[R] [External] Why does the print method fail for very small numbers?

[iuke-tier@ey m@iii@g oii uiow@@edu]

Values this small are denormalized numbers and only have a few binary
digits of accuracy.  The smallest positive value an IEEE double
precision number can have is 2^-1074.  The second smallest IEEE double
precision number is 2^-1073, and the third smallest is 2^-1073 +
2^-1074.

The numbers 2.48e-324 and 4.940656e-324 are both read in as 2^-1074:

> 2.48e-324 == 2^-1074
[1] TRUE
> 4.940656e-324 == 2^-1074
[1] TRUE

You can see the binary representations with

example(numToBits) ## for bitC
bitC(2^-1074)
bitC(2^-1073)
bitC(2^-1073 + 2^-1074)

Look up 'denormalized numbers' if you want to learn more.

Best,

luke

On Thu, 17 Feb 2022, Marius Hofert wrote:

> Dear expeRts,
>
> I'm familiar with IEEE 754. Is there an easy way to explain why even
> just printing of small numbers fails?
>
> 1e-317 # 1e-317 => fine
> 1e-318 # 9.999987e-319 => gets tricky; seems to call print() => as.character() => format() => paste()
> 1e-318 == 9.999987e-319 # TRUE
> 2.48e-324 # prints 4.940656e-324 for me
> 2.48e-324 == 4.940656e-324 # TRUE
> ## Relative error as a plot
> rel_error <- function(x)
>    plot(abs((as.numeric(as.character(x)) - x) / x), type = "l",
>         ylab = "Relative error between x and as.numeric(as.character(x))")
> rel_error(seq(0.001, 0.001 + .Machine$double.xmin, length.out = 1001)) # fine
> rel_error(seq(0, .Machine$double.xmin, length.out = 1001)) # printing breaks down
>
> Of course, [0,.Machine$double.xmin] is somewhat of a strange set of numbers to consider,
> and I expect things like "==" to be easily fooled there, but already the print method (?)
>
> Thanks & cheers,
> Marius
>
> sessionInfo()
> R version 4.1.2 (2021-11-01)
> Platform: x86_64-apple-darwin21.2.0 (64-bit)
> Running under: macOS Monterey 12.1
> ...
>
>
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-- 
Luke Tierney
Ralph E. Wareham Professor of Mathematical Sciences
University of Iowa                  Phone:             319-335-3386
Department of Statistics and        Fax:               319-335-3017
    Actuarial Science
241 Schaeffer Hall                  email:   luke-tierney using uiowa.edu
Iowa City, IA 52242                 WWW:  http://www.stat.uiowa.edu