(±1)^(±Inf)

[gusl]

julia> 1^Inf
1.0
julia> 1^(-Inf)
1.0
julia> (-1)^Inf
1.0
julia> (-1)^(-Inf)
1.0

Here's what R does:

> 1^Inf
> [1] 1
> 1^(-Inf)
> [1] 1
> (-1)^Inf
> [1] NaN
> (-1)^(-Inf)
> [1] NaN

Matlab returns NaN for all 4 possibilities.

Is there a standard that we're trying to follow?

[StefanKarpinski]

R's behavior seems the most sane. We're just returning what openlibm's exp function returns.

[johnmyleswhite]

This is a little weird. Here's the more direct analogue to R:

julia> 1.0^Inf
1.0

julia> 1.0^-Inf
1.0

julia> (-1.0)^(Inf)
1.0

julia> (-1.0)^(-Inf)
1.0

R's 1 is Julia's 1.0.

[jiahao]

I agree that R has saner behaviour. There is a well defined limit for the first two cases but not the last two.

[JeffBezanson]

The C libm pow does in fact specify (-1.0)^Inf == 1.0, but this seems crazy to me. NaN is better.

[StefanKarpinski]

I agree that this is clearly not right behavior. NaN is the right thing to return.

[jiahao]

This comment in openlibm/src/e_pow.c suggests that the intention is to return NaN in all four cases, which is still wrong.

[Keno]

Should this be fixed in openlibm then?

[JeffBezanson]

Tough call. I don't know if we want to break from standard behavior in openlibm. OTOH, the standard is flagrantly wrong here :)

[StefanKarpinski]

When a standard is this wrong, I'm ok with violating it by default. Maybe we can have a compile-time flag called STUPID_AND_STRICT.

[lindahua]

From math standpoint, even 1^Inf should be NaN. If you treat this as a binary function, the limit is not well defined.

We all seem to think about it as 1^b as b approaches Inf. What about we think about it as a^Inf as a approaches 1?

Obviously, approaching that point from different paths yields different results indicates that it is not well defined.

[StefanKarpinski]

Ok, well, I'm also fine with having these all be NaN, but the difference is that you don't need a limit for the value 1 while for functions that are only defined on finite values you need a limit to consider Inf.

[jiahao]

What about we think about it as a^Inf as a approaches 1?

That's a good point; the double limit lim_{a->1} \lim{b->∞} a^b doesn't exist. I guess all of these should be NaN.

IEEE 754-2008, p.44 (Section 9.2.1. Special values) says:

For the pow function (integral exponents get special treatment):
...
pow (−1, ±∞) is 1 with no exception
pow (+1, y) is 1 for any y (even a quiet NaN)

I'm not sure why this is mathematically justifiable, but it looks like we have a standards-compliant implementation.

[JeffBezanson]

That's really odd. Alan will have to call up Prof Kahan and get the story on that :)