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            The following is the most obvious way to compare two floating-point values
            u and v
            for being close at a given absolute tolerance epsilon:
          
abs(u - v) <= epsilon; // (1)
            However, in many circumstances, this is not what we want. The same absolute
            tolerance value 0.01 may
            be too small to meaningfully compare two values of magnitude 10e12 and at the same time too little to
            meaningfully compare values of magnitude 10e-12.
            For examples, see Squassabia.
          
            We do not want to apply the same absolute tolerance for huge and tiny
            numbers. Instead, we would like to scale the epsilon
            with u and v. The Unit Test Framework
            implements floating-point comparison algorithm that is based on the solution
            presented in Knuth:
          
abs(u - v) <= epsilon * abs(u) && abs(u - v) <= epsilon * abs(v)); // (2)
            defines a very close with tolerance epsilon
            relationship between u
            and v, while
          
abs(u - v) <= epsilon * abs(u) || abs(u - v) <= epsilon * abs(v); // (3)
            defines a close enough with tolerance epsilon
            relationship between u
            and v.
          
Both relationships are commutative but are not transitive. The relationship defined in (2) is stronger that the relationship defined in (3) since (2) necessarily implies (3).
            The multiplication in the right side of inequations may cause an unwanted
            underflow condition. To prevent this, the implementation is using modified
            version of (2) and (3),
            which scales the checked difference rather than epsilon:
          
abs(u - v)/abs(u) <= epsilon && abs(u - v)/abs(v) <= epsilon; // (4)
abs(u - v)/abs(u) <= epsilon || abs(u - v)/abs(v) <= epsilon; // (5)
            This way all underflow and overflow conditions can be guarded safely.
            The above however, will not work when v
            or u is zero. In such
            cases the solution is to resort to a different algorithm, e.g. (1).
          
            In case of absence of domain specific requirements the value of tolerance
            can be chosen as a sum of the predicted upper limits for "relative
            rounding errors" of compared values. The "rounding" is
            the operation by which a real value 'x' is represented in a floating-point
            format with 'p' binary digits (bits) as the floating-point value X. The "relative rounding error" is
            the difference between the real and the floating point values in relation
            to real value: abs(x-X)/abs(x).
            The discrepancy between real and floating point value may be caused by
            several reasons:
          
The first two operations proved to have a relative rounding error that does not exceed
half_epsilon = half of the 'machine epsilon value'
            for the appropriate floating point type FPT
            [9]. Conversion to binary presentation, sadly, does not have
            such requirement. So we can't assume that float(1.1)
            is close to the real number 1.1
            with tolerance half_epsilon
            for float (though for 11./10 we can). Non-arithmetic operations either
            do not have a predicted upper limit relative rounding errors.
          
| ![[Note]](../../../../../../../../doc/src/images/note.png) | Note | 
|---|---|
| Note that both arithmetic and non-arithmetic operations might also produce others "non-rounding" errors, such as underflow/overflow, division-by-zero or "operation errors". | 
            All theorems about the upper limit of a rounding error, including that
            of half_epsilon, refer
            only to the 'rounding' operation, nothing more. This means that the 'operation
            error', that is, the error incurred by the operation itself, besides
            rounding, isn't considered. In order for numerical software to be able
            to actually predict error bounds, the IEEE754
            standard requires arithmetic operations to be 'correctly or exactly rounded'.
            That is, it is required that the internal computation of a given operation
            be such that the floating point result is the exact result rounded to
            the number of working bits. In other words, it is required that the computation
            used by the operation itself doesn't introduce any additional errors.
            The IEEE754 standard does not require
            same behavior from most non-arithmetic operation. The underflow/overflow
            and division-by-zero errors may cause rounding errors with unpredictable
            upper limits.
          
            At last be aware that half_epsilon
            rules are not transitive. In other words combination of two arithmetic
            operations may produce rounding error that significantly exceeds 2*half_epsilon.
            All in all there are no generic rules on how to select the tolerance
            and users need to apply common sense and domain/ problem specific knowledge
            to decide on tolerance value.
          
To simplify things in most usage cases latest version of algorithm below opted to use percentage values for tolerance specification (instead of fractions of related values). In other words now you use it to check that difference between two values does not exceed x percent.
For more reading about floating-point comparison see references below.
Books
Donald. E. Knuth, 1998, Addison-Wesley Longman, Inc., ISBN 0-201-89684-2, Addison-Wesley Professional; 3rd edition. (The relevant equations are in §4.2.2, Eq. 36 and 37.)
Ulrich W. Kulisch, 2002, Springer, Inc., ISBN 0-201-89684-2, Springer; 1st edition
Periodicals
Alberto Squassabia, in C++ Report (March 2000)
Pete Becker, in C/C++ Users Journal (December 2000)
Publications
David Goldberg, pages 150-230, in Computing Surveys (March 1991), Association for Computing Machinery, Inc.
Philippe Langlois, Technical report, INRIA
Lots of information on floating point arithmetics.