paranoia.1bk (6875B)
1 Lest this program stop prematurely, i.e. before displaying 2 3 `END OF TEST', 4 5 try to persuade the computer NOT to terminate execution when an 6 error like Over/Underflow or Division by Zero occurs, but rather 7 to persevere with a surrogate value after, perhaps, displaying some 8 warning. If persuasion avails naught, don't despair but run this 9 program anyway to see how many milestones it passes, and then 10 amend it to make further progress. 11 12 Answer questions with Y, y, N or n (unless otherwise indicated). 13 14 15 Diagnosis resumes after milestone Number 0 Page: 1 16 17 Users are invited to help debug and augment this program so it will 18 cope with unanticipated and newly uncovered arithmetic pathologies. 19 20 Please send suggestions and interesting results to 21 Richard Karpinski 22 Computer Center U-76 23 University of California 24 San Francisco, CA 94143-0704, USA 25 26 In doing so, please include the following information: 27 Precision: double; 28 Version: 10 February 1989; 29 Computer: 30 31 Compiler: 32 33 Optimization level: 34 35 Other relevant compiler options: 36 37 Diagnosis resumes after milestone Number 1 Page: 2 38 39 Running this program should reveal these characteristics: 40 Radix = 1, 2, 4, 8, 10, 16, 100, 256 ... 41 Precision = number of significant digits carried. 42 U2 = Radix/Radix^Precision = One Ulp 43 (OneUlpnit in the Last Place) of 1.000xxx . 44 U1 = 1/Radix^Precision = One Ulp of numbers a little less than 1.0 . 45 Adequacy of guard digits for Mult., Div. and Subt. 46 Whether arithmetic is chopped, correctly rounded, or something else 47 for Mult., Div., Add/Subt. and Sqrt. 48 Whether a Sticky Bit used correctly for rounding. 49 UnderflowThreshold = an underflow threshold. 50 E0 and PseudoZero tell whether underflow is abrupt, gradual, or fuzzy. 51 V = an overflow threshold, roughly. 52 V0 tells, roughly, whether Infinity is represented. 53 Comparisions are checked for consistency with subtraction 54 and for contamination with pseudo-zeros. 55 Sqrt is tested. Y^X is not tested. 56 Extra-precise subexpressions are revealed but NOT YET tested. 57 Decimal-Binary conversion is NOT YET tested for accuracy. 58 59 Diagnosis resumes after milestone Number 2 Page: 3 60 61 The program attempts to discriminate among 62 FLAWs, like lack of a sticky bit, 63 Serious DEFECTs, like lack of a guard digit, and 64 FAILUREs, like 2+2 == 5 . 65 Failures may confound subsequent diagnoses. 66 67 The diagnostic capabilities of this program go beyond an earlier 68 program called `MACHAR', which can be found at the end of the 69 book `Software Manual for the Elementary Functions' (1980) by 70 W. J. Cody and W. Waite. Although both programs try to discover 71 the Radix, Precision and range (over/underflow thresholds) 72 of the arithmetic, this program tries to cope with a wider variety 73 of pathologies, and to say how well the arithmetic is implemented. 74 75 The program is based upon a conventional radix representation for 76 floating-point numbers, but also allows logarithmic encoding 77 as used by certain early WANG machines. 78 79 BASIC version of this program (C) 1983 by Prof. W. M. Kahan; 80 see source comments for more history. 81 82 Diagnosis resumes after milestone Number 3 Page: 4 83 84 Program is now RUNNING tests on small integers: 85 -1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K. 86 87 Searching for Radix and Precision. 88 Radix = 2.000000 . 89 Closest relative separation found is U1 = 1.1102230e-16 . 90 91 Recalculating radix and precision 92 confirms closest relative separation U1 . 93 Radix confirmed. 94 The number of significant digits of the Radix is 53.000000 . 95 Some subexpressions appear to be calculated extra 96 precisely with about 11 extra B-digits, i.e. 97 roughly 3.31133 extra significant decimals. 98 That feature is not tested further by this program. 99 100 Diagnosis resumes after milestone Number 30 Page: 5 101 102 Subtraction appears to be normalized, as it should be. 103 Checking for guard digit in *, /, and -. 104 *, /, and - appear to have guard digits, as they should. 105 106 Diagnosis resumes after milestone Number 40 Page: 6 107 108 Checking rounding on multiply, divide and add/subtract. 109 * is neither chopped nor correctly rounded. 110 / is neither chopped nor correctly rounded. 111 Addition/Subtraction neither rounds nor chops. 112 Sticky bit used incorrectly or not at all. 113 FLAW: lack(s) of guard digits or failure(s) to correctly round or chop 114 (noted above) count as one flaw in the final tally below. 115 116 Does Multiplication commute? Testing on 20 random pairs. 117 No failures found in 20 integer pairs. 118 119 Running test of square root(x). 120 Testing if sqrt(X * X) == X for 20 Integers X. 121 Test for sqrt monotonicity. 122 sqrt has passed a test for Monotonicity. 123 Testing whether sqrt is rounded or chopped. 124 Square root is neither chopped nor correctly rounded. 125 Observed errors run from -5.0000000e-01 to 5.0000000e-01 ulps. 126 127 Diagnosis resumes after milestone Number 90 Page: 7 128 129 Testing powers Z^i for small Integers Z and i. 130 ... no discrepancis found. 131 132 Seeking Underflow thresholds UfThold and E0. 133 Smallest strictly positive number found is E0 = 4.94066e-324 . 134 Since comparison denies Z = 0, evaluating (Z + Z) / Z should be safe. 135 What the machine gets for (Z + Z) / Z is 2.00000000000000000e+00 . 136 This is O.K., provided Over/Underflow has NOT just been signaled. 137 Underflow is gradual; it incurs Absolute Error = 138 (roundoff in UfThold) < E0. 139 The Underflow threshold is 2.22507385850720188e-308, below which 140 calculation may suffer larger Relative error than merely roundoff. 141 Since underflow occurs below the threshold 142 UfThold = (2.00000000000000000e+00) ^ (-1.02200000000000000e+03) 143 only underflow should afflict the expression 144 (2.00000000000000000e+00) ^ (-1.02200000000000000e+03); 145 actually calculating yields: 0.00000000000000000e+00 . 146 This computed value is O.K. 147 148 Testing X^((X + 1) / (X - 1)) vs. exp(2) = 7.38905609893065218e+00 as X -> 1. 149 Accuracy seems adequate. 150 Testing powers Z^Q at four nearly extreme values. 151 ... no discrepancies found. 152 153 154 Diagnosis resumes after milestone Number 160 Page: 8 155 156 Searching for Overflow threshold: 157 This may generate an error. 158 Can `Z = -Y' overflow? 159 Trying it on Y = -Inf . 160 Seems O.K. 161 Overflow threshold is V = 1.79769313486231571e+308 . 162 Overflow saturates at V0 = Inf . 163 No Overflow should be signaled for V * 1 = 1.79769313486231571e+308 164 nor for V / 1 = 1.79769313486231571e+308 . 165 Any overflow signal separating this * from the one 166 above is a DEFECT. 167 168 169 Diagnosis resumes after milestone Number 190 Page: 9 170 171 172 What message and/or values does Division by Zero produce? 173 Trying to compute 1 / 0 produces ... Inf . 174 175 Trying to compute 0 / 0 produces ... NaN . 176 177 Diagnosis resumes after milestone Number 220 Page: 10 178 179 180 The number of FLAWs discovered = 1. 181 182 The arithmetic diagnosed seems Satisfactory though flawed. 183 END OF TEST.