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Tolerance-based comparisons

Assertions in the Unit Test Framework use two kinds of comparison. For u being close to zero with absolute tolerance eps:

abs(u) <= eps; // (abs)

For u and v being close with relative tolerance eps:

   abs(u - v)/abs(u) <= eps
&& abs(u - v)/abs(v) <= eps; // (rel)

For rationale for choosing these formulae, see section Floating point comparison algorithms.

Assertion BOOST_TEST (when comparing floating-point numbers) uses the following algorithm:

[Note] Note

Therefore in order to check if a number is close to zero with tolerance, you need to type:

BOOST_TEST(v == T(0), tt::tolerance(eps));

The compatibility assertions BOOST_<level>_CLOSE and BOOST_<level>_CLOSE_FRACTION perform formula (rel).

The compatibility assertion BOOST_<level>_SMALL performs formula (abs).

The Unit Test Framework also provides unary predicate small_with_tolerance and binary predicate predicate close_at_tolerance that implement formula (abs) and (rel) respectively.

Tolerance in operator<

Tolerance-based computations also apply to operator< and other relational operators. The semantics are defined as follows:

[Note] Note

This implies that the exactly one of these: u < v, u == v, u > v, passes with BOOST_TEST at any given tolerance.

[Caution] Caution

Relation less-at-tolerance is not a Strict Weak Ordering as it lacks the transitivity of the equivalence; using it as predicate in std::map or any order-based STL algorithm would result in undefined behavior.