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1.1 root 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 = 1.3877788e-17 .
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 56.000000 .
95:
96: Diagnosis resumes after milestone Number 30 Page: 5
97:
98: Subtraction appears to be normalized, as it should be.
99: Checking for guard digit in *, /, and -.
100: *, /, and - appear to have guard digits, as they should.
101:
102: Diagnosis resumes after milestone Number 40 Page: 6
103:
104: Checking rounding on multiply, divide and add/subtract.
105: Multiplication appears to round correctly.
106: Division appears to round correctly.
107: Addition/Subtraction appears to round correctly.
108: Checking for sticky bit.
109: Sticky bit used incorrectly or not at all.
110:
111: Does Multiplication commute? Testing on 20 random pairs.
112: No failures found in 20 integer pairs.
113:
114: Running test of square root(x).
115: Testing if sqrt(X * X) == X for 20 Integers X.
116: Test for sqrt monotonicity.
117: sqrt has passed a test for Monotonicity.
118: Testing whether sqrt is rounded or chopped.
119: Square root is neither chopped nor correctly rounded.
120: Observed errors run from 0.0000000e+00 to 5.0000000e-01 ulps.
121:
122: Diagnosis resumes after milestone Number 90 Page: 7
123:
124: Testing powers Z^i for small Integers Z and i.
125: ... no discrepancis found.
126:
127: Seeking Underflow thresholds UfThold and E0.
128: Smallest strictly positive number found is E0 = 2.93874e-39 .
129: Since comparison denies Z = 0, evaluating (Z + Z) / Z should be safe.
130: What the machine gets for (Z + Z) / Z is 2.00000000000000000e+00 .
131: This is O.K., provided Over/Underflow has NOT just been signaled.
132:
133: Diagnosis resumes after milestone Number 120 Page: 8
134:
135:
136: FLAW: X = 4.04076183095161331e-39
137: is not equal to Z = 2.93873587705571877e-39 .
138: yet X - Z yields 0.00000000000000000e+00 .
139: Should this NOT signal Underflow, this is a SERIOUS DEFECT
140: that causes confusion when innocent statements like
141: if (X == Z) ... else ... (f(X) - f(Z)) / (X - Z) ...
142: encounter Division by Zero although actually
143: X / Z = 1 + 0.375 .
144: The Underflow threshold is 2.93873587705571877e-39, below which
145: calculation may suffer larger Relative error than merely roundoff.
146: SERIOUS DEFECT: Range is too narrow; U1^4 Underflows.
147: Since underflow occurs below the threshold
148: UfThold = (2.00000000000000000e+00) ^ (-1.28000000000000000e+02)
149: only underflow should afflict the expression
150: (2.00000000000000000e+00) ^ (-1.28000000000000000e+02);
151: actually calculating yields: 0.00000000000000000e+00 .
152: This computed value is O.K.
153:
154: Testing X^((X + 1) / (X - 1)) vs. exp(2) = 7.38905609893065007e+00 as X -> 1.
155: Accuracy seems adequate.
156: Testing powers Z^Q at four nearly extreme values.
157: ... no discrepancies found.
158:
159:
160: Diagnosis resumes after milestone Number 160 Page: 9
161:
162: Searching for Overflow threshold:
163: This may generate an error.
164:
165: * * * FLOATING-POINT ERROR * * *
166: Can `Z = -Y' overflow?
167: Trying it on Y = -8.50705917302346159e+37 .
168: Seems O.K.
169: Overflow threshold is V = 1.70141183460469229e+38 .
170: There is no saturation value because the system traps on overflow.
171: No Overflow should be signaled for V * 1 = 1.70141183460469229e+38
172: nor for V / 1 = 1.70141183460469229e+38 .
173: Any overflow signal separating this * from the one
174: above is a DEFECT.
175:
176:
177: Diagnosis resumes after milestone Number 190 Page: 10
178:
179:
180: What message and/or values does Division by Zero produce?
181: Trying to compute 1 / 0 produces ...
182: * * * FLOATING-POINT ERROR * * *
183:
184: Trying to compute 0 / 0 produces ...
185: * * * FLOATING-POINT ERROR * * *
186:
187: Diagnosis resumes after milestone Number 220 Page: 11
188:
189:
190: The number of SERIOUS DEFECTs discovered = 1.
191: The number of FLAWs discovered = 1.
192:
193: The arithmetic diagnosed has unacceptable Serious Defects.
194:
195: A total of 3 floating point exceptions were registered.
196: END OF TEST.
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