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    Subject[2.1.82] SERIOUS FLAWS in IEEE math handling on ALPHA.
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    [ NOTE: This is a cross-posting to several mailing lists, so please
    respect the reply-to. Thanks! -dbk ]

    Hi folks,

    it appears as if there is a serious flaw in the IEEE math handling on
    the Alpha. A colleague of mine complained to me about having serious
    problems with his numerical programs getting SIGFPE at times, even so
    he as compiled them with -mieee. That prompted me to investigate the
    problem and i fetched the paranoia IEEE conformance test and ran it on
    my UDB (using linux 2.1.82, egcs 1.0 and glibc 2.05c). As control we
    ran the same test on an UDB running NetBSD 1.3 (using gcc 2.7.2.1) and
    on an EB64-Clone running Digital Unix 4.0 (using Digitals c89).

    The results were somewhat unexpected:

    * Digital Unix was flawless. (As it should be, considering the price
    tag.)

    * NetBSD failed the Inf and NaN tests with SIGFPE, but was flawless
    otherwise.

    * Linux had 1 serious defect, 2 defects and 1 flaw (see the attached
    output of paranoia).

    I am now looking at the IEEE kernel code, hoping to find the bugs.
    But somebody with more than an idea of what he is doing might actually
    find it faster.

    Until these bugs are fixed, i would advise people who need to rely on
    the results of their numerical calculations to be very cautious when
    using Linux/Alpha.

    The paranoia outputs of OSF and NetBSD are available on request, should
    need be.

    Yours,
    Dominik Kubla
    Unix User Group ("UFO")
    Johannes Gutenberg-University, Mainz

    =============================== CUT HERE ================================

    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).


    To continue, press RETURN
    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: DEC Multia (VX-42), Linux 2.1.82, glibc 2.0.5c

    Compiler: egcs-1.0

    Optimization level: -O3

    Other relevant compiler options: -mieee -lm

    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 = 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 .

    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 apparently used correctly.

    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 appears to be correctly rounded.

    To continue, press RETURN
    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.

    FLAW: Underflow can stick at an allegedly positive
    value PseudoZero that prints out as 8.9003e-308 .
    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.
    Smallest strictly positive number found is E0 = 2.22507e-308 .
    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.

    To continue, press RETURN
    Diagnosis resumes after milestone Number 120 Page: 8


    The Underflow threshold is 2.22507385850720138e-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.


    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 = -8.98846567431157954e+307 .
    Seems O.K.
    SERIOUS DEFECT: overflow past -8.98846567431157954e+307
    shrinks to -0.00000000000000000e+00 .
    Overflow threshold is V = 8.98846567431157954e+307 .
    Overflow saturates at V0 = 8.98846567431157954e+307 .
    No Overflow should be signaled for V * 1 = 8.98846567431157954e+307
    nor for V / 1 = 8.98846567431157954e+307 .
    Any overflow signal separating this * from the one
    above is a DEFECT.

    DEFECT: Comparison alleges that what prints as Z = 2.22507385850720138e-308
    is too far from sqrt(Z) ^ 2 = 1.16204282018886510e-307 .
    DEFECT: Comparison alleges that what prints as Z = 2.22507385850720138e-308
    is too far from sqrt(Z) ^ 2 = 1.16204282018886510e-307 .

    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 ... Inf .

    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 = 1.
    The number of DEFECTs discovered = 2.
    The number of FLAWs discovered = 1.

    The arithmetic diagnosed has unacceptable Serious Defects.
    END OF TEST.

    -
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