RelativeResidualTolerance

A residual that is merely non-zero is not evidence. A root found numerically is
only as accurate as the search that found it, and the terms of the equation at
such a root cancel to within that accuracy rather than exactly. Judging the
residual on its own size threw away every root of
1/210 - 17x/210 + 101x^2/210 - 247x^3/210 + x^4, whose four roots come
back as decimals and leave 5.2e-6 against terms of about 1.2e-2, and answered
that quartic with the empty set.

The two cases are far apart, which is what makes a threshold possible at all.
A spurious root is not a near miss: it comes from a rewrite that widened the
domain, so the residual is a whole quantity in its own right -- 2*pi*i for the
logarithm sum, or the difference between -3 and 9 for an exponential. Over the
equations measured, every genuine root sits at 4.5e-4 or below and most at
1e-15, while every spurious one sits between 1.5 and 2. This is a hundredth:
twenty times above the worst genuine root and a hundred and fifty times below
the mildest spurious one.