Managing floating point accuracy

Question

I'm struggling with issues re. floating point accuracy, and could not find a solution.

Here is a short example:

aa<-c(99.93029, 0.0697122)
aa
[1] 99.9302900  0.0697122
aa[1]
99.93029
print(aa[1],digits=20)
99.930289999999999

It would appear that, upon storing the vector, R converted the numbers to something with a slightly different internal representation (yes, I have read circle 1 of the "R inferno" and similar material).

How can I force R to store the input values exactly "as is", with no modification?

In my case, my problem is that the values are processed in such a way that the small errors very quickly grow:

aa[2]/(100-aa[1])*100
[1] 100.0032 ## Should be 100, of course !
print(aa[2]/(100-aa[1])*100,digits=20)
[1] 100.00315593171625

So I need to find a way to get my normalization right.

Thanks

PS- There are many questions on this site and elsewhere, discussing the issue of apparent loss of precision, i.e. numbers displayed incorrectly (but stored right). Here, for instance: How to stop read.table from rounding numbers with different degrees of precision in R? This is a distinct issue, as the number is stored incorrectly (but displayed right).

(R version 3.2.1 (2015-06-18), win 7 x64)

Answer 1

Floating point precision has always generated lots of confusion. The crucial idea to remember is: when you work with doubles, there is no way to store each real number "as is", or "exactly right" -- the best you can store is the closest available approximation. So when you type (in R or any other modern language) something like x = 99.93029, you'll get this number represented by 99.930289999999999.

Now when you expect a + b to be "exactly 100", you're being inaccurate in terms. The best you can get is "100 up to N digits after the decimal point" and hope that N is big enough. In your case it would be correct to say 99.9302900 + 0.0697122 is 100 with 5 decimal points of accuracy. Naturally, by multiplying that equality by 10^k you'll lose additional k digits of accuracy.

So, there are two solutions here:

a. To get more precision in the output, provide more precision in the input.

bb <- c(99.93029, 0.06971) 
print(bb[2]/(100-bb[1])*100, digits = 20)
[1] 99.999999999999119

b. If double precision not enough (can happen in complex algorithms), use packages that provide extra numeric precision operations. For instance, package gmp.