Using R and SAS and other softwares, we some times need to pay attention to numerical error. A real number is represented in the software
as a binary, finite length (32 bit) discrete number. This lead to error, Or rounding error.  This error some times are not ignorable.

Example:

Mathematically, we have, for any x, the identity

                                         [ sqrt(x+1) - sqrt(x) ] =  1/ [ sqrt(x+1) + sqrt(x) ]

When x is a large number, for example x=500009999998, R calculates

> x<- 500009999998
> sqrt(x+1) - sqrt(x)
[1] 7.071067e-07
> 1/( sqrt(x+1) + sqrt(x))
[1] 7.070997e-07

Try other large x values, you get similar result (that answers are different). You may argue that this difference is small, until you try the following

> x <- 500009999998
> 666668888880*(sqrt(x+1) - sqrt(x))
[1] 471406
>
> 666668888880/(sqrt(x+1) + sqrt(x))
[1] 471401.4

The difference is 4.6.  

Question: which answer among the two is more acurate? (closer to the true value)?

We can use the R package Rmpfr to calculate the "TRUE" value. This package let you use arbitry length to represent a number, not 32 not 64, can
be any user defined length bit.  We of course are interested when using longer bits to represent a real number.  say   120 bits.


You will find out that the expression using dividing is more acurate, i.e.   666668888880/(sqrt(x+1)+sqrt(x))  is better.