Remove pi_interval

[dpsanders]

pi_interval(T) is the same as Interval{T}(pi).

cc @yashrajgupta

[dpsanders]

Also we can use oftype(a, pi) if a is an Interval.

[yashrajgupta]

Seems like something is different,

julia> pi_interval(Float64)
Interval(3.141592653589793, 3.1415926535897936)

julia> Interval{Float64}(pi)
Interval(3.141592653589793, 3.141592653589793)

[dpsanders]

Ah, oops. Could you fix this please?

[dpsanders]

(Please quote code inside triple backticks with jl after the first set.)

```jl
(code)
```

[yashrajgupta]

I think pi_interval is a function which we have defined but Interval{T}(pi) is a type conversion of a rational number pi in to an Interval of type T. So what do we want at the end? Is it that pi_interval should be interval of one absolute value of pi or should it an interval containing pi and nearby values.

[dpsanders]

Of course we want it to be the smallest interval containing pi, which is obtained currently from

julia> convert(Interval, pi)
Interval(3.141592653589793, 3.1415926535897936)

(although who knows how efficient that is).

[Please quote code inline as `(code)`.]

[dpsanders]

The idea is always that interval arithmetic should give results that are guaranteed to contain the true results. In this case pi is an irrational (not rational, as you wrote) number, which is not representable exactly in floating point. So any interval calculations with pi must use an interval containing pi, preferably the smallest possible one.

[dpsanders]

Hmmm. It seems that convert here is slow. Maybe we should retain pi_interval(T) and just calculate the interval once and for all statically.

[yashrajgupta]

Sorry about the typo. Yes we should retain pi_interval function until we find a smaller interval for pi moreover I think Interval{T}(pi) does not contain the pi value since it is just a single value interval.

[dpsanders]

I am saying that this is the smallest interval for Float64:

julia> x = convert(Interval{Float64}, pi)
Interval(3.141592653589793, 3.1415926535897936)

You can test this:

julia> nextfloat(x.lo) == x.hi
true

[dpsanders]

Ah, I see that pi_interval already does what I was suggesting -- defines it once and for all.

[dpsanders]

But we should define Interval{Float64}(pi) and Interval(pi) to do the same thing.

[lbenet]

I think pi_interval here is ok; the version for Float64 uses atomic(Interval{Float64}, pi) which is more efficient than convert. Perhaps this should be better explained in the documentation.

[Kolaru]

In #271 I plan to get rid of pi_interval by defining (for each flavor but here is the version for Interval)

Interval{T}(::Irrational{:π}) where T = atomic(Interval{T}, π)
Interval(::Irrational{:π}) = Interval{Float64}(π)

If further optimisation is required, the methods are specific enough to have them there.

[lbenet]

Closed by #338