A few notes to finish IEEE 754

• The gap is 2^e-23
• So the next number following 33.25

  33.25 = 100001.01 = 1.0000101x25
   so the gap is 25-23 = 2-18
   33.25+2-18
     

• Notice again that the gap grows as the number increases.
• Underflow is when a nonzero number is represented as 0
• Quad precision

Floating Point Co-Processors

• Provide floating point capacity to a processor without this capibility
• Exploit another dimension of parallelism.
• Generaly these have their own registers
• Some don't interact with memory, others do.
• None fetch their own instructions.
• Look at the pictures on page 300.
• There is a different set of instructions for floating point operations
• Calculate xⁿ x float, n int.

     pow = x;
     for(i=1;i<n;i++) {
      pow *= x;
     }
  
     ; assume x in in %fp0;
     ; assume n is in %l0
     ; pow is in %fp1
     ; i is in %l1
  
     set 0, %l1
     fset 1,%fp1
  
     subcc %l0,%l1,%r0
  
  1  top:   bne   out
  2         add  %l1,1,%l1
  
  3         fmpy %fp1, %fp0, %fp1
  4         b    top
  5         subcc %l0,%l1,%r0
  
     out:
     

• Notice here that we do 4 integer operations for the 1 floating point op
• This is after eliminating the nop in the bds.
• If the floating point multiply can be done in 4 integer operations or less, then this is wonderful.
• A simple view of execution:
• 
• A view where fp ops take longer:
• 
• Or if fp ops are in parallel:
• 
• Other advantages:
  • Reduce the cost of the basic chip
  • Save space on the cpu chip (Different chip for FPU)
• Another measurement of performance is megaflops (Mflops)
  • Or Teraflops
  • Or Petaflops
  • Good only for measuring programs with many floating point operations
  • Such as a scientific computation.
  • Not so good with business transactions.
  • Again, vendors can tune performance.