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   I have finally revised the flops.c program to version 2.0 which
   addresses the concerns brought out over the last year or so (version
   1.2c and earliar versions). Below is a discussion of the new flops.c
   program (flops20.c) and some results for the HP 9000/730 and IBM
   RS/6000 Model 550 systems.

   Flops.c is a 'c' program which attempts to estimate your systems
   floating-point 'MFLOPS' rating for the FADD, FSUB, FMUL, and FDIV
   operations based on specific 'instruction mixes' (discussed below).
   The program provides an estimate of PEAK MFLOPS performance by making
   maximal use of register variables with minimal interaction with main
   memory. The execution loops are all small so that they will fit in
   any cache. Flops.c can be used along with Linpack and the Livermore
   kernels (which exercise memory much more extensively) to gain further
   insight into the limits of system performance. The flops.c execution
   modules include various percent weightings of FDIV's (from 0% to 25%
   FDIV's) so that the range of performance can be obtained when using
   FDIV's. FDIV's, being computationally more intensive than FADD's or
   FMUL's, can impact performance considerably on some systems.
   
   Flops.c consists of 8 independent 'modules' which, except for module
   2, conduct numerical integration of various functions. Some of the
   functions (sin(x) and cos(x)) are approximated using a power series
   expansion accurate to 1.0e-14 over the integration interval. Module 2,
   estimates the value of pi based upon the Maclaurin series expansion of
   atan(1). MFLOPS ratings are provided for each module, but the programs
   overall results are summerized by the MFLOPS(1), MFLOPS(2), MFLOPS(3),
   and MFLOPS(4) outputs.

   The MFLOPS(1) result is identical to the result provided by all
   previous versions of flops.c (flops12c.c and earliar versions). It is
   based only upon the results from modules 2 and 3. Actually, on faster
   machines, MFLOPS(1) from flops.c V2.0 is expected to provide more
   accurate results since the number of iterations conducted (which is
   reflected in the timing accuracy) is more tightly controlled than in
   previous versions of flops.c.
   
   Two problems surfaced in using MFLOPS(1). First, it was difficult to
   completely 'vectorize' the result due to the recurrence of the 's'
   variable in module 2. This problem is addressed in the MFLOPS(2) result
   which does not use module 2, but maintains nearly the same weighting of
   FDIV's (9.2%) as in MFLOPS(1) (9.6%). For scalar machines the MFLOPS(2)
   results 'should' be similar to the MFLOPS(1) results. However, for
   vector machines the MFLOPS(1) and MFLOPS(2) results may differ
   considerably since the MFLOPS(2) result is expected to be more
   completely vectorizable. The second problem with MFLOPS(1) centers
   around the percentage of FDIV's (9.6%) which was viewed as too high for
   an important class of problems. This concern is addressed in the
   MFLOPS(3) result which does only 3.4% FDIV's, and the MFLOPS(4) result
   where NO FDIV's are conducted at all.
   
   The number of floating-point instructions per iteration (loop) is
   given below for each module executed.

   MODULE   FADD   FSUB   FMUL   FDIV   TOTAL  Comment
     1        7      0      6      1      14   7.1%  FDIV's
     2        3      2      1      1       7   difficult to vectorize.
     3        6      2      9      0      17   0.0%  FDIV's
     4        7      0      8      0      15   0.0%  FDIV's
     5       13      0     15      1      29   3.4%  FDIV's
     6       13      0     16      0      29   0.0%  FDIV's
     7        3      3      3      3      12   25.0% FDIV's
     8       13      0     17      0      30   0.0%  FDIV's
   
   A*2+3     21     12     14      5      52   A=5, MFLOPS(1), Same as
	    40.4%  23.1%  26.9%  9.6%          previous versions of the
					       flops.c program. Includes
					       only Modules 2 and 3.
   
   1+3+4     58     14     66     14     152   A=4, MFLOPS(2), New output
   +5+6+    38.2%  9.2%   43.4%  9.2%          does not include Module 2,
   A*7                                         but does 9.2% FDIV's.
   
   1+3+4     62      5     74      5     146   A=0, MFLOPS(3), New output
   +5+6+    42.5%  3.4%   50.7%  3.4%          does not include Module 2,
   7+8                                         but does 3.4% FDIV's.

   3+4+6     39      2     50      0      91   A=0, MFLOPS(4), New output
   +8       42.9%  2.2%   54.9%  0.0%          does not include Module 2,
					       and does NO FDIV's.

   I hope that flops.c V2.0 (flops20.c) proves more useful than earliar
   versions.


(1) HP 9000/730 flops.c V2.0 Results, cc +OS +O3 -W1-a,archive   

   Below are the HP 9000/730 results (provided by Bo Thide'). The minimum
   MFLOPS rating is 15.1 MFLOPS for module 7, which does 25% FDIV's. The
   maximum MFLOPS rating is 37.1 MFLOPS for module 6, which does 0.0%
   FDIV's. FDIV appears to be reasonably efficient on the HP 9000/730,
   as indicated by the overall MFLOPS(n) outputs. 

   The 'Runtime' output is the time in microseconds (usec) for one
   iteration (loop) through the module. The MFLOPS rating is obtained by
   dividing the number of floating-point instructions in the loop by the
   Runtime (in microseconds). For example for module 1 below:
   MFLOPS = 14.0 / 0.5978 = 23.42.

   The Runtime output has already been adjusted for an estimate of the
   time in microseconds to conduct one empty 'for' loop (NullTime). If
   NullTime is not calculated (that is, NullTime = 0.0), due to compiler
   optimization, it can produce a 3% to 5% lower MFLOPS rating than would
   otherwise be obtained.


   FLOPS C Program (Double Precision), V2.0 18 Dec 1992

   Module     Error        RunTime      MFLOPS
			    (usec)
     1     -4.6896e-13      0.5978     23.4187
     2      2.2160e-13      0.2447     28.6079
     3     -6.9944e-15      0.7412     22.9342
     4     -9.7256e-14      0.6906     21.7195
     5     -1.6542e-14      0.9200     31.5217
     6      4.3632e-14      0.7822     37.0755
     7     -4.9454e-11      0.7972     15.0529
     8      7.2164e-14      0.8275     36.2538

   Iterations      =   32000000
   NullTime (usec) =     0.0306
   MFLOPS(1)       =    26.4673  [same as flops12c.c, 9.6% FDIV's]
   MFLOPS(2)       =    21.9633  [9.2% FDIV's]
   MFLOPS(3)       =    27.2566  [3.4% FDIV's]
   MFLOPS(4)       =    29.9188  [0.0% FDIV's]


(2) IBM RS/6000 Model 550 flops.c V2.0 results, cc -DUNIX -O -Q

   The IBM RS/6000 Model 550 flops20.c results are shown below. Here,
   the minimum MFLOPS rating is 7.3 MFLOPS also for module 7 which does
   25.0% FDIV's. The maximum MFLOPS rating is 56.9 MFLOPS (!) also for
   module 6 which does 0.0% FDIV's. While the Model 550 works wonders
   with FADD's and FMULS's its performance falls off rapidly with FDIV's.


   FLOPS C Program (Double Precision), V2.0 18 Dec 1992

   Module     Error        RunTime      MFLOPS
			    (usec)
     1     -4.6896e-13      0.7028     19.9200
     2      2.2160e-13      0.5806     12.0560
     3     -7.0499e-15      0.4372     38.8849
     4     -9.7145e-14      0.4359     34.4086
     5     -1.6542e-14      0.9903     29.2837
     6      4.3632e-14      0.5100     56.8627
     7     -4.9454e-11      1.6456      7.2921
     8      7.2164e-14      0.5572     53.8418

   Iterations      =   32000000
   NullTime (usec) =     0.0484
   MFLOPS(1)       =    15.5674  [same as flops12c.c, 9.6% FDIV's]
   MFLOPS(2)       =    15.7370  [9.2% FDIV's]
   MFLOPS(3)       =    27.6568  [3.4% FDIV's]
   MFLOPS(4)       =    46.8997  [0.0% FDIV's]

Al Aburto
aburto@nosc.mil

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