How to correctly and standardly compare floats?

Question

Every time I start a new project and when I need to compare some float or double variables I write the code like this one:

if (fabs(prev.min[i] - cur->min[i]) < 0.000001 &&
    fabs(prev.max[i] - cur->max[i]) < 0.000001) {
        continue;
}

Then I want to get rid of these magic variables 0.000001(and 0.00000000001 for double) and fabs, so I write an inline function and some defines:

#define FLOAT_TOL 0.000001

So I wonder if there is any standard way of doing this? May be some standard header file? It would be also nice to have float and double limits(min and max values)

Solution 1

Thanks for your answers, they helped me a lot. I've read these materials:first and second

The answer is to use my own function for relative comparison:

bool areEqualRel(float a, float b, float epsilon) {
    return (fabs(a - b) <= epsilon * std::max(fabs(a), fabs(b)));
}

This is the most suitable solution for my needs. However I've wrote some tests and other comparison methods. I hope this will be useful for somebody. areEqualRel passes these tests, others don't.

#include <iostream>
#include <limits>
#include <algorithm>

using std::cout;
using std::max;

bool areEqualAbs(float a, float b, float epsilon) {
    return (fabs(a - b) <= epsilon);
}

bool areEqual(float a, float b, float epsilon) {
    return (fabs(a - b) <= epsilon * std::max(1.0f, std::max(a, b)));
}

bool areEqualRel(float a, float b, float epsilon) {
    return (fabs(a - b) <= epsilon * std::max(fabs(a), fabs(b)));
}

int main(int argc, char *argv[])
{
    cout << "minimum: " << FLT_MIN      << "\n";
    cout << "maximum: " << FLT_MAX      << "\n";
    cout << "epsilon: " << FLT_EPSILON  << "\n";

    float a = 0.0000001f;
    float b = 0.0000002f;
    if (areEqualRel(a, b, FLT_EPSILON)) {
        cout << "are equal a: " << a << " b: " << b << "\n";
    }
    a = 1000001.f;
    b = 1000002.f;
    if (areEqualRel(a, b, FLT_EPSILON)) {
        cout << "are equal a: " << a << " b: " << b << "\n";
    }
}

Solution 2

From The Floating-Point Guide:

This is a bad way to do it because a fixed epsilon chosen because it “looks small” could actually be way too large when the numbers being compared are very small as well. The comparison would return “true” for numbers that are quite different. And when the numbers are very large, the epsilon could end up being smaller than the smallest rounding error, so that the comparison always returns “false”.

The problem with the "magic number" here is not that it's hardcoded but that it's "magic": you didn't really have a reason for choosing 0.000001 over 0.000005 or 0.0000000000001, did you? Note that float can approximately represent the latter and still smaller values - it's just about 7 decimals of precision after the first nonzero digit!

If you're going to use a fixed epsilon, you should really choose it according to the requirements of the particular piece of code where you use it. The alternative is to use a relative error margin (see link at the top for details) or, even better, or compare the floats as integers.

Solution 3

The Standard provides an epsilon value. It's in <limits> and you can access the value by std::numeric_limits<float>::epsilon and std::numeric_limits<double>::epsilon. There are other values in there, but I didn't check what exactly is.

Solution 4

You can use std::nextafter for testing two double with the smallest epsilon on a value (or a factor of the smallest epsilon).

bool nearly_equal(double a, double b)
{
  return std::nextafter(a, std::numeric_limits<double>::lowest()) <= b
    && std::nextafter(a, std::numeric_limits<double>::max()) >= b;
}

bool nearly_equal(double a, double b, int factor /* a factor of epsilon */)
{
  double min_a = a - (a - std::nextafter(a, std::numeric_limits<double>::lowest())) * factor;
  double max_a = a + (std::nextafter(a, std::numeric_limits<double>::max()) - a) * factor;

  return min_a <= b && max_a >= b;
}

Solution 5

Here is a c++11 implementation of @geotavros 's solution. It makes use of the new std::numeric_limits<T>::epsilon() function and the fact that std::fabs() and std::fmax() now have overloads for float, double and long float.

template<typename T>
static bool AreEqual(T f1, T f2) { 
  return (std::fabs(f1 - f2) <= std::numeric_limits<T>::epsilon() * std::fmax(std::fabs(f1), std::fabs(f2)));
}

View more solutions

Comments

[rubenvb]

+1: nice, but a header copy-paste isn't the most helpful IMO.

[Steve Jessop]

Although beware that epsilon is not a straight replacement for the constant tolerance as used by the questioner. It represents out-by-1 in the least significant bit of the value 1.0, so if your values are approximately 2, then it's too small to provide any tolerance. It's quite difficult to use effectively.

[Yttrill]

Agree with @ddyer: OP needs to go and do a course on numerical analysis.

[0xbadf00d]

I just want to show that there are more interesting values in the numeric_limits<float> implementation.

[TonyK]

Surely you mean std::max(fabs(a), fabs(b)), unless all your floats are positive

[Dmitriy]

Thanks TonyK, you're correct, I haven't time to write a complete unit test to see it. I've fixed areEqualRel in my post

[Chris Frederick]

For what it's worth, Bruce Dawson has mentioned that his article on comparing floating-point numbers is now obsolete, and that readers should refer to the 2012 edition instead.

[Michael Borgwardt]

@Chris Frederick: thanks, I'll add a link to that edition to the website

[Pascal Cuoq]

Where did you get the comment “smallest float value such that 1.0+DBL_EPSILON != 1.0” from? This is the wrong phrase to define DBL_EPSILON. blog.frama-c.com/index.php?post/2013/05/09/FLT_EPSILON

[0xbadf00d]

I've got the comment from the implementation in Visual Studio 2012. Didn't think about until your post.

[Gauthier]

How about unit tests? If I'm testing an algorithm and I want to check that the result, with given input values, is close to the expected (float) value?

[ddyer]

Good question, with no simple answer. If you're just checking for gross errors in the algorithm, then I suppose a slop factor is a good place to start. Other tests would involve feeding data designed to trigger problems, such as using 2^32-1 as an integer input. More generally, you'd probably plot the differences between your implementation and a reference standard, looking for evidence of divergence.

[Rick Ramstetter]

This is true only for a subset of use cases. In my current case, float comparisons are being made to debounce values coming from an A2D with an epsilon chosen to reflect that A2D's properties.

[JHBonarius]

nice, but probably doesn't work for special floats like subnormal numbers.