`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, reached from an expression anyone might write. `{1}^n * {2}^n = ({1} * {2})^n` was applied unconditionally in two places -- once in PowerRules and once, separately, in FactorizeRules. Both now carry the same 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. Outside those two the sides can differ by a full turn of the argument. sqrt(x) * sqrt(y) sqrt(x * y) -> unchanged x ^ (3/2) * y ^ (3/2) gathered -> unchanged sqrt(2) * sqrt(3) sqrt(6) unchanged -- both bases positive x ^ 2 * y ^ 2 (x * y) ^ 2 unchanged -- 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 back an answer that leaned 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 it, because it generates single-variable expressions and this needs two, both negative. **The quotient form is deliberately left, and is #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` has no limit at all, which is the whole of #739 and #740. Measured: twelve tests fail, two of them limits. Which to prefer is a maintainer's call, so it is filed with the options rather than taken here, and pinned by a test that asserts the disagreement so the day it is fixed that test fails and says where to look. Unit 5544 pass 0 fail; F# 130/130; casbench 113/117 0 wrong; rootcheck 596/596; simpsweep 10463/10463; propcheck 0 failures. Co-authored-by: Claude Opus 5 (1M context) <noreply@anthropic.com>

Cannot collapse 1+sqrt(2x)+x/2 into (1+sqrt(2x)/2)^2 #176

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WhiteBlackGoose commented on Jul 31, 2020

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