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Does the 12th digit really matter or is it more important to know the capability and limitations of the tool you are using to solve the problem at hand?

This is where I was heading

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11's sqrt does not have a an simple exact floating point representation.

Neither does 99's sqrt. There is a natural loss of information in rounding. The difference here is that when converted to a numerical answer immediately, there is one rounding. However when simplified symbolically first, the loss of information from the rounding gets multiplied by 3.

Looking at another example (very simplistic, but it shows the point):

RPN: 1 ENTER pi SQUARED / SQRT (& -> NUM as required).

Hugh's first setting (symbolic approx.): 0.318309886183

Hugh's second setting (symbolic exact): 0.318309886184

Your setting (numeric exact): 0.318309886183

Something with more significant digits (Wolfram Alpha, windows calculator, Hugh's ExactCalc) etc.: 0.318309886183790671...

_{.}So in this case the symbolic simplification gives a better answer.

^{}The example of the original post is very good in that it shows how the loss of accuracy accumulates. Anyone doing a series of calculations (e.g. a formula or program) and thinks they are getting an answer "correct to the last digit of the calculator's display capability" is fooling themselves. It is therefore always important to assess the influence of the elements of a formula on rounding & accuracy. Computers have made too easy to just put in numbers and take a result. Too often I see people using a result to many digits when they have not even considered the order of magnitude (and I can see the old slide-rulists jumping in here "we had to work that out, so you always had a sense of what order of magnitude to expect" :-) ) .Another interesting read is Prof. Kahan's

A Logarithm Too Clever by Half, and many other articles he wrote (available on his webpage http://www.cs.berkeley.edu/~wkahan/).

So, my point in this is: Hugh's implication that the 50g is "not so accurate in accurate mode" is

**A BIG RED HERRING**, which I am sure he knows as he is well versed in computer mathematics (Hugh, did you post this just to see where it would go? :-) ).