avoid catastrophic cancellation in divided differences

[araujoms]

There's a catastrophic cancellation happening in the computation of the divided differences matrix, as @antoine-levitt told me about on Slack. The problem is when we don't approximate the divided difference by the derivative of the mean, and the values are still close to eachother. Currently the threshold is set to be abs(x-y) < sqrt(eps(T)). Then the error of (f(x)-f(y))/(x-y) will be roughly eps(T)/sqrt(eps(T)) = sqrt(eps(T)), which is pretty bad. Things become catastrophic when we look at the second-order divided difference, then the error is roughly eps(T)/(sqrt(eps(T)) * sqrt(eps(T))) = 1, and the third order that gives us error 1/sqrt(eps(T)).

Luckily there's a simple fix: the error incurred by using the derivative of the mean is roughly tol^2, so we can use it for x,y that are further apart instead of the naïve formula. Equating both errors gives us eps(T) / tol = tol^2 => tol = cbrt(eps(T)). Substituting that into the formula for the errors of the second-order divided difference gives us eps(T)/ (tol * tol_2) = tol_2^2 => tol_2 = eps(T)^(2/9), and doing the same for the third order gives us eps(T)^(4/27).

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 96.73%. Comparing base (e350ede) to head (e8f31ad).
⚠️ Report is 3 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #858      +/-   ##
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+ Coverage   96.68%   96.73%   +0.04%     
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  Files          56       56              
  Lines        9092     9103      +11     
==========================================
+ Hits         8791     8806      +15     
+ Misses        301      297       -4     

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[lkapelevich]

Can we please rename rteps, maybe thresh? Otherwise I trust you and good by me to merge.

Note to self, eps(T) / tol = tol^2 equates the cancellation error to the truncation error of the mean approximation.