------- 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 -------