AbstractFloat interface functions

[kalmarek]

to make Arbs slightly easier to use in generic numerical code

that includes low level functions such as:

• eps
• exponent
• you name it;)

and high level that could be composed of Arblib functions

• expm1 (needs to be added to

  Arblib.jl/src/arithmetic.jl

  Line 56 in bc9dc58

  :exp,

  ) (see fixed by add expm1 #102)
  ...

[lrnv]

exp1m is the first thing i came into. Maybe you could look at how ArbNumerics made it ?

[kalmarek]

@lrnv locally you can define it as Base.expm1(x::T) where T<:Arblib.ArbOrRef = Arblib.expm1!(T(prec=precision(x)), x), proper fix is to add it to the list mentioned above.

for Base.eps you may try

Base.eps(x::Arblib.ArbOrRef) = Arb(Arblib.set_ulp!(Mag(), Arblib.midref(x), prec=precision(x)))

(it is necessary to pass precision explicitly here as Mags don't carry it). This will allocate additional struct, and I'm not even sure this is what we want ;) Another possibility is

function Base.eps(x::Arblib.ArbOrRef)
    res = Arb(prec = precision(x))
    Arblib.set_ulp!(Arblib.radref(res), Arblib.midref(x), prec=precision(x))
    return res
end

the first one results in

julia> eps(rand(Arb))
[1.7272337110188889250772703725600799142232000728872562770047406940337183606325e-77 +/- 1.46e-154]

the second with

julia> eps(rand(Arb))
[+/- 1.73e-77]

or we could marry both

function Base.eps(x::Arblib.ArbOrRef)
    res = Arb(prec = precision(x))
    Arblib.set_ulp!(Arblib.radref(res), Arblib.midref(x), prec=precision(x))
    Arblib.midref(res)[] = Arblib.radref(res)
    Arblib.zero!(Arblib.radref(res))
    return res
end

Hmm... interesting:

julia> a = exp(Arb(rand())); Arblib.zero!(Arblib.radref(a)); (a, Arblib.radref(a))
([2.1545763189042442095875868401977617041872092787378076208404894100210187829306 +/- 3.03e-77], (0))

@Joel-Dahne could you please explain why this printing of a which has zero radius? is is again about guaranteed digits?

[Joel-Dahne]

Many good ideas here! I'll have to get back when I have more time. But I can at least comment about the printing.

Yes, it's related to the guaranteed digits and comes from the fact that we determine the number of digits to print based on the precision (using digits_prec(prec::Integer) = floor(Int, prec * log(2) / log(10))). This number is exactly representable in base 10 but requires many more digits to do that. We can print the full number by printing more digits

julia> Arblib.string_nice(eps(one(Arb))) # Default version
"[1.7272337110188889250772703725600799142232000728872562770047406940337183606325e-77 +/- 1.46e-154]"

julia> Arblib.string_nice(eps(one(Arb)), 200) # More digits
"1.7272337110188889250772703725600799142232000728872562770047406940337183606324854115943015006944576453121094587892299327193990197893663893387306007554116149549372494220733642578125000000000000000000000e-77"

It seems like choosing a default printing is a non-trivial task to do... We should probably experiment a bit with it and see what we like in the end. At the very least we should mention something about it in the documentation.