A x = b
A => QR # QR decomposition
Q R x = b
R x = Q' b # Q is inverted by transposing
x = backsubstitute(R, Q' b)
Q was the transpose of it's inverse, and so you don't get any rounding error when you transpose it. R didn't need to be inverted because it was in a form where back substitution lets you apply the inverse of R to be without needing the inverse of R.
In 20 years or so, if it turns out that all of the dire predictions were significantly overblown, would you at least have the courtesy to blush in embarrassment for the current alarmism?