Consider adding "compensated" evaluation via error-free transformations

[dhermes]

There are three applications I have in mind:

• Evaluating B(s) for a Bézier curves and producing Bhat, err where Bhat is the value computed and err is a running error.
  • See Accurate evaluation of a polynomial and its derivative in Bernstein form (EFT for de Casteljau algorithm)
• Evaluating B1(s) - B2(t) for two Bézier curves and producing Dhat, err. It may be more beneficial to produce the tensor product surface implied by this difference, i.e. if B1(s) = SUM_i pi B{i,m}(s) and B2(t) = SUM_j qj B{j,n}(t) then the tensor product surface will have control net pi - qj.
  • See Accurate evaluation of Bézier curves and (tensor product) surfaces and the Bernstein-Fourier algorithm (Conclusion: use de Casteljau)
• Evaluating B(s, t) for a Bézier surface (or "triangle" or "bivariate") and producing Bhat, err.
  • See Accurate evaluation algorithm for bivariate polynomial in Bernstein-Bézier form (i.e. a "Bézier triangle")

I have started some investigation of this.

[dhermes]

This issue may also help find roots in the presence of poorly conditioned roots, as in #21. In that problem, the issue isn't that finding p(s) = 0 is challenging, but that finding p(s) = 2.125 is difficult because so many nearby values to the true one produce 2.125 due to rounding.