Here is a technical question. I know that people always talk about
ulp's in the context of how good a function implementation is. I think
the ulp is the number of base 2 digits at the end of the mantissa that
we cannot rely on.
So if one were to write a naive implementation of lexp(x) that used
Taylor's series if x is positive, and 1/lexp(-x) is x is negative - well
one could fairly easily estimate an upper bound on the ulp, but it
wouldn't be low (like ulp=1 or 2), but probably rather higher (ulp of
the order of 10 or 20).
So do people really work hard to get that last drop of ulp out of their
calculations? Would a ulp=10 be considered unacceptable?
Also, looking through the source code for the FreeBSD implementation of
exp, I saw that they used some rather smart rational function. (I don't
know how they came up with it.)
Presumably a big part of the issue is to make the functions work rather
fast. And a naive implementation of Taylor's series wouldn't be fast.
But if people want lexp rather than exp, they must have already decided
that accuracy is more important than speed.
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