1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
|
Lest this program stop prematurely, i.e. before displaying
`END OF TEST',
try to persuade the computer NOT to terminate execution when an
error like Over/Underflow or Division by Zero occurs, but rather
to persevere with a surrogate value after, perhaps, displaying some
warning. If persuasion avails naught, don't despair but run this
program anyway to see how many milestones it passes, and then
amend it to make further progress.
Answer questions with Y, y, N or n (unless otherwise indicated).
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:
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.
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.
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 = 1.1102230e-16 .
Recalculating radix and precision
confirms closest relative separation U1 .
Radix confirmed.
The number of significant digits of the Radix is 53.000000 .
Some subexpressions appear to be calculated extra
precisely with about 11 extra B-digits, i.e.
roughly 3.31133 extra significant decimals.
That feature is not tested further by this program.
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.
Diagnosis resumes after milestone Number 40 Page: 6
Checking rounding on multiply, divide and add/subtract.
* is neither chopped nor correctly rounded.
/ is neither chopped nor correctly rounded.
Addition/Subtraction neither rounds nor chops.
Sticky bit used incorrectly or not at all.
FLAW: lack(s) of guard digits or failure(s) to correctly round or chop
(noted above) count as one flaw in the final tally below.
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 -5.0000000e-01 to 5.0000000e-01 ulps.
Diagnosis resumes after milestone Number 90 Page: 7
Testing powers Z^i for small Integers Z and i.
... no discrepancis found.
Seeking Underflow thresholds UfThold and E0.
Smallest strictly positive number found is E0 = 4.94066e-324 .
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 2.22507385850720188e-308, below which
calculation may suffer larger Relative error than merely roundoff.
Since underflow occurs below the threshold
UfThold = (2.00000000000000000e+00) ^ (-1.02200000000000000e+03)
only underflow should afflict the expression
(2.00000000000000000e+00) ^ (-1.02200000000000000e+03);
actually calculating yields: 0.00000000000000000e+00 .
This computed value is O.K.
Testing X^((X + 1) / (X - 1)) vs. exp(2) = 7.38905609893065218e+00 as X -> 1.
Accuracy seems adequate.
Testing powers Z^Q at four nearly extreme values.
... no discrepancies found.
Diagnosis resumes after milestone Number 160 Page: 8
Searching for Overflow threshold:
This may generate an error.
Can `Z = -Y' overflow?
Trying it on Y = -Inf .
Seems O.K.
Overflow threshold is V = 1.79769313486231571e+308 .
Overflow saturates at V0 = Inf .
No Overflow should be signaled for V * 1 = 1.79769313486231571e+308
nor for V / 1 = 1.79769313486231571e+308 .
Any overflow signal separating this * from the one
above is a DEFECT.
Diagnosis resumes after milestone Number 190 Page: 9
What message and/or values does Division by Zero produce?
Trying to compute 1 / 0 produces ... Inf .
Trying to compute 0 / 0 produces ... NaN .
Diagnosis resumes after milestone Number 220 Page: 10
The number of FLAWs discovered = 1.
The arithmetic diagnosed seems Satisfactory though flawed.
END OF TEST.
|
