From: already5chosen@yahoo.com   
      
   On Thu, 8 Jan 2026 21:35:16 +0100   
   Terje Mathisen  wrote:   
      
   > MitchAlsup wrote:   
   > >   
   > >   
   > > But a single incorrect rounding is 1 ULP all by itself.   
   >   
   > :-)   
   >   
   > Yeah, there are strong forces who want to have, at least as a   
   > suggested/recommended option, a set of transcendental functions which   
   > are exactly rounded.   
   >   
      
   I wonder who are those forces and what is the set they push for.   
      
   I would guess that they are mostly software and hardware verification   
   people rather than people that use transcendental functions in   
   engineering and physical calculations.   
      
   The majority of the latter crowd would likely have no objections   
   against 0.75 ULP RMS as long as implementation is both fast and   
   preserving few invariant, of which I can primarily think of two:   
      
   1. Evenness/odness   
   If precise function F(x) is even or odd then approximate functions f(x)   
   is also even or odd.   
      
   2. Weak preservation of sign of delta.   
   If F(x) < F(x+ULP) then f(x) <= f(x+ULP)   
   If F(x) > F(x+ULP) then f(x) >= f(x+ULP)   
      
      
   In practice, it's probably unlikely to have these invariant preserved   
   when RMS error is 0.75 ULP, but for RMS error = 0.60-0.65 ULP I don't   
   see difficulties.   
   For many transcendental functions there will be only few problematic   
   values of x near edges of implementation-specific ranges where one   
   has to be careful.   
      
   > It is provably doable for float, in very close to the same cycle   
   > count as the best libraries in current use, double is "somewhat"   
   > harder to totally verify/prove.   
   >   
   > Terje   
   >   
      
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