From: fateman@gmail.com   
      
   On Wednesday, January 20, 2021 at 9:31:47 AM UTC-8, Nasser M. Abbasi wrote:   
      
   >   
   > The difference is the use of the backtick ` at end   
   > of the real number.   
   >   
   > This tells mathematica to use machine precision avoiding the issue.   
      
   Using machine precision has other issues of course.  The computation without   
   the backtick is   
   done in HIGHER (software implemented) precision by Mathematica.  What you are   
   pointing   
   out is that in this example, lower precision implemented better by the   
   hardware IEEE floating-point   
   arithmetic is apparently better than the higher-precision Mathematica   
   software.  Also   
   note the very terrible result, 0.*10^37,  which compares EQUAL to zero.    
   Imagine having   
   written an iterative algorithm that checks for convergence, and "succeeds" not   
   when   
   a result is zero, but when a result has no significance.   it is possible to   
   program around   
   such issues, if you are aware that this is an issue at all, and are   
   sufficiently clever.  But   
   if you write a program f[x_]:= Do[x=2*x-x,120],  you might not know what kind   
   of number you   
   are given.  I suppose you could convert every input to machine float, but is   
   that what you want?   
   RJF   
      
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