The base 2 representation of 0.4 repeats the digit sequence 1001
infinitely, hence must be rounded. The problem occurs in C the same as it
does in R.
bill@Bill-T490:~$ cat a.c
#include <stdio.h>
int main(int argc, char* argv[])
{
double d = 0.4 + 0.3 + 0.2 + 0.1;
printf("0.4+0.3+0.2+0.1 -> %24.17g\n", d);
printf("0.4+0.3+0.2+0.1 == 1.0 -> %s\n", d == 1.0 ? "true" : "false");
return 0;
}
bill@Bill-T490:~$ gcc a.c
bill@Bill-T490:~$ ./a.out
0.4+0.3+0.2+0.1 -> 0.99999999999999989
0.4+0.3+0.2+0.1 == 1.0 -> false
-Bill
On Tue, Feb 1, 2022 at 7:01 PM Nathan Boeger <nboe...@gmail.com> wrote:
> Thank you for this explanation!
>
> I have a long background in C/C++ and never realized this was such an issue
> with some languages. At least, with trivial single digit decimals. I
> understand accuracy issues with very large decimals, repeating or
> non-terminating rationals and I have handled them in the past. It makes me
> worried about all the R scripts I have written before (yikes!).
>
> Cheers
>
> -nb
>
> On Wed, 2 Feb 2022 at 02:44, Richard M. Heiberger <r...@temple.edu> wrote:
>
> > RShowDoc('FAQ')
> >
> > then search for 7.31
> >
> >
> > This statement
> > "If you stop at a 5 or 7 or 8 and back up to the previous digit, you
> round
> > up. Else you leave the previous result alone."
> > is not quite right. The recommendation in IEEE 754, and this is how R
> > does arithmetic, is to Round Even.
> >
> > I ilustrate here with decimal, even though R and other programs use
> binary.
> >
> > > x <- c(1.4, 1.5, 1.6, 2.4, 2.5, 2.6, 3.4, 3.5, 3.6, 4.4, 4.5, 4.6)
> > > r <- round(x)
> > > cbind(x, r)
> > x r
> > [1,] 1.4 1
> > [2,] 1.5 2
> > [3,] 1.6 2
> > [4,] 2.4 2
> > [5,] 2.5 2
> > [6,] 2.6 3
> > [7,] 3.4 3
> > [8,] 3.5 4
> > [9,] 3.6 4
> > [10,] 4.4 4
> > [11,] 4.5 4
> > [12,] 4.6 5
> > >
> >
> > Numbers whose last digit is not 5 (when in decimal) round to the nearest
> > integer.
> > Numbers who last digit is 5 (1.5, 2.5, 3.5, 4.5 above)
> > round to the nearest EVEN integer.
> > Hence 1.5 and 3.5 round up to the even numbers 2 and 4.
> > 2.5 and 4.5 round down do the even numbers 2 and 4.
> >
> > This way the round ups and downs average out to 0. If we always went up
> > from .5 we would have
> > an updrift over time.
> >
> > For even more detail click on the link in FAQ 7.31 to my appendix
> > https:// link.springer.com/content/pdf/bbm%3A978-1-4939-2122-5%2F1.pdf
> > and search for "Appendix G".
> >
> > Section G.5 explains Round to Even.
> > Sections G.6 onward illustrate specific examples, such as the one that
> > started this email thread.
> >
> > Rich
>
> [[alternative HTML version deleted]]
>
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