R help - Oct 2025 - A very small p-value

Thanks for your answers.

I was not aware of the R function expm1(). 

I?m completely aware that 1 == 1 - 5e-19. But I was wondering why pt() returns
something smaller than double.eps.

For students who will use this exercise, it is disturbing to find 0 or 5e-19 :
yet it will be a good exercise to find that these quantities are equalled.

Regards, Christophe 
> Le 25 oct. 2025 ? 12:14, Ivan Krylov <ikrylov at disroot.org> a ?crit
:
> 
> ? Sat, 25 Oct 2025 11:45:42 +0200
> Christophe Dutang <dutangc at gmail.com> ?????:
> 
>> Indeed, the p-value is lower than the epsilon machine
>> 
>>> pt(t_score, df = n-2, lower=FALSE) < .Machine$double.eps  
>> [1] TRUE
> 
> Which means that for lower=TRUE, there will not be enough digits in R's
> numeric() type to represent the 5*10^-19 subtracted from 1 and
> approximately 16 zeroes.
> 
> Instead, you can verify your answer by asking for the logarithm of the
> number that is too close to 1, thus retaining more significant digits:
> 
> print(
> -expm1(pt(t_score, df = n-2, lower=TRUE, log.p = TRUE)),
> digits=16
> )
> # [1] 2.539746620181249e-19
> print(pt(t_score, df = n-2, lower=FALSE), digits=16)
> # [1] 2.539746620181248e-19
> 
> expm1(.) computes exp(.)-1 while retaining precision for numbers that
> are too close to 0, for which exp() would otherwise return 1.
> 
> See the links in
>
https://cran.r-project.org/doc/FAQ/R-FAQ.html#Why-doesn_0027t-R-think-these-numbers-are-equal_003f
> for a more detailed explanation.
> 
> -- 
> Best regards,
> Ivan
> (flipping the "days since referring to R FAQ 7.31" sign back to
0)

No, 0 and 5-19 are not "equalled".  THey are quite distinct.
As for pt() returning something smaller than double.eps, why wouldn't it?
If I calculate 10^-30, I get 1e-30, which is much smaller than  double.eps,
but is still correct.  It would be a serious error to return 0 for 10^-30.
Welcome to the wonderful world of floating-point arithmetic.
This really has nothing to do with R.

On Sun, 26 Oct 2025 at 09:38, Christophe Dutang <dutangc at gmail.com>
wrote:
>
> Thanks for your answers.
>
> I was not aware of the R function expm1().
>
> I?m completely aware that 1 == 1 - 5e-19. But I was wondering why pt()
returns something smaller than double.eps.
>
> For students who will use this exercise, it is disturbing to find 0 or
5e-19 : yet it will be a good exercise to find that these quantities are
equalled.
>
> Regards, Christophe
>
> > Le 25 oct. 2025 ? 12:14, Ivan Krylov <ikrylov at disroot.org> a
?crit :
> >
> > ? Sat, 25 Oct 2025 11:45:42 +0200
> > Christophe Dutang <dutangc at gmail.com> ?????:
> >
> >> Indeed, the p-value is lower than the epsilon machine
> >>
> >>> pt(t_score, df = n-2, lower=FALSE) < .Machine$double.eps
> >> [1] TRUE
> >
> > Which means that for lower=TRUE, there will not be enough digits in
R's
> > numeric() type to represent the 5*10^-19 subtracted from 1 and
> > approximately 16 zeroes.
> >
> > Instead, you can verify your answer by asking for the logarithm of the
> > number that is too close to 1, thus retaining more significant digits:
> >
> > print(
> > -expm1(pt(t_score, df = n-2, lower=TRUE, log.p = TRUE)),
> > digits=16
> > )
> > # [1] 2.539746620181249e-19
> > print(pt(t_score, df = n-2, lower=FALSE), digits=16)
> > # [1] 2.539746620181248e-19
> >
> > expm1(.) computes exp(.)-1 while retaining precision for numbers that
> > are too close to 0, for which exp() would otherwise return 1.
> >
> > See the links in
> >
https://cran.r-project.org/doc/FAQ/R-FAQ.html#Why-doesn_0027t-R-think-these-numbers-are-equal_003f
> > for a more detailed explanation.
> >
> > --
> > Best regards,
> > Ivan
> > (flipping the "days since referring to R FAQ 7.31" sign back
to 0)
>
> ______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
https://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

? Sat, 25 Oct 2025 22:37:57 +0200
Christophe Dutang <dutangc at gmail.com> ?????:
> But I was wondering why pt() returns something smaller than
> double.eps.
That's possible because the significant digits are stored separately
from the exponent of the number (which goes approximately from 10^-308
to 10^+308). Thus 5., followed by the rest of the digits, times 10^-19
can be distinguished from zero, despite the same number "vanishes"
when
subtracted from 1.

If this answers the wrong "why", could you please rephrase the
question?

-- 
Best regards,
Ivan