A very small p-value

Dear list,

I'm computing a p-value for the Student test and discover some inconsistencies with the cdf pt().

The observed statistic is 11.23995 for 95 observations, so the p-value is very small

t_score <- 11.23995
n <- 95
print(pt(t_score, df = n-2, lower=FALSE), digits=22)

[1] 2.539746620181247991746e-19

print(integrate(dt, lower=t_score, upper=Inf, df=n-2)$value, digits = 22)

[1] 2.539746631161970791961e-19

But if I compute with pt(lower=TRUE), I got 0

print(1-pt(t_score, df = n-2, lower=TRUE), digits=22)

[1] 0

Indeed, the p-value is lower than the epsilon machine

pt(t_score, df = n-2, lower=FALSE) < .Machine$double.eps

[1] TRUE

Using the square of t statistic which follows a Fisher distribution, I got the same issue:

print(pf(z, 1, n-2, lower=FALSE), digits=22) 

[1] 5.079493240362495983491e-19

print(integrate(df, lower=z, upper=Inf, df1=1, df2=n-2)$value, digits = 22) 

[1] 5.079015231299358486828e-19

print(1-pf(z, 1, n-2, lower=TRUE), digits=22) 

[1] 0
	
When using the t.test() function, the p-value is naturally printed : p-value < 2.2e-16.

Any comment is welcome. 

Christophe

R.version

_                           
platform       aarch64-apple-darwin20      
arch           aarch64                     
os             darwin20                    
system         aarch64, darwin20           
status                                     
major          4                           
minor          5.1                         
year           2025                        
month          06                          
day            13                          
svn rev        88306                       
language       R                           
version.string R version 4.5.1 (2025-06-13)
nickname       Great Square Root 
-------------------------------------------------
Christophe DUTANG
LJK, Ensimag, Grenoble INP, UGA, France
ILB research fellow
Web: http://dutangc.free.fr
-------------------------------------------------