# Gradual underflow, IEEE Exceptional handling

## Gradual underflow, IEEE Exceptional handling

Dear Irena,
Presently i am at the University of Iowa in the state of Iowa USA. I will be grateful to if you could put me in touch with people who look at implementation of IEEE Exceptional handling [in particular gradual overflow] in IA-64. Recently i have read the IA 64 document on software assisted exceptional handling and other operations in IA-64 and i wish to discuss it.

The problem of underflow in computations is troublesome. The IEEE standard has recommended the use of gradual underflow to deal with it. Gradual underflow requires the use of denormalized floating point numbers. As i understand in Intel strives to meet the IEEE standard hence has made provisions for denormalized numbers and impletation of gradual underflow in IA-64. However denormalized numbers are a costly options and it will be desirable to have way of high level implemention gradual underflow with out denormalized numbers.
Recently i have been working on Bounded arithmetic. This is an arithmetic of real numbers which have finite Infinity and a finite zero. Specifically in this arithmetic real numbers and the results of their mathematical operations are bounded form above and below by finite real numbers. I have recently shown that this arithmetic can be used to implement gradual underflow by software without using denormalized numbers. This arithmetic also shows the concept of "gradual overflow" where numbers lose precision gradually as the overflow thershold is approached. This could be used to handle the overflow problem. With this mail i am sending the abstract of a paper which i have recently written on this subject but have not communicated as yet for publication.

title{Computer arithmetic and exceptions via Bounded arithmetic}

abstract-The use of Bounded Arithmetic in numerical computation is
proposed. This is an arithmetic of real numbers with finite
`infinity' and finite `zero', i.e., a finite number behaves like
infinity and another finite number masquerades as zero.
Specifically, in it the real numbers and the results of their
mathematical operations are bounded from above and below by finite
real numbers. Application of this arithmetic to numerical
computation shows interesting possibilities e.g., an
over/underflow free arithmetic, high level implementation of
where numbers overflow gradually with sliding precision, extended
floating point range etc. The `Gradual overflow' may increase the
accuracy of overflowed results obtained by the infinity or the
largest number substitutions. These possibilities are demonstrated
by numerical examples.

I would like to know whether anyone in Intel is interested in these ideas and would like to open a discussion and dialoge.

K Avinash
The University of Iowa
Iowa city IA 52242 USA

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Hi K Avinash-

Thank you so much for joinging our forum and writing. Welcome. I will forward your question internally to some people and see if we can spur some dialogue. Hopefully others will join in on this discussion.
-irena

Is the full paper available? If not, could you
perhaps describe your idea in more detail? I didn't
follow how you intend to use bounded arithmetic to