From: minforth@gmx.net   
      
   Am 10.07.2025 um 06:32 schrieb Paul Rubin:   
   > minforth  writes:   
   >> You don't need 64-bit doubles for signal or image processing.   
   >> Most vector/matrix operations on streaming data don't require   
   >> them either. Whether SSE2 is adequate or not to handle such data   
   >> depends on the application.   
   >   
   > Sure, and for that matter, AI inference uses 8 bit and even 4 bit   
   > floating point.   
      
   Or fuzzy control for instance.   
      
   > Kahan on the other hand was interested in engineering   
   > and scientific applications like PDE solvers (airfoils, fluid dynamics,   
   > FEM, etc.).  That's an area where roundoff error builds up after many   
   > iterations, thus extended precision.   
      
      
   That's why I use Kahan summation for dot products. It is slow but   
   rounding error accumulation remains small. A while ago I read an   
   article about this issue in which the author(s) performed extensive tests   
   of different dot product calculation algorithms on many serial   
   data sets from finance, geology, oil industry, meteorology etc.   
   Their target criterion was to find an acceptable balance between   
   computational speed and minimal error.   
      
   The 'winner' was a chained fused-multiply-add algorithm (many   
   CPUs/GPUs can perform FMA in hardware) which makes for shorter code   
   (good for caching). And it supports speed improvement by   
   parallelization (recursive halving of the sets until manageable   
   vector size followed by parallel computation).   
      
   I don't do parallelization, but I was still surprised by the good   
   results using FMA. In other words, increasing floating-point number   
   size is not always the way to go. Anyhow, first step is to select   
   the best fp rounding method ....   
      
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