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74 lines
3.8 KiB
Plaintext
74 lines
3.8 KiB
Plaintext
[section:relative_error Relative Error]
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Given an actual value /a/ and a found value /v/ the relative error can be
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calculated from:
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[equation error2]
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However the test programs in the library use the symmetrical form:
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[equation error1]
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which measures /relative difference/ and happens to be less error
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prone in use since we don't have to worry which value is the "true"
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result, and which is the experimental one. It guarantees to return a value
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at least as large as the relative error.
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Special care needs to be taken when one value is zero: we could either take the
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absolute error in this case (but that's cheating as the absolute error is likely
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to be very small), or we could assign a value of either 1 or infinity to the
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relative error in this special case. In the test cases for the special functions
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in this library, everything below a threshold is regarded as "effectively zero",
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otherwise the relative error is assigned the value of 1 if only one of the terms
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is zero. The threshold is currently set at `std::numeric_limits<>::min()`:
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in other words all denormalised numbers are regarded as a zero.
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All the test programs calculate /quantized relative error/, whereas the graphs
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in this manual are produced with the /actual error/. The difference is as
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follows: in the test programs, the test data is rounded to the target real type
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under test when the program is compiled,
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so the error observed will then be a whole number of /units in the last place/
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either rounded up from the actual error, or rounded down (possibly to zero).
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In contrast the /true error/ is obtained by extending
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the precision of the calculated value, and then comparing to the actual value:
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in this case the calculated error may be some fraction of /units in the last place/.
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Note that throughout this manual and the test programs the relative error is
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usually quoted in units of epsilon. However, remember that /units in the last place/
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more accurately reflect the number of contaminated digits, and that relative
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error can /"wobble"/ by a factor of 2 compared to /units in the last place/.
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In other words: two implementations of the same function, whose
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maximum relative errors differ by a factor of 2, can actually be accurate
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to the same number of binary digits. You have been warned!
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[h4:zero_error The Impossibility of Zero Error]
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For many of the functions in this library, it is assumed that the error is
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"effectively zero" if the computation can be done with a number of guard
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digits. However it should be remembered that if the result is a
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/transcendental number/
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then as a point of principle we can never be sure that the result is accurate
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to more than 1 ulp. This is an example of what
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[@http://en.wikipedia.org/wiki/William_Kahan] called
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[@http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma]:
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consider what happens if the first guard digit is a one, and the remaining guard digits are all zero.
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Do we have a tie or not? Since the only thing we can tell about a transcendental number
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is that its digits have no particular pattern, we can never tell if we have a tie,
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no matter how many guard digits we have. Therefore, we can never be completely sure
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that the result has been rounded in the right direction. Of course, transcendental
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numbers that just happen to be a tie - for however many guard digits we have - are
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extremely rare, and get rarer the more guard digits we have, but even so....
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Refer to the classic text
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[@http://docs.sun.com/source/806-3568/ncg_goldberg.html What Every Computer Scientist Should Know About Floating-Point Arithmetic]
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for more information.
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[endsect][/section:relative_error Relative Error]
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[/
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Copyright 2006, 2012 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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