doc/background/error.qbk

[section:relative_error Relative Error]

Given an actual value /a/ and a found value /v/ the relative error can be
calculated from:

[equation error2]

However the test programs in the library use the symmetrical form:

[equation error1]

which measures /relative difference/ and happens to be less error
prone in use since we don't have to worry which value is the "true"
result, and which is the experimental one.  It guarantees to return a value
at least as large as the relative error.

Special care needs to be taken when one value is zero: we could either take the
absolute error in this case (but that's cheating as the absolute error is likely
to be very small), or we could assign a value of either 1 or infinity to the
relative error in this special case.  In the test cases for the special functions
in this library, everything below a threshold is regarded as "effectively zero",
otherwise the relative error is assigned the value of 1 if only one of the terms
is zero.  The threshold is currently set at `std::numeric_limits<>::min()`:
in other words all denormalised numbers are regarded as a zero.

All the test programs calculate /quantized relative error/, whereas the graphs
in this manual are produced with the /actual error/.  The difference is as
follows: in the test programs, the test data is rounded to the target real type
under test when the program is compiled,
so the error observed will then be a whole number of /units in the last place/
either rounded up from the actual error, or rounded down (possibly to zero).
In contrast the /true error/ is obtained by extending
the precision of the calculated value, and then comparing to the actual value:
in this case the calculated error may be some fraction of /units in the last place/.

Note that throughout this manual and the test programs the relative error is
usually quoted in units of epsilon. However, remember that /units in the last place/
more accurately reflect the number of contaminated digits, and that relative
error can /"wobble"/ by a factor of 2 compared to /units in the last place/.
In other words: two implementations of the same function, whose
maximum relative errors differ by a factor of 2, can actually be accurate
to the same number of binary digits.  You have been warned!

[h4:zero_error The Impossibility of Zero Error]

For many of the functions in this library, it is assumed that the error is
"effectively zero" if the computation can be done with a number of guard
digits.  However it should be remembered that if the result is a
/transcendental number/
then as a point of principle we can never be sure that the result is accurate
to more than 1 ulp.  This is an example of what
[@http://en.wikipedia.org/wiki/William_Kahan] called
[@http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma]:
consider what happens if the first guard digit is a one, and the remaining guard digits are all zero.
Do we have a tie or not?  Since the only thing we can tell about a transcendental number
is that its digits have no particular pattern, we can never tell if we have a tie,
no matter how many guard digits we have.  Therefore, we can never be completely sure
that the result has been rounded in the right direction.  Of course, transcendental
numbers that just happen to be a tie - for however many guard digits we have - are
extremely rare, and get rarer the more guard digits we have, but even so....

Refer to the classic text
[@http://docs.sun.com/source/806-3568/ncg_goldberg.html What Every Computer Scientist Should Know About Floating-Point Arithmetic]
for more information.

[endsect][/section:relative_error Relative Error]

[/
  Copyright 2006, 2012 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]