## Math::Interval

- viz. https://en.wikipedia.org/wiki/Interval_arithmetic
- viz. https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node45.html

### A elementary implementation of Interval Arithmetic using raku Ranges

Check out the Wikipedia page first - sub and div may confound your expectations:

## Synopsis

#!/usr/bin/env raku use lib '../lib'; use Data::Dump::Tree; use Math::Interval; # Range-Range Operations use interval math say (1..2) + (2..4); #3..6 say (2..4) - (1..2); #0..3 (yes - this is weird!) say (2..4) * (1..2); #2..8 say (2..4) / (1..2); #1.0..4.0 # strings work too! (must be numbers) say (1..2) + ('1'..'5'); #2..7 # an Interval is returned ddt ((1..2) + (2..4)) ~~ Interval; #True #| Interval is a child of class Range where endpoints are always Numeric #| No cats ears, not Positional, not Iterable, no .elems my Interval $i1 .= new(range => 2.5^..^8.5); #3.5..7.5 my Interval $i2 .= new(2..^8); #2e0..7e0 my Interval $i4 .= new(1,2); #1..2 my Interval $i5 .= new($i1); #2e0..7e0 my Interval $i8 .= new(4.5,6.5); # '~~' checks if y contains x say 2 ~~ $i2; #True say $i8 ~~ $i1; #True say $i1 ~~ $i8; #False # cmp checks Order say $i1 cmp $i1; #Same say $i1 cmp $i4; #More say $i4 cmp $i1; #Less say $i1 cmp $i2; #Nil overlaps are not ordered say $i2 cmp $i1; #Nil "" # union ∪ [(|)] and intersection ∩ [(&)] say $i8 ∪ $i4; say $i4 ∩ $i8; #∅ the null Set() # gotchas #say +$i1; #fails - Interval has no .elems #say $i1.Set; #fails - raku Sets must contain discrete items say $i1.Range.Set; #coerce to Range to discretize an Interval ## the -Ofun bit # a divisor that spans 0 produces a disjoint multi-interval my $j1 = (2..4)/(-2..4); ddt $j1; #any(-Inf..-1.0, 0.5..Inf).Junction say 3 ~~ $j1; #True # Junction[Interval] can still be used my $j2 = $j1 + 2; ddt $j2; #any(-Inf..1.0, 2.5..Inf).Junction say 5 ~~ $j2; #True # but this can only go so far... my $j3 = $j1 * (-2..4); ddt $j3; #any(-Inf..Inf, -Inf..Inf).Junction say 3 ~~ $j3; #True (but meaningless!)

## Explanation

We split the use cases of the built-in Range type as follows:

### class Range

- Use case
- to generate lists of consecutive numbers or strings
- to act as a matcher to check if a Numeric or Stringy or Range is within a certain Range

- endpoints with/out cats ears (±1)
- does Positional, does Iterable
- arithmetic +-*/ operators with scalars are distributed to endpoints like Junction with 2 elems, then each endpoint coerced to .Int
- prefix '+' special cased to .elems
- use
`~~`

to check containment`say 3 ~~ 1..12;`

#True x1 <= a <= x2`say 2..3 ~~ 1..12;`

#True x1 >= y1 && x2 <= y2`say 1..12 ~~ 2..3;`

#False (y must contain x)

- cmp works for Range op Range (for Real op Range .elems is used)
`(0..2) cmp (0..12)`

#Less x1 < x2 || x1 == x2 && y1 < y2`(0..2) cmp (0..2)`

#Same x1 == x2 && y1 == y2`(1..2) cmp (0..12)`

#More x1 > x2 || x1 == x2 && y1 > y2

- Set operations work for Any op Range and Range op Range by auto coercing both args via
`.Set`

### class Interval

- Use case
- to act as a matcher to check if a Numeric or Interval is within a certain range

- endpoints will ingest cats ears, x1 <= x2
- not Iterable nor Positional
- arithmetic +-*/ operators with scalars are distributed to endpoints like Junction with 2 elems
- Rangy op Rangy --> Interval arithmetic +-*/ operators implemented
- Rangy ** N --> Interval operator implemented
- prefix '+' will fail (Interval has no .elems)
- use
`~~`

to check containment`say 3 ~~ 1..12;`

#True x1 <= a <= x2`say 2..3 ~~ 1..12;`

#True x1 >= y1 && x2 <= y2`say 1..12 ~~ 2..3;`

#False (y must contain x)

- cmp works for Interval op Interval
- UNLIKE Range, overlapping intervals are not ordered and yet not equal
`(1..2) cmp (3..4)`

#Less x2 < y1`(1..2) cmp (2..4)`

#Nil x2 !< y1 !!`(0..2) cmp (0..2)`

#Same x1 == x2 && y1 == y2`(0..3) cmp (0..2)`

#Nil x1 !> y2 !!`(3..4) cmp (1..2)`

#More x1 > y2

Set operations
`#say $i1.Set;`

#fails - in general Sets contain discrete items and Intervals are continuous
`say $i1.Range.Set;`

#coerce to Range first to use all Set operators (which discretizes the Interval)

However, the following two Set operations are implemented for Intervals:

intersection (&) ∩
`∅`

if x1 > y2 || y1 > x2
`max(x1,y1)..min(x2,y2)`

union (|) ∪ (of two intersecting intervals)
`min(x1,y1)..max(x2,y2)`

other
`(3..4).abs`

#4 (always x2)
`(3..4).width`

#1 (x2-x1)

which leads to these design points:

`class Interval is Range {...}`

- so can be used wherever a Range is used
- noting lack of Iterator or Positional support

`Interval: new $range`

- rejects non-Real endpoints (eg. Str)
- adjusts endpoints (±1) to strip cats ears

`.Range`

coerces to Range`subset Rangy of Any where * ~~ Range|Interval;`

No provision is (yet) made for Rounded Interval Arithmetic

No provision is (yet) made for complex intervals

Only a handful of all possible Interval operations (eg. log, exp, trig)

*Please feel free to submit any of these as a PR (see TODOs below)*

## TODOs

### Additional arithmetic operations

- power (even / odd)
- log / exp
- trig
- Newton

### Copyright

copyright(c) 2023 Henley Cloud Consulting Ltd.