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Lest this program stop prematurely, i.e. before displaying
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`END OF TEST',
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try to persuade the computer NOT to terminate execution when an
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error like Over/Underflow or Division by Zero occurs, but rather
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to persevere with a surrogate value after, perhaps, displaying some
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warning. If persuasion avails naught, don't despair but run this
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program anyway to see how many milestones it passes, and then
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amend it to make further progress.
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Answer questions with Y, y, N or n (unless otherwise indicated).
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Diagnosis resumes after milestone Number 0 Page: 1
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Users are invited to help debug and augment this program so it will
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cope with unanticipated and newly uncovered arithmetic pathologies.
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Please send suggestions and interesting results to
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Richard Karpinski
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Computer Center U-76
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University of California
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San Francisco, CA 94143-0704, USA
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In doing so, please include the following information:
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Precision: double;
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Version: 10 February 1989;
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Computer:
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Compiler:
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Optimization level:
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Other relevant compiler options:
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Diagnosis resumes after milestone Number 1 Page: 2
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Running this program should reveal these characteristics:
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Radix = 1, 2, 4, 8, 10, 16, 100, 256 ...
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Precision = number of significant digits carried.
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U2 = Radix/Radix^Precision = One Ulp
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(OneUlpnit in the Last Place) of 1.000xxx .
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U1 = 1/Radix^Precision = One Ulp of numbers a little less than 1.0 .
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Adequacy of guard digits for Mult., Div. and Subt.
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Whether arithmetic is chopped, correctly rounded, or something else
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for Mult., Div., Add/Subt. and Sqrt.
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Whether a Sticky Bit used correctly for rounding.
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UnderflowThreshold = an underflow threshold.
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E0 and PseudoZero tell whether underflow is abrupt, gradual, or fuzzy.
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V = an overflow threshold, roughly.
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V0 tells, roughly, whether Infinity is represented.
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Comparisions are checked for consistency with subtraction
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and for contamination with pseudo-zeros.
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Sqrt is tested. Y^X is not tested.
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Extra-precise subexpressions are revealed but NOT YET tested.
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Decimal-Binary conversion is NOT YET tested for accuracy.
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Diagnosis resumes after milestone Number 2 Page: 3
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The program attempts to discriminate among
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FLAWs, like lack of a sticky bit,
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Serious DEFECTs, like lack of a guard digit, and
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FAILUREs, like 2+2 == 5 .
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Failures may confound subsequent diagnoses.
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The diagnostic capabilities of this program go beyond an earlier
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program called `MACHAR', which can be found at the end of the
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book `Software Manual for the Elementary Functions' (1980) by
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W. J. Cody and W. Waite. Although both programs try to discover
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the Radix, Precision and range (over/underflow thresholds)
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of the arithmetic, this program tries to cope with a wider variety
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of pathologies, and to say how well the arithmetic is implemented.
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The program is based upon a conventional radix representation for
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floating-point numbers, but also allows logarithmic encoding
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as used by certain early WANG machines.
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BASIC version of this program (C) 1983 by Prof. W. M. Kahan;
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see source comments for more history.
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Diagnosis resumes after milestone Number 3 Page: 4
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Program is now RUNNING tests on small integers:
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-1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K.
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Searching for Radix and Precision.
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Radix = 2.000000 .
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Closest relative separation found is U1 = 1.1102230e-16 .
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Recalculating radix and precision
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confirms closest relative separation U1 .
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Radix confirmed.
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The number of significant digits of the Radix is 53.000000 .
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Diagnosis resumes after milestone Number 30 Page: 5
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Subtraction appears to be normalized, as it should be.
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Checking for guard digit in *, /, and -.
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*, /, and - appear to have guard digits, as they should.
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Diagnosis resumes after milestone Number 40 Page: 6
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Checking rounding on multiply, divide and add/subtract.
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Multiplication appears to round correctly.
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Division appears to round correctly.
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Addition/Subtraction appears to round correctly.
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Checking for sticky bit.
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Sticky bit apparently used correctly.
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Does Multiplication commute? Testing on 20 random pairs.
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No failures found in 20 integer pairs.
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Running test of square root(x).
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Testing if sqrt(X * X) == X for 20 Integers X.
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Test for sqrt monotonicity.
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sqrt has passed a test for Monotonicity.
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Testing whether sqrt is rounded or chopped.
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Square root appears to be correctly rounded.
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Diagnosis resumes after milestone Number 90 Page: 7
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Testing powers Z^i for small Integers Z and i.
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... no discrepancis found.
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Seeking Underflow thresholds UfThold and E0.
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Smallest strictly positive number found is E0 = 2.22507e-308 .
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Since comparison denies Z = 0, evaluating (Z + Z) / Z should be safe.
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What the machine gets for (Z + Z) / Z is 2.00000000000000000e+00 .
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This is O.K., provided Over/Underflow has NOT just been signaled.
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Diagnosis resumes after milestone Number 120 Page: 8
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FLAW: X = 3.05947655544740190e-308
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is not equal to Z = 2.22507385850720140e-308 .
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yet X - Z yields 0.00000000000000000e+00 .
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Should this NOT signal Underflow, this is a SERIOUS DEFECT
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that causes confusion when innocent statements like
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if (X == Z) ... else ... (f(X) - f(Z)) / (X - Z) ...
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encounter Division by Zero although actually
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X / Z = 1 + 0.375 .
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The Underflow threshold is 2.22507385850720140e-308, below which
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calculation may suffer larger Relative error than merely roundoff.
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Since underflow occurs below the threshold
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UfThold = (2.00000000000000000e+00) ^ (-1.02200000000000000e+03)
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only underflow should afflict the expression
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(2.00000000000000000e+00) ^ (-1.02200000000000000e+03);
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actually calculating yields: 0.00000000000000000e+00 .
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This computed value is O.K.
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Testing X^((X + 1) / (X - 1)) vs. exp(2) = 7.38905609893065220e+00 as X -> 1.
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Accuracy seems adequate.
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Testing powers Z^Q at four nearly extreme values.
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... no discrepancies found.
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Diagnosis resumes after milestone Number 160 Page: 9
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Searching for Overflow threshold:
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This may generate an error.
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* * * FLOATING-POINT ERROR * * *
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Can `Z = -Y' overflow?
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Trying it on Y = -8.98846567431157950e+307 .
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Seems O.K.
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Overflow threshold is V = 1.79769313486231570e+308 .
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There is no saturation value because the system traps on overflow.
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No Overflow should be signaled for V * 1 = 1.79769313486231570e+308
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nor for V / 1 = 1.79769313486231570e+308 .
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Any overflow signal separating this * from the one
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above is a DEFECT.
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Diagnosis resumes after milestone Number 190 Page: 10
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What message and/or values does Division by Zero produce?
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Trying to compute 1 / 0 produces ...
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* * * FLOATING-POINT ERROR * * *
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Trying to compute 0 / 0 produces ...
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* * * FLOATING-POINT ERROR * * *
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Diagnosis resumes after milestone Number 220 Page: 11
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No failures, defects nor flaws have been discovered.
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Rounding appears to conform to the proposed IEEE standard P754,
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except for possibly Double Rounding during Gradual Underflow.
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The arithmetic diagnosed appears to be Excellent!
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A total of 3 floating point exceptions were registered.
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END OF TEST.
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