Accuracy and error analysis (was Re: [Maxima] primes)

ok, long explanation.
Consider the maxima definition

f[i,n]:=f[i-1,n]-  (f[i-1,n]^2-n)/(2*f[i-1,n]);

This is a newton iteration for square root of n
Compute the square root of 1+d this way..
f[0,3]:1;
f[9,3];  --> 9 iterations towards sqrt(3).

next.. sqrt(1+d)

f[0,1+d]:1   /* initial guess */

f[2,1+d];   /*after 2 iterations */

(d/2) - (( - d + ((d/2) + 1)^2 - 1)/(2 * ((d/2) + 1))) + 1

which you can understand by

taylor(%,d,0,3)  --> 1 +d/2 -d^2/8 + ....

so it is getting what I said it should get.

But look again at the formula. It takes advantage of
the fact that each occurrence of "d" is the same "d", and
thus  d-d/2 is  d/2.  and other formulas might
make use of simplifications like (1-d)*(1+d) = 1-d^2.

  In general, if maxima is doing
interval arithmetic with uncertainties, there is not just
one d.  Each d is different.  e.g. if you are talking about
one unit in the last place, then d might be a shorthand for something
like any number in [-1.0e-18, +1.0e-18]

In such a case  d[1] -d[2]/2  is  not d[3]/2  but a number in
an interval like  [d/2, 3d/2].

and (1-d)^2  is   (1-d[1])*(1-d[2]).  Instead of getting
1-2d+d^2,  you get something like +-(1+2|d| +d^2), which is
cruder than you'd like.

I hope I've conveyed a little of the problem. 
Oh, and solving the newton iteration problem
so that you just solve that
one example does not automate all of error analysis...