From: user5857@newsgrouper.org.invalid   
      
   Terje Mathisen  posted:   
      
   > 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.   
      
   https://guillaume.melquiond.fr/doc/05-imacs17_1-article.pdf   
      
   > 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   
   >   
      
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