default show for Arb might be confusing

[kalmarek]

This is an interval containing 0:

julia> let x = sin(Arb(π))
           @info "" x contains(x, 0) Arblib.midref(x) Float64(Arblib.radref(x))
       end
┌ Info: 
│   x = [+/- 2.83e-77]
│   contains(x, 0) = true
│   Arblib.midref(x) = 1.0969174409793520767073235159733943661910882381141135060401716459409529154108e-77
└   Float64(Arblib.radref(x)) = 1.730607217577946e-77

This is an interval containing 1:

julia> let x = sin(Arb(π)/2)
           @info "sin(Arb(π))" x contains(x, 1) Arblib.midref(x) Float64(Arblib.radref(x))
       end
┌ Info: sin(Arb(π))
│   x = [1.000000000000000000000000000000000000000000000000000000000000000000000000000 +/- 1.74e-77]
│   contains(x, 1) = true
│   Arblib.midref(x) = 0.99999999999999999999999999999999999999999999999999999999999999999999999999999
└   Float64(Arblib.radref(x)) = 8.737900781332737e-78

This is an interval containing 1.5:

julia> let x = 1.5sin(Arb(π)/2)
           @info "1.5sin(Arb(π)/2)" x contains(2x, 3) Arblib.midref(x) Float64(Arblib.radref(x))
       end
┌ Info: 1.5sin(Arb(π)/2)
│   x = [1.5000000000000000000000000000000000000000000000000000000000000000000000000000 +/- 4.77e-77]
│   contains(2x, 3) = true
│   Arblib.midref(x) = 1.5
└   Float64(Arblib.radref(x)) = 3.0379188354575523e-77

let's try the same with set_interval!:

julia> let x = Arblib.set_interval!(Arb(), Arf(-0.5), Arf(0.5))
           @info "[-0.5, 0.5]" x contains(x, 0) Arblib.midref(x) Float64(Arblib.radref(x))
       end
┌ Info: [-0.5, 0.5]
│   x = [+/- 0.501]
│   contains(x, 0) = true
│   Arblib.midref(x) = 0
└   Float64(Arblib.radref(x)) = 0.5000000009313226

julia> let x = Arblib.set_interval!(Arb(), Arf(0.5), Arf(1.5))
           @info "[0.5, 1.5]" x contains(x, 1) Arblib.midref(x) Float64(Arblib.radref(x))
       end
┌ Info: [0.5, 1.5]
│   x = [1e+0 +/- 0.501]
│   contains(x, 1) = true
│   Arblib.midref(x) = 1
└   Float64(Arblib.radref(x)) = 0.5000000009313226

julia> let x = Arblib.set_interval!(Arb(), Arf(1.0), Arf(2.0))
           @info "[1.0, 2.0]" x contains(2x, 3) Arblib.midref(x) Float64(Arblib.radref(x))
       end
┌ Info: [1.0, 2.0]
│   x = [+/- 2.01]              ← this is unexpected
│   contains(2x, 3) = true
│   Arblib.midref(x) = 1.5
└   Float64(Arblib.radref(x)) = 0.5000000009313226

moreover:

julia> let x = Arblib.set_interval!(Arb(), Arf(1.0), Arf(2.0))
           @info "[1.0, 2.0]" Arblib.string_decimal(x) Arblib.string_nice(x)
       end
┌ Info: [1.0, 2.0]
│   Arblib.string_decimal(x) = "1.5 +/- 0.5"
└   Arblib.string_nice(x) = "[+/- 2.01]"

[saschatimme]

What is exactly the bug? My hunch is that in the second and third case the midpoint is not exactly what is printed. Printed is the rounded version. The first I guess is a concise printing. In the other cases the midpoint has an exact representation and is therefore abbreviated.

[kalmarek]

the bug is that

julia> Arblib.set_interval!(Arb(), Arf(-2.0), Arf(2.0))
[+/- 2.01]

julia> Arblib.set_interval!(Arb(), Arf(1.0), Arf(2.0))
[+/- 2.01]

julia> Arblib.string_decimal(Arblib.set_interval!(Arb(), Arf(1.0), Arf(2.0)))
"1.5 +/- 0.5"

julia> Arblib.string_decimal(Arblib.set_interval!(Arb(), Arf(-2.0), Arf(2.0)))
"0 +/- 2"

the interval [-2.0, 2.0] has the same string_nice (which is used for show) as [1.0, 2.0]

[saschatimme]

oh I see

[Joel-Dahne]

I don't think this is a bug but a result of the fact that string_nice only shows correct digits. The Interval [1.0, 2.0] contains both 1 and 2 so not a single digit of the number is known, hence it give anything before the +/- and has to resort to using the radius alone for the result. We get the same result with for example

julia> Arblib.set_interval!(Arb(), Arf(80000), Arf(90000))
[+/- 9.01e+4]

[kalmarek]

thanks for the explanation @Joel-Dahne; I found it confusing, but it makes perfect sense after explanation; however: given that this requires an explanation should we switch to string_decimal for show?

[Joel-Dahne]

That is a good point actually! I have actually been bitten a couple of times by this behaviour myself. The main risk I see with this is that reading that string back with arb_get_str is not guaranteed to give an enclosure since for string_decimal, as written in the documentation, "The printed value of the radius is not adjusted to compensate for the fact that the binary-to-decimal conversion of both the midpoint and the radius introduces additional error."

[kalmarek]

I see, but given the problems with string_nice users should be discouraged from parsing the readable string representation of Arbs:

julia> x = Arblib.set_interval!(Arb(), Arf(1.0), Arf(2.0))
[+/- 2.01]

julia> y = Arb(Arblib.string_nice(x))
[+/- 2.02]

julia> Arblib.midref(y)
0

julia> z = Arb(Arblib.string_decimal(x)) # ← but not guaranteed to enclose x...
[+/- 2.01]

julia> Arblib.midref(z)
1.5

the functions for loading and storing are dump_string and load_string!. This requires additional entry in the docs though.

EDIT:
is there a function in julia for serialization into strings? I thought about repr, but reading the docs it doesn't seem to be the one...