Precision Tail-off?

[Richard Damon]

On 2/17/23 5:27 AM, Stephen Tucker wrote:
> Thanks, one and all, for your reponses.
>
> This is a hugely controversial claim, I know, but I would consider this
> behaviour to be a serious deficiency in the IEEE standard.
>
> Consider an integer N consisting of a finitely-long string of digits in
> base 10.
>
> Consider the infinitely-precise cube root of N (yes I know that it could
> never be computed unless N is the cube of an integer, but this is a
> mathematical argument, not a computational one), also in base 10. Let's
> call it RootN.
>
> Now consider appending three zeroes to the right-hand end of N (let's call
> it NZZZ) and NZZZ's infinitely-precise cube root (RootNZZZ).

The key factor here is IEEE floating point is storing numbers in BINARY, 
not DECIMAL, so a multiply by 1000 will change the representation of the 
number, and thus the possible resolution errors.

Store you numbers in IEEE DECIMAL floating point, and the variations by 
multiplying by powers of 10 go away.

>
> The *only *difference between RootN and RootNZZZ is that the decimal point
> in RootNZZZ is one place further to the right than the decimal point in
> RootN.

No, since the floating point number is stored as a fraction times a 
power of 2, the fraction has changed as well as the power of 2.

>
> None of the digits in RootNZZZ's string should be different from the
> corresponding digits in RootN.

Only if the storage format was DECIMAL.

>
> I rest my case.
>
> Perhaps this observation should be brought to the attention of the IEEE. I
> would like to know their response to it.

That is why they have developed the Decimal Floating point format, to 
handle people with those sorts of problems.

They just aren't common enough for many things to have adopted the use 
of it.

>
> Stephen Tucker.

-- 
Richard Damon