From: johnl@taugh.com   
      
   According to MitchAlsup  :   
   >   
   >John Levine  posted:   
   >   
   >> According to David Schultz  :   
   >> >This reminds me of when I took a numerical analysis course. (The many   
   >> >ways that computer calculations can go wrong and how to deal with it.)   
   >> >The professor said that the schools IBM (360 or 370, ca. 1980) was   
   >> >perfect for the course because of the defects in its floating point   
   >> >system. Guard digits and rounding sorts of things as near as I can recall.   
   >>   
   >> The 360's floating point is a famous and somewhat puzzling failure,   
   considering   
   >> how much else they got right.   
   >>   
   >> It does hex normalization rather than binary. They assumed that   
   >> leading digits are evenly distributed so there's be on average one   
   >> zero bit, but in fact they're geometrically distributed, so on average   
   >> there's two. They got one bit back by making the exponent units of 16   
   >> rather than 2, but that's still one bit gone. It truncated rather than   
   >> rounding, another bit gone.  They also truncated rather than rounding   
   >> results.   
      
   Oh I forgot that using hex exponents meant there was no hidden bit, so   
   in practice it lost three bits of precision on every operation.  There was   
   a great deal of grumbling that people with 709x Fortran codes had to   
   make everything double precision to keep getting reasonably good results.   
      
   >> Originally there wre no guard digits which made the results comically   
   >> bad but IBM retrofitted them at great cost to all the installed machines.   
   >>   
   >> IEEE floating point can be seen as a reaction to that, how do you use   
   >> the same number of bits but get good results.   
   >   
   >VAX got this correct too (the VAX format not the one inherited from   
   >PDP-11/45; PDP-11/40* FP was worse). ...   
      
   The VAX is the first machine I know that used the hidden bit trick to   
   get an extra bit of significance.  The PDP-6/10 was pretty close but   
   their format was two's complement which meant no hidden bit but they   
   could use integer comparisons on normalized floating point numbers.   
      
   --   
   Regards,   
   John Levine, johnl@taugh.com, Primary Perpetrator of "The Internet for   
   Dummies",   
   Please consider the environment before reading this e-mail. https://jl.ly   
      
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    * Origin: you cannot sedate... all the things you hate (1:229/2)