From: fateman@cs.berkeley.edu   
      
   On 12/28/2015 2:02 AM, Axel Vogt wrote:   
      
   >   
   > BTW it is markting's fairytale that a CAS guarantees arbitrary   
   > precision for expressions, already for x - y that may fail, the   
   > 'catastrophic cancellation'.   
   >   
   >   
      
   I recommend that you be clearer on the use of the terms precision   
   and accuracy.   
      
   You can always perform a subtraction to a specified precision.   
   Just follow the rules.  Even the hardware can do it.   
      
   The result might not be accurate. (= high relative error,   
   compared to the exact correct result)   
      
   It is possible to guarantee results of arithmetic   
   within computed bounds using (validated) interval   
   arithmetic. Usually these bounds are too pessimistic   
   and computed too slowly to be of great interest.   
      
   Mathematica's default significance arithmetic does something   
   similar in intent but not the same.   
      
   If you want a program to evaluate an expression numerically   
   with an absolute or relative error less than some given value you can   
   try various approaches like doing the same calculation in   
   higher and higher precision until the answer does not change.   
   There are well known examples where this does not work.   
      
   See   
   http://sourceforge.net/p/maxima/mailman/message/34666077/   
      
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