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RE: Transcendental function tables: comments welcome!
Just a few minor points on Eric Postpischil's comments.
Notes for Table D.2 and Table D.3 define "n" as "Any nonzero
positive integer." "Nonzero" is redundant (+0 notwithstanding).
The standard addresses an international audience, and in some parts of
the world "positive" includes zero ("strictly positive" does not, but
sounds silly to the other parts of the world). So let's be unambiguous.
Did we discuss previously the possibility that cos could signal
underflow in some as-yet unimplemented format, if a representable
number happened to be extraordinarily close to an odd multiple of
pi/2? Conceivably, it could happen in some non-degenerate format.
The Continued Fraction expansion of pi is known sufficiently far out
that the format degeneracy should be quite extreme.
It is even conceivable that logp1 could underflow -- maybe
there is a huge sequence of zeroes or ones somewhere in the
binary representation of 1/e.
Wait a moment: logp1(x) is log(1+x), not log(x)+1, which you seem
to think here. Nevertheless, what follows would still apply.
The Continued Fraction expansion of e is completely known, and has
regularly increasing partial quotients, so you should be able to
compute just what kind of a format is needed to cause trouble.
Actually, I just realised that the CF expansion of pi or e is pretty
irrelevant unless there is a partial convergent whose denominator
is a power of the base -- which is highly unlikely, and probably has
been verified. In that case, no floating point number will be a
best approximation, and the lower bound on cos or logp1 for n-bit
precision (regardless of exponent range) would be about 1/2**(2n+1).
So emax would have to be less than twice the precision in bits, and
that is pretty degenerate, or rather, it would not qualify as an
extended format in the IEEE 754 sense, I think.
Michel.
Sent: 2007-05-28 21:06:39 UTC