simplify("(x^2)^0.5") should be "abs(x)"

[paulftw]

While there are genuine use cases when returning "x" would be better (that's what most humans would expect), strictly speaking it is incorrect.

To reproduce in the mathnotepad:

derivative("(x^2)^0.5", "x").evaluate({x:-3})
    1

derivative("(x^2)^0.5", "x", {simplify:false}).evaluate({x:-3})
    -1

Here I evaluate the same derivative at the same point x=-3 but get two different results.
Same problem for x^4 and I assume all other even powers.

[josdejong]

Thanks that is a very good and valid point.

Anyone able to improve simplify? Help would be welcome.

[gwhitney]

Well, the relation (x^2)^0.5 = abs(x) is not an identity for complex numbers, e.g, (i^2)^0.5 = i but abs(i) = 1. But I completely agree that (x^2)^0.5 = x is not an identity in general either, for example with negative numbers as Paul points out. I think the only tenable approach is to further restrict use of such an identity, or eliminate it altogether from simplify, rather than replacing it with (x^2)^0.5 = abs(x); or possibly having both with different restrictions (e.g., if x is known to be a real number, then I guess (x^2)^0.5 = abs(x) is OK.

[josdejong]

Hm that makes sense Glen. I hadn't thought about complex numbers. It makes sense to restrict this simplification, and only simplify to abs(x) for example when the value is known to be a real number.

[gwhitney]

The other tricky bit here is that it is not clear which direction is "simpler," as is often the case with simplification. For example, for all real numbers ((x^2)^0.5)^2 = x^2, but if we were to simplify the inner part first, we would get abs(x)^2, and then we might need to add another simplification rule to know that abs(x)^2 = x^2 (again, only true for real numbers, not complex numbers, as abs(i)^2 = 1 but i^2 = -1).