[Cfp-interest 2222] Re: quantum

[Vincent Lefevre]

On 2021-10-07 13:53:09 +0100, Mike Cowlishaw wrote:
> We're going to have to agree to differ, here :-). To me, the
> representation is defined by the three integers.

Well, 3 components, because there are 2 ways to express the
representation: either with exponent e or with exponent q.
The representation itself can be regarded as abstract data.

> > What matters is that the data that are manipulated are FP 
> > numbers (Level 2), FP representations (Level 3) or FP 
> > encodings (Level 4).
> > So it makes sense to say which level some property depends on.
> > For the quantum, it would depend on FP representations (Level 3).
> 
> Which in turn is those three integers; hence "used when the significand is
> regarded as an integer" (which refers to p18 of 754-2019).

These integers are just a was to express the representation. What
is manipulated is a floating-point number with some representation.
See also 5.3.2 in IEEE 754-2019:

  sourceFormat quantum(source)

  If x is a finite number, the operation quantum(x) is the number
  represented by (0, q, 1) where q is the exponent of x. If x is
  infinite, quantum(x) is +∞ with no exception.

x is here a floating-point number, so it is the quantum that is
obtained from the floating-point number at Level 3.

> > For the ulp, it would depend on FP numbers only (Level 2), 
> > but the ulp function can be extended to Level 1 for practical reasons.
> 
> I'll take your word for it -- I've never found a use for 'ulp' although I
> probably (and confusingly) used 'ulp' to mean what is now called 'quantum'
> about 25 years ago.  It's probably best for it to rest in peace, like
> 'mantissa' in similar contexts...

Unlike quantum, the ulp is useful in many places, such as error
analysis. You often cannot use the quantum, as this would mean that
you should also specify the representation, which is not practical
and is useless in this context. For instance, ulp(1) = β^(1−p),
whatever the representation of 1, while quantum(1) will depend on
the representation of 1.

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Vincent Lefèvre <vincent at vinc17.net> - Web: <https://www.vinc17.net/>
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