ... 5.0)

let (a, b) = bisect (x :: Interval Double) in sup a == inf b

let (a, b) = bisect (x :: Interval Double) in inf a == inf x

let (a, b) = bisect (x :: Interval Double) in sup b == sup x

Magnitude

>>> magnitude (1 ... 20)
20

>>> magnitude (-20 ... 10)
20

>>> magnitude (singleton 5)
5

0 <= magnitude x

"mignitude"

>>> mignitude (1 ... 20)
1

>>> mignitude (-20 ... 10)
0

>>> mignitude (singleton 5)
5

0 <= mignitude x

Check if interval X totally contains interval Y

>>> (20 ... 40 :: Interval Double) `contains` (25 ... 35 :: Interval Double)
True

>>> (20 ... 40 :: Interval Double) `contains` (15 ... 35 :: Interval Double)
False

Flipped version of contains. Check if interval X a subset of interval Y

>>> (25 ... 35 :: Interval Double) `isSubsetOf` (20 ... 40 :: Interval Double)
True

>>> (20 ... 40 :: Interval Double) `isSubsetOf` (15 ... 35 :: Interval Double)
False

For all x in X, y in Y. x op y

For all x in X, y in Y. x < y

>>> (5 ... 10 :: Interval Double) <! (20 ... 30 :: Interval Double)
True

>>> (5 ... 10 :: Interval Double) <! (10 ... 30 :: Interval Double)
False

>>> (20 ... 30 :: Interval Double) <! (5 ... 10 :: Interval Double)
False

For all x in X, y in Y. x <= y

>>> (5 ... 10 :: Interval Double) <=! (20 ... 30 :: Interval Double)
True

>>> (5 ... 10 :: Interval Double) <=! (10 ... 30 :: Interval Double)
True

>>> (20 ... 30 :: Interval Double) <=! (5 ... 10 :: Interval Double)
False

For all x in X, y in Y. x == y

Only singleton intervals or empty intervals can return true

>>> (singleton 5 :: Interval Double) ==! (singleton 5 :: Interval Double)
True

>>> (5 ... 10 :: Interval Double) ==! (5 ... 10 :: Interval Double)
False

For all x in X, y in Y. x /= y

>>> (5 ... 15 :: Interval Double) /=! (20 ... 40 :: Interval Double)
True

>>> (5 ... 15 :: Interval Double) /=! (15 ... 40 :: Interval Double)
False

For all x in X, y in Y. x >= y

>>> (20 ... 40 :: Interval Double) >=! (10 ... 20 :: Interval Double)
True

>>> (5 ... 20 :: Interval Double) >=! (15 ... 40 :: Interval Double)
False

For all x in X, y in Y. x > y

>>> (20 ... 40 :: Interval Double) >! (10 ... 19 :: Interval Double)
True

>>> (5 ... 20 :: Interval Double) >! (15 ... 40 :: Interval Double)
False

Does there exist an x in X, y in Y such that x op y?

Does there exist an x in X, y in Y such that x < y?

Does there exist an x in X, y in Y such that x <= y?

Does there exist an x in X, y in Y such that x == y?

Does there exist an x in X, y in Y such that x /= y?

Does there exist an x in X, y in Y such that x >= y?

Does there exist an x in X, y in Y such that x > y?

The nearest value to that supplied which is contained in the interval.

(clamp xs y) `elem` xs

Inflate an interval by enlarging it at both ends.

>>> inflate 3 (-1 ... 7)
-4 ... 10

>>> inflate (-2) (0 ... 4)
-2 ... 6

inflate x i `contains` i

Deflate an interval by shrinking it from both ends. Note that in cases that would result in an empty interval, the result is a singleton interval at the midpoint.

>>> deflate 3.0 (-4.0 ... 10.0)
-1.0 ... 7.0

>>> deflate 2.0 (-1.0 ... 1.0)
0.0 ... 0.0

Scale an interval about its midpoint.

>>> scale 1.1 (-6.0 ... 4.0)
-6.5 ... 4.5

>>> scale (-2.0) (-1.0 ... 1.0)
-2.0 ... 2.0

abs x >= 1 ==> (scale (x :: Double) i) `contains` i

forAll (choose (0,1)) $ \x -> abs x <= 1 ==> i `contains` (scale (x :: Double) i)

Construct a symmetric interval.

>>> symmetric 3
-3 ... 3

>>> symmetric (-2)
-2 ... 2

x `elem` symmetric x

0 `elem` symmetric x

id function. Useful for type specification

>>> :t idouble (1 ... 3)
idouble (1 ... 3) :: Interval Double

id function. Useful for type specification

>>> :t ifloat (1 ... 3)
ifloat (1 ... 3) :: Interval Float

an interval containing all x quot y prop> forAll (memberOf xs) $ x -> forAll (memberOf ys) $ y -> 0 notMember ys ==> (x quot y) member (xs iquot ys) prop> 0 member ys ==> ioProperty $ do z <- try (evaluate (xs iquot ys)); return $ z === Left DivideByZero

an interval containing all x rem y prop> forAll (memberOf xs) $ x -> forAll (memberOf ys) $ y -> 0 notMember ys ==> (x rem y) member (xs irem ys) prop> 0 member ys ==> ioProperty $ do z <- try (evaluate (xs irem ys)); return $ z === Left DivideByZero

an interval containing all x div y prop> forAll (memberOf xs) $ x -> forAll (memberOf ys) $ y -> 0 notMember ys ==> (x div y) member (xs idiv ys) prop> 0 member ys ==> ioProperty $ do z <- try (evaluate (xs idiv ys)); return $ z === Left DivideByZero

an interval containing all x mod y prop> forAll (memberOf xs) $ x -> forAll (memberOf ys) $ y -> 0 notMember ys ==> (x mod y) member (xs imod ys) prop> 0 member ys ==> ioProperty $ do z <- try (evaluate (xs imod ys)); return $ z === Left DivideByZero