From: terje.mathisen@tmsw.no   
      
   MitchAlsup wrote:   
   >   
   > Robert Finch  posted:   
   >> Improves the accuracy? of algorithms, but seems a bit specific to me.   
   >   
   > It is down in the 1% footprint area.   
   >   
   >> Are there other instruction sequence where double-rounding would be good   
   >> to avoid?   
   >   
   > Back when I joined Moto (1983) there was a lot of talk about double   
   > roundings and how it could screw up various algorithms but mainly in   
   > the 64-bit versus 80-bit stuff of 68881, where you got 11-more bits   
   > of precision and thus took a change of 2/2^10 of a double rounding.   
   > Today with 32-bit versus 64-bit you take a chance of 2/2^28 so the   
   > problem is greatly ameliorated although technically still present.   
      
   Actually, for the five required basic operations, you can always do the   
   op in the next higher precision, then round again down to the target,   
   and get exactly the same result.   
      
   This is because the mantissa lengths (including the hidden bit) increase   
   to at least 2n+2:   
      
   f16 1:5:10 (1+10=11, 11*2+2 = 22)   
   f32 1:8:23 (1+23=24, 24*2+2 = 50)   
   f64 1:11:52 (1+52=53, 53*2+2 = 108)   
   f128 1:15:112 (1+112=113)   
      
   You can however NOT use f128 FMUL + FADD to emulate f64 FMAC, since that   
   would require a triple sized mantissa.   
      
   The Intel+Motorola 80-bit format was a bastard that made it effectively   
   impossible to produce bit-for-bit identical results even when the FPU   
   was set to 64-bit precision.   
      
   Terje   
      
   --   
   -    
   "almost all programming can be viewed as an exercise in caching"   
      
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