Re: after-rounding tinyness detection and directed rounding

[Guillaume Melquiond <guillaume.melquiond@xxxxxxxxxxx>]

Le lundi 15 octobre 2007 à 11:21 -0400, Michel Hack a écrit :
> The returned result when rounding a positive tiny-before-rounding value
> in roundTowardPositive is Nmin, regardless of how small the true value is.
> The rounded result appears to have a full count of bits, but most or all
> of them are meaningless.  That's ok, we asked for a bound and got one.
> The result is not wrong.
>
> Nevertheless, the information that the true value is very small is lost.

I cannot agree with your assertion. Consider a format with no exponent
bounds, either the rounded result is Nmin, or it is not.

1. Case where the extended rounded result is Nmin. All the digits
(except the last one) are correct, since the relative error is bounded
by 2^{1-p}. So it means that the Nmin rounded result in the bounded
format has all its digits correct too, as it is exactly the same value.

2. Case where the extended rounded result is not Nmin. If it is bigger,
it is away from the underflow range, so it is impossible for it to be a
tiny-before-rounding value. So the extended rounded result is in the
underflow range, so the implementation has to signal underflow in
after-rounding mode.

So either the result is the same as if no underflow had happened, or the
implementation signals underflow. No information was lost. If you
disagree, please provide an example with explicit numerical values, so
that I see where my interpretation is mistaken.

Best regards,

Guillaume