What is the correct/standard way to check if difference is smaller than machine precision?

Question

I often end up in situations where it is necessary to check if the obtained difference is above machine precision. Seems like for this purpose R has a handy variable: .Mac

Answer 1

Definition of a machine.eps: it is the lowest value eps for which 1+eps is not 1

As a rule of thumb (assuming a floating point representation with base 2):
This eps makes the difference for the range 1 .. 2,
for the range 2 .. 4 the precision is 2*eps
and so on.

Unfortunately, there is no good rule of thumb here. It's entirely determined by the needs of your program.

In R we have all.equal as a built in way to test approximate equality. So you could use maybe something like (x)

i <- 0.1
 i <- i + 0.05
 i
if(isTRUE(all.equal(i, .15))) { #code was getting sloppy &went to multiple lines
    cat("i equals 0.15\n") 
} else {
    cat("i does not equal 0.15\n")
}
#i equals 0.15

Google mock has a number of floating point matchers for double precision comparisons, including DoubleEq and DoubleNear. You can use them in an array matcher like this:

ASSERT_THAT(vec, ElementsAre(DoubleEq(0.1), DoubleEq(0.2)));

Update:

Numerical Recipes provide a derivation to demonstrate that using a one-sided difference quotient, sqrt is a good choice of step-size for finite difference approximations of derivatives.

The Wikipedia article site Numerical Recipes, 3rd edition, Section 5.7, which is pages 229-230 (a limited number of page views is available at http://www.nrbook.com/empanel/).

all.equal(target, current,
           tolerance = .Machine$double.eps ^ 0.5, scale = NULL,
           ..., check.attributes = TRUE)

These IEEE floating point arithmetic is a well known limitation of computer arithmetic and is discussed in several places:

• The FAQ in R has a whole question dedicated to it: R FAQ 7.31
  • The R Inferno by Patrick Burns dedicated the first "Circle" to this problem (page 9 onward)
  • Arithmetic Sum Proof Problem asked in Math Meta
  • David Goldberg, "What Every Computer Scientist Should Know About Floating-point Arithmetic," ACM Computing Surveys 23, 1 (1991-03), 5-48 doi>10.1145/103162.103163 (revision also available)
  • The Floating-Point Guide - What Every Programmer Should Know About Floating-Point Arithmetic
  • 0.30000000000000004.com compares floating point arithmetic across programming languages
• Canonical duplicate for "floating point is inaccurate" (a meta discussion about a canonical answer for this issue)
• Several Stack Overflow questions including
  • Why can't decimal numbers be represented exactly in binary?
  • Why are floating point numbers inaccurate?
  • Is floating point math broken?
• Math Tricks Explained by Arthur T. Benjamin

. dplyr::near() is another option for testing if two vectors of floating point numbers are equal.

The function has a built in tolerance parameter: tol = .Machine$double.eps^0.5 that can be adjusted. The default parameter is the same as the default for all.equal().