...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

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 inequalities 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:

- Type promotion
- Arithmetic operations
- Conversion from a decimal presentation to a binary presentation
- Non-arithmetic operation

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 | |
---|---|

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**

- The art of computer programming (vol II)
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.)

- Rounding near zero, in Advanced Arithmetic for the Digital Computer
Ulrich W. Kulisch, 2002, Springer, Inc., ISBN 0-201-89684-2, Springer; 1st edition

**Periodicals**

- Comparing Floats: How To Determine if Floating Quantities Are Close Enough Once a Tolerance Has Been Reached
Alberto Squassabia, in C++ Report (March 2000)

- The Journeyman's Shop: Trap Handlers, Sticky Bits, and Floating-Point Comparisons
Pete Becker, in C/C++ Users Journal (December 2000)

**Publications**

- What Every Computer Scientist Should Know About Floating-Point Arithmetic
David Goldberg, pages 150-230, in Computing Surveys (March 1991), Association for Computing Machinery, Inc.

- From Rounding Error Estimation to Automatic Correction with Automatic Differentiation
Philippe Langlois, Technical report, INRIA

- William Kahan home page
Lots of information on floating point arithmetics.