Precision Tail-off?

[Stephen Tucker]

As a follow-up to my previous message, I have just produced the following
log on IDLE, for your information:
------------------------------
>>> math.e ** (math.log
(123456789000000000000000000000000000000000000000000) / 3)
4.979338592181741e+16
>>> 10 ** (math.log10 (123456789000000000000000000000000000000000000000000)
/ 3)
4.979338592181736e+16
>>> 123456789000000000000000000000000000000000000000000 ** (1.0 / 3.0)
4.979338592181734e+16
>>> 123456789e42 ** (1.0 / 3.0)
4.979338592181734e+16
------------------------------

Stephen Tucker.


On Fri, Feb 17, 2023 at 10:27 AM Stephen Tucker <stephen_tucker at sil.org>
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 *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.
>
> None of the digits in RootNZZZ's string should be different from the
> corresponding digits in RootN.
>
> 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.
>
> Stephen Tucker.
>
>
> On Thu, Feb 16, 2023 at 6:49 PM Peter Pearson <pkpearson at nowhere.invalid>
> wrote:
>
>> On Tue, 14 Feb 2023 11:17:20 +0000, Oscar Benjamin wrote:
>> > On Tue, 14 Feb 2023 at 07:12, Stephen Tucker <stephen_tucker at sil.org>
>> wrote:
>> [snip]
>> >> I have just produced the following log in IDLE (admittedly, in Python
>> >> 2.7.10 and, yes I know that it has been superseded).
>> >>
>> >> It appears to show a precision tail-off as the supplied float gets
>> bigger.
>> [snip]
>> >>
>> >> For your information, the first 20 significant figures of the cube
>> root in
>> >> question are:
>> >>    49793385921817447440
>> >>
>> >> Stephen Tucker.
>> >> ----------------------------------------------
>> >> >>> 123.456789 ** (1.0 / 3.0)
>> >> 4.979338592181744
>> >> >>> 123456789000000000000000000000000000000000. ** (1.0 / 3.0)
>> >> 49793385921817.36
>> >
>> > You need to be aware that 1.0/3.0 is a float that is not exactly equal
>> > to 1/3 ...
>> [snip]
>> > SymPy again:
>> >
>> > In [37]: a, x = symbols('a, x')
>> >
>> > In [38]: print(series(a**x, x, Rational(1, 3), 2))
>> > a**(1/3) + a**(1/3)*(x - 1/3)*log(a) + O((x - 1/3)**2, (x, 1/3))
>> >
>> > You can see that the leading relative error term from x being not
>> > quite equal to 1/3 is proportional to the log of the base. You should
>> > expect this difference to grow approximately linearly as you keep
>> > adding more zeros in the base.
>>
>> Marvelous.  Thank you.
>>
>>
>> --
>> To email me, substitute nowhere->runbox, invalid->com.
>> --
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>>
>