On Sep 13, 2005, at 4:38 AM, Martin Bonner wrote:
Doug Gregor [mailto:dgregor@cs.indiana.edu] wrote:
I've traced through this, and the right "exp" is getting called. The machine epsilon for long double is:
4.9406564584124654417656879286822137236505980261432476442558568250067 550 727020875e-324
Are you sure that's right? That implies a 1074-bit mantissa. That's a VERY long double! (It is also a rather odd sized one. In such a large type, I would expect the mantissa to be an exact number of bytes - or mantissa+sign to be an exact number of bytes).
That value looks much more reasonable for the smallest non-zero value of long double.
(I am assuming that by epsilon you mean the smallest values such that 1+epsilon != 1, rather than the smallest value such that epsilon > 0).
I discussed this with Marshall Clow offline and we eventually talked with an Apple engineer. Marshall summarized the issue as: "gcc 4 uses a "doubled double" format for long double (why? I don't know), where there are two double values inside the long double. For numbers that can be represented exactly (like, say 0 or 1), the "next value" is very, very close, because (I'm assuming) they can denormalize the lower value, and get 1e-324 type ranges. However, for numbers that cannot be represented exactly, the "next value" is on the order of 2^-112 greater. Since epsilon is defined as the difference between 1 and the "next value greater than one", they can return a very, very small number here - and it is correct (but not useful)." AFAICT, for this floating-point format, the machine epsilon is truly a useless value; we shouldn't be comparing against it in our tests because it doesn't really reflect the precision of the floating-point format. Doug