The principal root of a negative real carries a 1e-102 real part, and its square then lies a hair below the negative axis, where ln takes the wrong branch

[Rafael-SOWNet]

Measured on master (2b4a5256), with DowncastingEnabled off, as a numerical verification evaluates:

(-0.473)^(1/2)          -> -6.1e-102 + 0.68775i        (the real part should be 0)
(-4)^(1/2)              -> -1.8e-101 + 2i
((-0.473)^(1/2))^2      -> -0.473 - 8.4e-102 i         (should be -0.473 exactly)
ln(((-0.473)^(1/2))^2)  -> -0.7487 - 3.14159 i         (ln(-0.473) is -0.7487 + 3.14159 i)

Two causes, both in Number.Pow:

1. A negative real to a half power goes through the polar form, rho^p (cos(p theta) + i sin(p theta)) with theta = pi from Arctan2, and cos(pi/2) at a hundred digits is 6e-102, not zero. The principal root of a negative real is the root of the modulus times i, exactly, and the same for every power that is a whole number of halves ((-4)^(3/2) is -8i; (-8)^(1/3) is not affected — a third is not a multiple of a right angle, and this library's real-root convention for it stays).
2. With the downcasting off, the literal 2 arrives as the decimal 2, not the Integer, so z^2 takes the polar form too instead of the exact multiplication z * z — which is what turns (0 + 0.68775i)^2 into -0.473 - 8.4e-102 i: a hair below the negative axis, where ln reads -i pi for the +i pi the number has.

Together they made a right antiderivative read as wrong: u ln(u^2) for u = sqrt(-1 + a x) is sqrt(-1 + a x) ln(((-1 + a x)^(1/2))^2) written back, whose numerical derivative at -1 + a x < 0 disagreed with the integrand by exactly the 2 pi i of the branch — the corpus harness reported the answer wrong at 3 of 6 points while its symbolic derivative is the integrand.

Fix in the PR that found it: a negative real base to a real power that is a whole number of halves is the modulus to that power times i^k, exactly; and a real power that is a whole number is read as the integer it is, so the exact multiplication is used.

[Rafael-SOWNet]

On master (40b7e587) both causes are gone, with the downcasting off:

(-0.473)^(1/2)           0 + 0.68774995456…i     (was -6.1e-102 + 0.68775i)
(-4)^(3/2)               -8i exactly
((-0.473)^(1/2))^2       -0.473, imaginary part zero
ln(((-0.473)^(1/2))^2)   -0.74866 + 3.14159i     (+i pi, the principal branch)

and with it on, the root and the logarithm come out the same. A half power of a negative real is i times the root of the modulus now, a decimal power with nothing after the point is taken as the whole power it is (#1379), and a zero imaginary part is on the real axis whatever its sign (#1433). Closing.