Rational powers and coversion

[pwl]

There is something wrong with Rational powers:

julia> BigFloat(2)^(BigFloat(1)/BigFloat(3))-BigFloat(2)^(1//3)
1.615949454734010502584703185359054123712092705937300412204179527347802183365638e-17

where there shouldn't be any difference (1//3 is of infinite precision). The issue comes from the definition of ^ for rationals, where convert is applied to the result of a division of two integers, which is basically a Float64, the result is then converted back to BigFloat

^{T<:AbstractFloat}(x::T, y::Rational) = x^(convert(T, y.num / y.den))
^{T<:AbstractFloat}(x::Complex{T}, y::Rational) = x^(convert(T, y.num / y.den))

Redefining ^ like this

^{T<:AbstractFloat}(x::T, y::Rational) = x^(convert(T, y.num) / convert(T, y.den))
^{T<:AbstractFloat}(x::Complex{T}, y::Rational) = x^(convert(T, y.num) / convert(T, y.den))

fixes the problem:

julia> BigFloat(2)^(BigFloat(1)/BigFloat(3))-BigFloat(2)^(1//3)
0.000000000000000000000000000000000000000000000000000000000000000000000000000000

In summary, convert doesn't commute with division (nor with other similar operations) and convert should be applied as early as possible (directly to the integers in this case) to avoid precision loss. There might be more cases like this that I am not aware of.

EDIT: This bug is new in 0.5 and it was introduced in #17530.

[simonbyrne]

Would you mind opening a pull request?