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The three ways to compare floating-point numbers

 

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Kevin Reid
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The three ways to compare floating-point numbers

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One bit of advice sometimes given to the novice programmer is don't ever compare floating-point numbers for equality, the reason being that floating-point calculations are inexact, and one should use a small epsilon, allowable error, instead, e.g. if (abs(value - 1.0) < 0.0001).

This advice is actually wrong, or rather, overly strong. There is a situation in which it is 100% valid to compare floats, and that is an cache or anything else which is comparing a float with, not a specific constant (in which case the epsilon notion is appropriate), but rather a previous value from the same source; floating-point numbers may be approximations of exact arithmetic, but that doesn't mean you won't get the same result from the same inputs.

So, don't get any bright ideas about outlawing aFloat == anotherFloat.

Unfortunately, there's a case in which the common equality on floats isn't what you want for previous-value comparison anyway: for most definitions of ==, NaN ≠≠ NaN. This definition makes sense for numerics (and is conformant to IEEE floating point specifications), because NaN is “not a number”; it's an error marker, provided as an alternative to exceptions (or rather, floating point error signals/traps/whateveryoucallit) which propagates to the end of your calculation rather than aborting it and requiring immediate error handling, which can be advantageous in both code simplicity and efficiency. So if you think about calculating within the space of “numbers”, then NaN is outside of that. But if you're working in the space of “results of calculations”, then you probably want to see NaN == NaN, but that may not be what you get.

Mathematically, the floating-point comparison is not an equivalence relation, because it is not reflexive on NaN.

(It's also typically the case that 0 == -0, even though positive and negative zero are distinct values. Oh, and NaNs carry data, but I'm not talking about that.)

What to do about it, in a few languages:

JavaScript

Even the === operator does not compare identities rather than numeric values, so if you want to compare NaN you have to do it as a special case. Google Caja handles it this way:

/**
 * Are x and y not observably distinguishable?
 */
function identical(x, y) {
  if (x === y) {
    // 0 === -0, but they are not identical
    return x !== 0 || 1/x === 1/y;
  } else {
    // NaN !== NaN, but they are identical.
    // NaNs are the only non-reflexive value, i.e., if x !== x,
    // then x is a NaN.
    return x !== x && y !== y;
  }
}
Common Lisp

The = operator generally follows the IEEE comparison (if the implementation has NaN at all) and the eql operator does the identical-object comparison.

E

The == operator is guaranteed to be reflexive, and return false for distinguishable objects, so it is appropriate for the “cache-like” use cases, and the <=> operator does conventional !(NaN <=> NaN), 0.0 <=> -0.0 floating-point comparison.

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