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Diagnosis resumes after milestone Number 0 Page: 1 Users are invited to help debug and augment this program so it will cope with unanticipated and newly uncovered arithmetic pathologies. Please send suggestions and interesting results to Richard Karpinski Computer Center U-76 University of California San Francisco, CA 94143-0704, USA In doing so, please include the following information: Precision: double; Version: 10 February 1989; Computer: Compiler: Optimization level: Other relevant compiler options: To continue, press RETURN Diagnosis resumes after milestone Number 1 Page: 2 Running this program should reveal these characteristics: Radix = 1, 2, 4, 8, 10, 16, 100, 256 ... Precision = number of significant digits carried. U2 = Radix/Radix^Precision = One Ulp (OneUlpnit in the Last Place) of 1.000xxx . U1 = 1/Radix^Precision = One Ulp of numbers a little less than 1.0 . Adequacy of guard digits for Mult., Div. and Subt. Whether arithmetic is chopped, correctly rounded, or something else for Mult., Div., Add/Subt. and Sqrt. Whether a Sticky Bit used correctly for rounding. UnderflowThreshold = an underflow threshold. E0 and PseudoZero tell whether underflow is abrupt, gradual, or fuzzy. V = an overflow threshold, roughly. V0 tells, roughly, whether Infinity is represented. Comparisions are checked for consistency with subtraction and for contamination with pseudo-zeros. Sqrt is tested. Y^X is not tested. Extra-precise subexpressions are revealed but NOT YET tested. Decimal-Binary conversion is NOT YET tested for accuracy. To continue, press RETURN Diagnosis resumes after milestone Number 2 Page: 3 The program attempts to discriminate among FLAWs, like lack of a sticky bit, Serious DEFECTs, like lack of a guard digit, and FAILUREs, like 2+2 == 5 . Failures may confound subsequent diagnoses. The diagnostic capabilities of this program go beyond an earlier program called `MACHAR', which can be found at the end of the book `Software Manual for the Elementary Functions' (1980) by W. J. Cody and W. Waite. Although both programs try to discover the Radix, Precision and range (over/underflow thresholds) of the arithmetic, this program tries to cope with a wider variety of pathologies, and to say how well the arithmetic is implemented. The program is based upon a conventional radix representation for floating-point numbers, but also allows logarithmic encoding as used by certain early WANG machines. BASIC version of this program (C) 1983 by Prof. W. M. Kahan; see source comments for more history. To continue, press RETURN Diagnosis resumes after milestone Number 3 Page: 4 Program is now RUNNING tests on small integers: -1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K. Searching for Radix and Precision. Radix = 2.000000 . Closest relative separation found is U1 = 5.9604645e-08 . Recalculating radix and precision confirms closest relative separation U1 . Radix confirmed. The number of significant digits of the Radix is 24.000000 . To continue, press RETURN Diagnosis resumes after milestone Number 30 Page: 5 Subtraction appears to be normalized, as it should be. Checking for guard digit in *, /, and -. *, /, and - appear to have guard digits, as they should. To continue, press RETURN Diagnosis resumes after milestone Number 40 Page: 6 Checking rounding on multiply, divide and add/subtract. Multiplication appears to round correctly. Division appears to round correctly. Addition/Subtraction appears to round correctly. Checking for sticky bit. Sticky bit used incorrectly or not at all. Does Multiplication commute? Testing on 20 random pairs. No failures found in 20 integer pairs. Running test of square root(x). Testing if sqrt(X * X) == X for 20 Integers X. Test for sqrt monotonicity. sqrt has passed a test for Monotonicity. Testing whether sqrt is rounded or chopped. Square root is neither chopped nor correctly rounded. Observed errors run from 0.0000000e+00 to 5.0000000e-01 ulps. To continue, press RETURN Diagnosis resumes after milestone Number 90 Page: 7 Testing powers Z^i for small Integers Z and i. DEFECT: computing (3.00000000000000000e+00) ^ (5.00000000000000000e+00) yielded 2.42998046875000000e+02; which compared unequal to correct 2.43000000000000000e+02 ; they differ by -1.95312500000000000e-03 . Errors like this may invalidate financial calculations involving interest rates. Similar discrepancies have occurred 49 times. To continue, press RETURN Diagnosis resumes after milestone Number 100 Page: 8 Seeking Underflow thresholds UfThold and E0. Smallest strictly positive number found is E0 = 1.4013e-45 . Since comparison denies Z = 0, evaluating (Z + Z) / Z should be safe. What the machine gets for (Z + Z) / Z is 2.00000000000000000e+00 . This is O.K., provided Over/Underflow has NOT just been signaled. Underflow is gradual; it incurs Absolute Error = (roundoff in UfThold) < E0. The Underflow threshold is 1.17549467086791992e-38, below which calculation may suffer larger Relative error than merely roundoff. Since underflow occurs below the threshold UfThold = (2.00000000000000000e+00) ^ (-1.26000000000000000e+02) only underflow should afflict the expression (2.00000000000000000e+00) ^ (-1.26000000000000000e+02); actually calculating yields: nan . SERIOUS DEFECT: this is not between 0 and underflow threshold = 1.17549467086791992e-38 . Testing X^((X + 1) / (X - 1)) vs. exp(2) = 7.38905572891235352e+00 as X -> 1. DEFECT: Calculated 4.00000000000000000e+00 for (1 + (-5.96046447753906250e-08) ^ (-3.35544320000000000e+07); differs from correct value by -3.38905572891235352e+00 . This much error may spoil financial calculations involving tiny interest rates. Testing powers Z^Q at four nearly extreme values. ... no discrepancies found. To continue, press RETURN Diagnosis resumes after milestone Number 160 Page: 9 Searching for Overflow threshold: This may generate an error. Can `Z = -Y' overflow? Trying it on Y = -inf . Seems O.K. SERIOUS DEFECT: overflow past -inf shrinks to nan . Overflow threshold is V = 3.40282350000000000e+38 . Overflow saturates at V0 = inf . No Overflow should be signaled for V * 1 = 3.40282350000000000e+38 nor for V / 1 = 3.40282350000000000e+38 . Any overflow signal separating this * from the one above is a DEFECT. DEFECT: Comparison alleges that what prints as Z = 1.40129899978637695e-45 is too far from sqrt(Z) ^ 2 = 0.00000000000000000e+00 . DEFECT: Comparison alleges that Z = inf is too far from sqrt(Z) ^ 2 (nan) . To continue, press RETURN Diagnosis resumes after milestone Number 190 Page: 10 What message and/or values does Division by Zero produce? This can interupt your program. You can skip this part if you wish. Do you wish to compute 1 / 0? Trying to compute 1 / 0 produces ... nan . Do you wish to compute 0 / 0? Trying to compute 0 / 0 produces ... nan . To continue, press RETURN Diagnosis resumes after milestone Number 220 Page: 11 The number of SERIOUS DEFECTs discovered = 2. The number of DEFECTs discovered = 4. The arithmetic diagnosed has unacceptable Serious Defects. END OF TEST.