Numbers type on which intervals are defined.

The type of intervals.

Neutral element for addition.

Neutral element for multiplication.

π with bounds properly rounded.

2π with bounds properly rounded.

π/2 with bounds properly rounded.

e (Euler's constant) with bounds properly rounded.

The entire set of numbers.

• since 1.5

v a b returns the interval [a, b]. BEWARE that, unless you take care, if you use v a b with literal values for a and/or b, the resulting interval may not contain these values because the compiler will round them to binary numbers before passing them to v.

• raises Invalid_argument

  if the interval [a, b] is equal to [-∞,-∞] or [+∞,+∞] or one of the bounds is NaN.

low t returns the lower bound of the interval.

high t returns the higher bound of the interval.

Returns the interval containing the conversion of an integer to the number type.

to_string i return a string representation of the interval i.

• parameter fmt

  is the format used to print the two bounds of i. Default: "%g" for float numbers.

Print the interval to the channel. To be used with Printf format "%a".

Print the interval to the formatter. To be used with Format format "%a".

fmt number_fmt returns a format to print intervals where each component is printed with number_fmt.

Example: Printf.printf ("%s = " ^^ fmt "%.10f" ^^ "\n") name i.

compare_f a x returns

• 1 if high(a) < x,
• 0 if low(a) ≤ x ≤ high(a), i.e., if x ∈ a, and
• -1 if x < low(a).

is_bounded x says whether the interval is bounded, i.e., -∞ < low(x) and high(x) < ∞.

• since 1.5

is_entire x says whether x is the entire interval.

• since 1.5

equal a b says whether the two intervals are the same.

• since 1.5

Synonym for equal.

• since 1.5

subset x y returns true iff x ⊆ y.

• since 1.5

x <= y says whether x is weakly less than y i.e., ∀ξ ∈ x, ∃η ∈ y, ξ ≤ η and ∀η ∈ y, ∃ξ ∈ x, ξ ≤ η.

• since 1.5

x >= y says whether x is weakly greater than y i.e., ∀ξ ∈ x, ∃η ∈ y, ξ ≥ η and ∀η ∈ y, ∃ξ ∈ x, ξ ≥ η.

• since 1.5

precedes x y returns true iff x is to the left but may touch y.

• since 1.5

interior x y returns true if x is interior to y in the topological sense. For example interior entire entire is true.

• since 1.5

x < y says whether x is strictly weakly less than y i.e., ∀ξ ∈ x, ∃η ∈ y, ξ < η and ∀η ∈ y, ∃ξ ∈ x, ξ < η.

• since 1.5

x > y iff y < x.

• since 1.5

strict_precedes x y returns true iff x is to the left and does not touch y.

• since 1.5

disjoint x y returns true iff x ∩ y = ∅.

• since 1.5

size a returns an interval containing the true width of the interval high a - low a.

size_high a returns the width of the interval high a - low a rounded up.

size_low a returns the width of the interval high a - low a rounded down.

mag x returns the magnitude of x, that is sup{ |x| : x ∈ x }.

mig x returns the mignitude of x, that is inf{ |x| : x ∈ x }.

sgn a returns the sign of each bound, e.g., for floats [float (compare (low a) 0.), float (compare (high a) 0.)].

truncate a returns the integer interval containing a, that is [floor(low a), ceil(high a)].

abs a returns the absolute value of the interval, that is

• a if low a ≥ 0.,
• ~- a if high a ≤ 0., and
• [0, max (- low a) (high a)] otherwise.

hull a b returns the smallest interval containing a and b, that is [min (low a) (low b), max (high a) (high b)].

inter_exn x y returns the intersection of x and y.

• raises Domain_error

  if the intersection is empty.

• since 1.5

inter_exn x y returns Some z where z is the intersection of x and y if it is not empty and None if the intersection is empty.

• since 1.5

max a b returns the "maximum" of the intervals a and b, that is [max (low a) (low b), max (high a) (high b)].

min a b returns the "minimum" of the intervals a and b, that is [min (low a) (low b), min (high a) (high b)].

a + b returns [low a +. low b, high a +. high b] properly rounded.

a +. x returns [low a +. x, high a +. x] properly rounded.

x +: a returns [a +. low a, x +. high a] properly rounded.

a - b returns [low a -. high b, high a -. low b] properly rounded.

a -. x returns [low a -. x, high a -. x] properly rounded.

x -: a returns [x -. high a, x -. low a] properly rounded.

~- a is the unary negation, it returns [-high a, -low a].

a * b multiplies a by b according to interval arithmetic and returns the proper result. If a=zero or b=zero then zero is returned.

x *. a multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero is returned.

a *. x multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero is returned.

a / b divides the first interval by the second according to interval arithmetic and returns the proper result. Raise Interval.Division_by_zero if b=zero.

a /. x divides a by x according to interval arithmetic and returns the proper result. Raise Interval.Division_by_zero if x=0.0.

x /: a divides x by a according to interval arithmetic and returns the result. Raise Interval.Division_by_zero if a=zero.

inv a returns 1. /: a but is more efficient. Raise Interval.Division_by_zero if a=zero.

invx a is the extended division. When 0 ∉ a, the result is One(inv a). If 0 ∈ a, then the two natural intervals (properly rounded) Two([-∞, 1/(low a)], [1/(high a), +∞]) are returned. Raise Interval.Division_by_zero if a=zero.

cancelminus x y returns the tightest interval z such that x ⊆ z + y. If no such z exists, it returns entire.

• since 1.5

cancelplus x y returns the tightest interval z such that x ⊆ z - y. If no such z exists, it returns entire.

• since 1.5

a**n returns interval a raised to nth power according to interval arithmetic. If n=0 then one is returned.

• raises Domain_error

  if n < 0 and a=zero.