I found Joseph Darcy's recent post "Notions of Floating-Point Equality" interesting, informative, and even entertaining. If you've done scientific programming, you're probably familiar with the problems of floating point uncertainties, significant digits, divisions by zero that really wouldn't happen if you were doing the calculation with pencil and paper, tests of whether the values stored in two memory locations are equal, etc. It's a pain in the neck, really, dealing with these issues in a program.
So, what does the == operator really mean?
I'm kind of reminded of a famous quote from a former United States president. When asked a particular question, his response was: "It depends on what the meaning of the word 'is' is."
So what is the meaning of is? Or of ==?
Joe gets right to the point, in saying:
the "==" operator defined by IEEE 754 and used by Java for comparing floating-point values (JLSv3 §15.21.1) is not an equivalence relation. Equivalence relations satisfy three properties, reflexivity (something is equivalent to itself), symmetry (if a is equivalent to b, b is equivalent to a), and transitivity (if a is equivalent to b and b is equivalent to c, then a is equivalent to c).
One of the simplest tests in scientific programming for detecting numeric anomalies is to check if a value is equal to itself. Why is this useful? Because NaN (Not a Number) is not equal to itself. If variable X is NaN, then a test in Java (and other languages that implement the IEEE 754 standard) that compares X to itself returns a false conditional result:
x = [complicated equation involving divisions by the difference
between large numbers that are close to one another and
trigonometric operations involving of angles that are very
close to one another, etc.]
if (x == x) {
// keep processing
} else {
System.out.println("Cancelling processing: looks like we have a NaN");
}
So, why is Joe Darcy writing about floating point equality? It's related to a BigDecimal discussion that took place on the Project Coin mailing list:
While subtle, the different notions of numerical equality each serve a useful purpose and knowing which notion is appropriate for a given task is an important factor in writing correct programs.
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