在R中使用sqrt（x）和x ^ 0.5时结果不同

Question

I have an issue where there is a difference in the results when I use sqrt(x) rather than x^0.5.

The calculations are being carried out on floating point numbers such as:

0.002296438

Trouble is this truncated version as displayed in Rstudio does not replicate the problem. However the non-truncated version does (any idea how I can get the non-truncated version to display so I can show a working example)?

The errors are indeed small of the order of e^-18 which are not so worrying in themselves. However over even moderately large data sets (10,000 date points) these errors compound to give errors in the variance estimate at the 4th decimal place which is more concerning!

I realize that R is only accurate to 16 decimal places, see answer from nullglob below but these errors seems to be systematic? Every time you run sqrt(x) and x^0.5 they both produce the same answer each time. However these answers are still different from each other.

Formatting Decimal places in R

Is one version considered to be more accurate than the other?

Baz

OK here is the example

[1] 0.002296437934635199226707
> test4=sqrt(0.002296437934635199226707)
> test5=0.002296437934635199226707^0.5
> test6=test5-test4
test6
[1] 6.938894e-18

Answer 1

Confirmed (more or less) the suggestion of different C calls with the following C code, which calls the sqrt() and pow() functions from the system math library.

#include <math.h>
#include <stdio.h>

int main(int argc, char **argv) {
    double x = 0.002296437934635199226707;
    long double y = 0.002296437934635199226707;

    printf("%1.22g\n",sqrt(x)-pow(x,0.5));
    printf("%1.22Lg\n",sqrtl(y)-powl(y,0.5));

    return(0);
}

and prints

-6.938893903907228377648e-18
-3.388131789017201356273e-21

on my system (32-bit Ubuntu 12.04), ie the results for long double are slightly closer than the results for double. The commenters above are correct, though, that if you're worrying about this level of precision you probably have larger problems; is the rest of your algorithm stable to this level of precision?

In order to dig deeper (which probably isn't worth it except for intellectual curiosity) you'd have to find out more about the implementation of these two functions in the system libraries for your particular OS.