Guard the product of two powers against the branch cut (#801)

[Rafael-SOWNet]

Fixes #801. Splits out #802.

sqrt(x) * sqrt(y) was simplified to sqrt(x * y) — and at x = y = -1 the first is i * i = -1 while the second is sqrt(1) = 1. A wrong answer, from an expression anyone might write.

Two copies of the same unguarded rule

{1}^n * {2}^n = ({1} * {2})^n was applied unconditionally in two places — once in PowerRules, and separately in FactorizeRules. Guarding only the first changed nothing, because Simplify runs both.

Both now carry the condition the ({}^{})^{} rule directly above the first already had, and were missing for the same reason: #752 fixed that rule and did not look at its neighbours. True for a whole n whatever the signs, since a^3 b^3 is (ab)(ab)(ab), and for positive real bases whatever the exponent.

	before	after
sqrt(x) * sqrt(y)	sqrt(x * y)	unchanged
x ^ (3/2) * y ^ (3/2)	gathered	unchanged
sqrt(2) * sqrt(3)	sqrt(6)	sqrt(6) — both bases positive
x ^ 2 * y ^ 2	(x * y) ^ 2	(x * y) ^ 2 — whole exponent

How it was found

Not by a harness. The perfect-square collapse for #176 asked Simplify whether a cross term was twice the product of two roots, and got an answer leaning on this rewrite — so the collapse fired on x + 2*sqrt(x*y) + y, which is 0 at x = y = -1 where its "square" is -4. That fix gates its symbolic match on a numeric check, so the wrong collapse never shipped; this is the cause underneath it.

simpsweep cannot reach this: it generates single-variable expressions, and this needs two variables taken negative at once.

The quotient form is deliberately left — #802

It has the same hole (sqrt(2)/sqrt(-3) is -0.8165i where (2/-3)^(1/2) is +0.8165i), but guarding it costs answers rather than shapes. It is what lets the limit machinery read a 1^oo out of a quotient, and with it guarded:

(x ^ 2 + 1) ^ x / (x ^ 2) ^ x    limit x -> +oo    1  ->  no longer answered
(x ^ 3 + 1) ^ x / (x ^ 3) ^ x    limit x -> +oo    1  ->  no longer answered

That is the whole of #739 and #740. I measured it — twelve tests fail, two of them limits — and it is a maintainer's call rather than mine, so #802 carries the counterexample, the measured cost and three options (leave it; add sign analysis so x^2 is known non-negative; or condition it on MathS.Settings.Codomain).

It is pinned by TheQuotientFormIsStillWrongAndThatIsRecorded, which asserts the disagreement — so on the day #802 is fixed, that test fails and points at it.

Harnesses

• unit 5544 pass, 0 fail; F# 130/130
• casbench 113/117, 0 wrong; rootcheck 596/596; simpsweep 10463/10463; propcheck 0 failures

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