R FAQ

7.31 Why doesn’t R think these numbers are equal?

The only numbers that can be represented exactly in R’s numeric type are integers and 
fractions whose denominator is a power of 2. All other numbers are internally rounded 
to (typically) 53 binary digits accuracy. As a result, two floating point numbers will 
not reliably be equal unless they have been computed by the same algorithm, and not 
always even then. For example

R> a <- sqrt(2)
R> a * a == 2
[1] FALSE
R> a * a - 2
[1] 4.440892e-16
R> print(a * a, digits = 18)
[1] 2.00000000000000044

The function all.equal() compares two objects using a numeric tolerance of .Machine$double.eps ^ 0.5. 
If you want much greater accuracy than this you will need to consider error propagation 
carefully.

A discussion with many easily followed examples is in Appendix G “Computational Precision 
and Floating Point Arithmetic”, pages 753–771 of Statistical Analysis and Data Display: 
An Intermediate Course with Examples in R, Richard M. Heiberger and Burt Holland 
(Springer 2015, second edition). This appendix is a free download from 
https://link.springer.com/content/pdf/bbm%3A978-1-4939-2122-5%2F1.pdf.

For more information, see e.g. David Goldberg (1991), “What Every Computer Scientist 
Should Know About Floating-Point Arithmetic”, ACM Computing Surveys, 23/1, 5–48, also 
available via https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html.

Here is another example, this time using addition:

R> .3 + .6 == .9
[1] FALSE
R> .3 + .6 - .9
[1] -1.110223e-16
R> print(matrix(c(.3,  .6,  .9, .3 + .6)), digits = 18)
                     [,1]
[1,] 0.299999999999999989
[2,] 0.599999999999999978
[3,] 0.900000000000000022
[4,] 0.899999999999999911