Hi Peter, I forwarded your excellent email to a workmate of mine, who
had the following to say about it; I've Cc'ed him on the email. I'll
put it inline in the email so I'm not top-posting :). (Note:
Everything down here are Michael's thoughts, not mine! Hope you
don't mind, Mike...)
On 16/11/2006, at 11:27 AM, Peter Miller wrote:
On Thu, 2006-11-16 at 10:46 +1100, Jamie Wilkinson wrote:
What's better:
if expected - epsilon <= computation()<= expected + epsilon:
versus:
if abs(computation() - expected) >= epsilon:
Epsilon if a function of the number of bits in the mantissa, but
not the
number of bits of exponent. So a better test tends to be
if (fabs(computation()/expected) > 1 + epsilon)
blah
From Michael Anderson:
The second equation is wrong! -
if (fabs(computation()/expected) > 1 + epsilon)
blah
With this test computation=-1 and expected=+1 passes for
epsilon=0 ... not good.
but that makes matters worse, because the division loses more
precision,
so the better but less obvious C code is
int expected_exponent;
frexp(expected, &expected_exponent);
double scaled_epsilon = ldexp(unscaled_epsilon, expected_exponent);
if (fabs(computation() - expected) >= scaled_epsilon)
blah
From Michael Anderson:
Since I'd written my thoughts on the interval case I thought I'd pass
them on too...
For the interval case...
Usually when I'm doing this kind of test I don't care about the
"exact" result, its usually a stopping criteria of some algorithm - a
convergence test or similar.
What I've always used in those cases is similar to
approxError = fabs(computation() - expected) / (1.0 + fabs(expected)
+fabs(computation()) ) >= epsilon
This is slower, but probably more robust over a wide range of
epsilon, expected and computation.
Essentially its a trade of between absolute error (for small values)
and relative error (for large values).
Which is usually more appropriate than a direct comparison for my
algorithms.
--
% Andre Pang (on behalf of Michael Anderson) <http://
www.algorithm.com.au/>
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