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I have realized that my problem here was of mathematical nature rather than computational. I was assuming that

$-\frac{a}{b} =\frac{a}{-b}$ $\implies$ $\sqrt{-\frac{a}{b}}=\sqrt{\frac{a}{-b}}$ $\implies$ $\frac{\sqrt{-a}}{\sqrt{b}}=\frac{\sqrt{a}}{\sqrt{-b}}$

However, if we consider only the positive branches of the square roots, the last equality does not hold.