[Cfp-interest 2229] Re: quantum

[Standards]

FWIW: a consistent, rigorous set of definitions on floating-point 
parameters (including a definition of ulp) is given in the LIA1 
(Language-Independent Arithmetic part 1, ISO 10967-1) standard 
(mentioned in Annex H of the C standard). ulp is subsequently used in 
the specification of the error bounds in many mathematical operations 
and functions (LIA-2) and complex operations and functions (LIA-3).

- Willem Wakker


Op 08-10-2021 om 15:03 schreef Mike Cowlishaw:
>   
>> On 2021-10-07 20:01:51 +0100, Mike Cowlishaw wrote:
>>>> 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) = ß^(1-p),
>>>> whatever the representation of 1, while quantum(1) will depend on
>>>> the representation of 1.
>>> I think we are going around in circles.  754 does not define
>>> 'representation' -- that word is used in its ordinary
>>> English sense.
>>> It is not defined in Clause 2.1 for that very reason.
>> It *is* defined in Clause 2.1:
>>
>>    floating-point representation: An unencoded member of a
>>    floating-point format, representing a finite number, a signed
>>    infinity, a quiet NaN, or a signaling NaN. A representation of a
>>    finite number has three components: a sign, an exponent, and a
>>    significand; its numerical value is the signed product of its
>>    significand and its radix raised to the power of its exponent.
> I'd forgotten that -- thanks (although one could argue that 'representation'
> need not be a 'floating-point representation').  Thankfully, that says the
> same as that which we have been discussing: "A representation of a finite
> number has three components: a sign, an exponent, and a significand".
>
> The quantum is simply the radix raised to the power of the exponent, as we
> already discussed, so for any given floating-point representation the
> quantum is well-defined.
>
>>> Although you imply otherwise, the quantum of a number is both
>>> important and useful (especially in radix 10, but also in
>>> radix 2 if you live in the USA
>>> -- where, for example, woodworking measurements are often given in
>>> 1/32 inch, etc.).
>> Perhaps in some contexts. But unlike the ulp, there is no way
>> to define the quantum on general real numbers without an
>> additional parameter (providing information about the
>> quantum). So this is off-topic here.
> Hmm, so how does one define the ulp without some additional parameter?  What
> is the last place in an irrational real number?
>
> Mike
>
>
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-- 
Willem Wakker, ACE Consulting bv
Mob: +31 625 026561
mailto:willemw at ace.nl