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Testing Floating-Point Applications Connie R. Masters & Alan A. Jorgensen PNSQC 2020 #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint Error in IEEE standard floating point Error detection Bounded

  1. Testing Floating-Point Applications Connie R. Masters & Alan A. Jorgensen PNSQC 2020 #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  2. • Error in IEEE standard floating point • Error detection • Bounded floating point #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  3. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  4. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  5. Representation of real numbers in a fixed space #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  6. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  7. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  8. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  9. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  10. • No indication of floating-point error • No efficient way to test for error • Calculation may be totally erroneous #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  11. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  12. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  13. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  14. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  15. Problems with Previous Testing Methods #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  16. Previous Testing Methods 1. Recomputation - With Redirected Rounding  Round to nearest (default)  Round to zero  Round to +infinity  Round to -infinity 2. Recomputation - With Higher Precision  32-bit  64- bit  128-bit 3. Temporary Exception Handling  Divide by zero  Not a Number  Overflow  Underflow #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  17. Kahan’s improved thin triangle equation for finding the area of the triangle, where A ≥ B ≥ C and difference in length of C and the length of B is δ. Area = SQRT(A+(B+C))(C-(A-B))(C+(A-B))(A+(B-C)))/4 Tested with 3 values of δ #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  18. “We can produce a good solution for the problem if we assume that A, B, and C are given exactly as numbers in [floating point].” Pat H. Sterbenz #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  19. Kahan’s improved thin triangle equation for finding the area of the triangle, where A ≥ B ≥ C and difference in length of C and the length of B is δ. Area = SQRT(A+(B+C))(C-(A-B))(C+(A-B))(A+(B-C)))/4 Tested with 3 values of δ #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  20. Kahan’s Thin Triangle Stress Test δ is embedded within the B value Required Area Using Bounded Area Using Double Precision Area Using Quad Precision Significant Floating Point Digits For Triangle One, A= 1.6, B=0.8001, and C=0.8 10 0.00715552931654077180000 0.00715552931654910837456 0.0071555293165 11 0.00715552931654077180000 0.00715552931654910837456 0.0071555293165 12 0.00715552931654077180000 0.00715552931654910837456 qNaN.sig For Triangle Two, A=1.6, B=0.80000001, and C=0.8 6 0.00007155417437995153200 0.00007155417539179666768 0.00007155417 7 0.00007155417437995153200 0.00007155417539179666768 0.00007155417 8 0.00007155417437995153200 0.00007155417539179666768 qNaN.sig For Triangle Three, A=1.6, B=0.800000000001, and C=0.8 2 0.00000071545439186697031 0.00000071554175280004451 0.000000715 3 0.00000071545439186697031 0.00000071554175280004451 0.000000715 4 0.00000071545439186697031 0.00000071554175280004451 qNaN.sig #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  21. • Muller’s recursion X N+1 := 111 – ( 1130 – 3000/X N– 1 )/X N for N = 1, 2, 3, ... Where X 0 := 2 and X 1 := –4 • Demonstration of rounding error • Converges to 100 • Correct answer is 6 • No numerical extremes • Test: Can BFP detect the precise failure point? #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  22. Bounded Floating Point 64- Bit Standard Floating Point X[0] = 2.000000000000000 2.00000000000000000 X[1] = - 4.000000000000000 - 4.00000000000000000 X[2] = 18.50000000000007 18.50000000000000000 X[3] = 9.37837837837878 9.37837837837837900 X[4] = 7.801152737756 7.80115273775216790 X[5] = 7.15441448103 7.15441448097533340 X[6] = 6.8067847377 6.80678473692481220 X[7] = 6.592632780 6.59263276872180270 X[8] = 6.44946611 6.44946593405408120 X[9] = 6.348454 6.34845206074892940 X[10] = 6.27448 6.27443866276447610 X[11] = 6.2193 6.21869676916201720 X[12] = 6.187 6.17585386514404090 X[13] = sNaN.sig 6.14262732158489030 X[14] = 6.12025116507937560 X[15] = 6.16612674271767690 X[16] = 7.23566541701194320 X[17] = 22.06955915453103100 X[18] = 78.58489258126825000 X[19] = 98.35041655134628500 X[20] = 99.89862634218410200 X[21] = 99.99387444125312600 X[22] = 99.99963059549460800 X[23] = 99.99997774322417900 X[24] = 99.99999865997196500 X[25] = 99.99999991936725500 #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  23. Applications of BFP • Weather Modeling • Missile Guidance Testing • Dynamic Internal Stress • Explosion Analysis • Mission-Critical Results #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  24. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  25. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  26. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  27. #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

  28. Testing Using Bounded Floating Point Thank you for coming! #PNSQC2020 #BoundedFloatingPoint #TrueNorthFloatingPoint

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