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Underflow for cos(x)
The best-known CF convergent for pi, 22/7, leads to an example of a
BFP format that might make sense for very coarse data when space is
at a premium -- namely binary8, with 1 sign bit, 4 exponent bits and
a 4-bit significand (one hidden bit). Then cos(11) = 0.0044 would
indeed underflow to 5*2^-10 (Nmin is 2^-6 = 0.0156).
Every partial convergent of pi with an even numerator 2n provides a
candidate for a very small cos(n), and if Nmax > n > Nmax/2 we will
get underflow; in fact, sometimes Nmax can even exceed 2n, as in the
example above.
Among the first 50 partial convergents of pi there are two where the
corresponding format has an emax that exceeds precision by 3 resp. 8,
i.e. the argument leaves the consecutive-integer range.
Michel.
Sent: 2007-06-01 05:26:14 UTC