[Cfp-interest 2228] Re: quantum

[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