dynarmic/docs/comparing-floating-point-numbers.md

Comparing floating point numbers with Catch2

If you are not deeply familiar with them, floating point numbers can be unintuitive. This also applies to comparing floating point numbers for (in)equality.

This page assumes that you have some understanding of both FP, and the meaning of different kinds of comparisons, and only goes over what functionality Catch2 provides to help you with comparing floating point numbers. If you do not have this understanding, we recommend that you first study up on floating point numbers and their comparisons, e.g. by reading this blog post.

Floating point matchers

#include <catch2/matchers/catch_matchers_floating_point.hpp>

Matchers are the preferred way of comparing floating point numbers in Catch2. We provide 3 of them:

• WithinAbs(double target, double margin),
• WithinRel(FloatingPoint target, FloatingPoint eps), and
• WithinULP(FloatingPoint target, uint64_t maxUlpDiff).

WithinRel matcher was introduced in Catch2 2.10.0

As with all matchers, you can combine multiple floating point matchers in a single assertion. For example, to check that some computation matches a known good value within 0.1% or is close enough (no different to 5 decimal places) to zero, we would write this assertion:

    REQUIRE_THAT( computation(input),
        Catch::Matchers::WithinRel(expected, 0.001)
     || Catch::Matchers::WithinAbs(0, 0.000001) );

WithinAbs

WithinAbs creates a matcher that accepts floating point numbers whose difference with target is less-or-equal to the margin. Since float can be converted to double without losing precision, only double overload exists.

REQUIRE_THAT(1.0, WithinAbs(1.2, 0.2));
REQUIRE_THAT(0.f, !WithinAbs(1.0, 0.5));
// Notice that infinity == infinity for WithinAbs
REQUIRE_THAT(INFINITY, WithinAbs(INFINITY, 0));

WithinRel

WithinRel creates a matcher that accepts floating point numbers that are approximately equal to the target with a tolerance of eps. Specifically, it matches if |arg - target| <= eps * max(|arg|, |target|) holds. If you do not specify eps, std::numeric_limits<FloatingPoint>::epsilon * 100 is used as the default.

// Notice that WithinRel comparison is symmetric, unlike Approx's.
REQUIRE_THAT(1.0, WithinRel(1.1, 0.1));
REQUIRE_THAT(1.1, WithinRel(1.0, 0.1));
// Notice that inifnity == infinity for WithinRel
REQUIRE_THAT(INFINITY, WithinRel(INFINITY));

WithinULP

WithinULP creates a matcher that accepts floating point numbers that are no more than maxUlpDiff ULPs away from the target value. The short version of what this means is that there is no more than maxUlpDiff - 1 representable floating point numbers between the argument for matching and the target value.

When using the ULP matcher in Catch2, it is important to keep in mind that Catch2 interprets ULP distance slightly differently than e.g. std::nextafter does.

Catch2's ULP calculation obeys these relations:

• ulpDistance(-x, x) == 2 * ulpDistance(x, 0)
• ulpDistance(-0, 0) == 0 (due to the above)
• ulpDistance(DBL_MAX, INFINITY) == 1
• ulpDistancE(NaN, x) == infinity

Important: The WithinULP matcher requires the platform to use the IEEE-754 representation for floating point numbers.

REQUIRE_THAT( -0.f, WithinULP( 0.f, 0 ) );

Approx

#include <catch2/catch_approx.hpp>

We strongly recommend against using Approx when writing new code. You should be using floating point matchers instead.

Catch2 provides one more way to handle floating point comparisons. It is Approx, a special type with overloaded comparison operators, that can be used in standard assertions, e.g.

REQUIRE(0.99999 == Catch::Approx(1));

Approx supports four comparison operators, ==, !=, <=, >=, and can also be used with strong typedefs over doubles. It can be used for both relative and margin comparisons by using its three customization points. Note that the semantics of this is always that of an or, so if either the relative or absolute margin comparison passes, then the whole comparison passes.

The downside to Approx is that it has a couple of issues that we cannot fix without breaking backwards compatibility. Because Catch2 also provides complete set of matchers that implement different floating point comparison methods, Approx is left as-is, is considered deprecated, and should not be used in new code.

The issues are

• All internal computation is done in doubles, leading to slightly different results if the inputs were floats.
• Approx's relative margin comparison is not symmetric. This means that Approx( 10 ).epsilon(0.1) != 11.1 but Approx( 11.1 ).epsilon(0.1) == 10.
• By default, Approx only uses relative margin comparison. This means that Approx(0) == X only passes for X == 0.

Approx details

If you still want/need to know more about Approx, read on.

Catch2 provides a UDL for Approx; _a. It resides in the Catch::literals namespace, and can be used like this:

using namespace Catch::literals;
REQUIRE( performComputation() == 2.1_a );

Approx has three customization points for the comparison:

• epsilon - epsilon sets the coefficient by which a result can differ from Approx's value before it is rejected. Defaults to std::numeric_limits<float>::epsilon()*100.

Approx target = Approx(100).epsilon(0.01);
100.0 == target; // Obviously true
200.0 == target; // Obviously still false
100.5 == target; // True, because we set target to allow up to 1% difference

• margin - margin sets the absolute value by which a result can differ from Approx's value before it is rejected. Defaults to 0.0.

Approx target = Approx(100).margin(5);
100.0 == target; // Obviously true
200.0 == target; // Obviously still false
104.0 == target; // True, because we set target to allow absolute difference of at most 5

• scale - scale is used to change the magnitude of Approx for the relative check. By default, set to 0.0.

Scale could be useful if the computation leading to the result worked on a different scale than is used by the results. Approx's scale is added to Approx's value when computing the allowed relative margin from the Approx's value.

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