sqrt(.Machine$double.xmax)^2 == Inf, but only on Windows in R

Pavel Krivitsky via R-devel
     on Mon, 28 Apr 2025 05:13:41 +0000 writes:
     > Hello, Under R 4.5.0 on Windows (x86-64), I get:

     >> sqrt(.Machine$double.xmax)^2
     > [1] Inf
     >> sqrt(.Machine$double.xmax)*sqrt(.Machine$double.xmax)
     > [1] Inf

     > On other hand on other platforms, including Debian Linux
     > (x86-64), I get:
     d> sqrt(.Machine$double.xmax)^2
     > [1] 1.797693134862315508561e+308
     d> sqrt(.Machine$double.xmax)*sqrt(.Machine$double.xmax)
     > [1] 1.797693134862315508561e+308

     > Windows is running inside a VirtualBox instance on the
     > same host as Linux. I don't have direct results from the
     > MacOS platforms, but based on the symptoms that had led me
     > to investigate, the behaviour is as the Linux.

     > Adding to the mystery, if I implement the same operation in C, e.g.,

     > library(inline)
     > sqrsqrt <- cfunction(sig = c(x = "numeric"), language = "C", "
     > double sqrtx = sqrt(Rf_asReal(x));
     > return Rf_ScalarReal(sqrtx*sqrtx);
     > ")

     > R on Linux yields:
     d> sqrsqrt(.Machine$double.xmax)
     > [1] 1.797693134862315508561e+308

     > i.e., the same number, whereas R on Windows yields:
     d> sqrsqrt(.Machine$double.xmax)
     > [1] 1.797693134862315508804e+308

     > which is not Inf but not the same as Linux either.

     > Lastly, on both platforms,
     d> sqrsqrt(.Machine$double.xmax) < .Machine$double.xmax
     > [1] TRUE

     > I am not sure if this is a bug, intended behaviour, or something else.
"something else":  It is not a bug, nor intended, but should
also *not* be surprising nor a mistery:  The largest possible
double precision number is by definition "very close to
infinity" (in the space of double precision numbers)
[R 4.5.0 patched on Linux (Fedora 40; x86_64)] :

(M <- .Machine$double.xmax)
[1] 1.797693e+308
M+1 == M
[1] TRUE
M*(1 + 2^-52)
[1] Inf
print(1 + 2^-52, digits= 16)
[1] 1
print(1 + 2^-52, digits= 17)
[1] 1.0000000000000002
What you see, I'd classify as quite related to R FAQ 7.31,
  := "the most frequently asked question about R
      among all the other frequently asked questions"

A nice reading is the README you get here
   https://github.com/ThinkR-open/seven31
which does link also to the R FAQ at
  https://cran.r-project.org/doc/FAQ/R-FAQ.html#Why-doesn_0027t-R-think-these-numbers-are-equal_003f

Of tangential interest only:
You mention that it is R 4.5.0 you use on Windows.
Would you (or anybody else) know if this is new behaviour or it
also happened e.g. in R 4.4.x versions on  Windows?
This depends on C math function sqrt. With sqrt implemented in mingw-w64 
(mingwex), which is still the case of mingw-w64 v11, so still of 
Rtools45, the result of sqrt(.Machine$double.xmax) is

"0x1p+512"

with mingw-w64 v12 (so with UCRT, and also on Linux, and also using mpfr 
library) it is

"0x1.fffffffffffffp+511"

It is not required by any standard for the math functions in the C math 
library to be precise (correctly rounded). But still, in this case, the 
correctly rounded value would be returned once Rtools can switch to 
mingw-w64 v12 (which could be no sooner than Rtools46, as mingw-w64 is a 
core component of the toolchain).

A bit of advertising -? I used Martin's Rmpfr package to compute this 
using mpfr.
And there is a related blog post: 
https://blog.r-project.org/2025/04/24/sensitivity-to-c-math-library-and-mingw-w64-v12

Best
Tomas

Best regards,
Martin

--
Martin Maechler
ETH Zurich  and  R Core team

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sqrt(.Machine$double.xmax)^2 == Inf, but only on Windows in R

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