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From: Douglas Gregor (doug.gregor_at_[hidden])
Date: 2005-09-13 08:08:24

On Sep 13, 2005, at 4:38 AM, Martin Bonner wrote:

> Doug Gregor [mailto:dgregor_at_[hidden]] 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

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


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