Inconsistent behaviors when evaluating negative exponents

[HarrisonGrodin]

Working with negative exponents has some very odd behaviors, to say the least. Running the internals of the ^ function gives different results than ^ itself, evaluation success changes based on whether a variable is replaced with its value, and @code_* forms are convinced that the result should always be of type Int64. Tested on v1.0.0 and v1.1.0-DEV.

julia> 2 ^ -5
0.03125

julia> Base.power_by_squaring(2, -5)
ERROR: DomainError with -5:
Cannot raise an integer x to a negative power -5.
Make x a float by adding a zero decimal (e.g., 2.0^-5 instead of 2^-5), or write 1/x^5, float(x)^-5, or (x//1)^-5
Stacktrace:
 [1] throw_domerr_powbysq(::Int64, ::Int64) at ./intfuncs.jl:176
 [2] power_by_squaring(::Int64, ::Int64) at ./intfuncs.jl:196
 [3] top-level scope at none:0

julia> @edit 2 ^ -5
# ^(x::T, p::T) where {T<:Integer} = power_by_squaring(x,p)

julia> x = -2;

julia> x ^ x
ERROR: DomainError with -2:
Cannot raise an integer x to a negative power -2.
Make x a float by adding a zero decimal (e.g., 2.0^-2 instead of 2^-2), or write 1/x^2, float(x)^-2, or (x//1)^-2
Stacktrace:
 [1] throw_domerr_powbysq(::Int64, ::Int64) at ./intfuncs.jl:176
 [2] power_by_squaring(::Int64, ::Int64) at ./intfuncs.jl:196
 [3] ^(::Int64, ::Int64) at ./intfuncs.jl:220
 [4] top-level scope at none:0

julia> x ^ (-2)
0.25

julia> (-2) ^ x
ERROR: DomainError with -2:
Cannot raise an integer x to a negative power -2.
Make x a float by adding a zero decimal (e.g., 2.0^-2 instead of 2^-2), or write 1/x^2, float(x)^-2, or (x//1)^-2
Stacktrace:
 [1] throw_domerr_powbysq(::Int64, ::Int64) at ./intfuncs.jl:176
 [2] power_by_squaring(::Int64, ::Int64) at ./intfuncs.jl:196
 [3] ^(::Int64, ::Int64) at ./intfuncs.jl:220
 [4] top-level scope at none:0

julia> (-2) ^ (-2)
0.25

julia> @code_warntype x ^ x
Body::Int64
220 1 ─ %1 = invoke Base.power_by_squaring(_2::Int64, _3::Int64)::Int64                                          │
    └──      return %1                                                                                           │

[affans]

This is intentional. See associated issue and discourse forum post.

#28685

[fredrikekre]

Closing as duplicate of #28685