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<title>Relative Error</title>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error">Relative Error</a>
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</h2></div></div></div>
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<p>
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Given an actual value <span class="emphasis"><em>a</em></span> and a found value <span class="emphasis"><em>v</em></span>
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the relative error can be calculated from:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../equations/error2.svg"></span>
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</p>
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<p>
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However the test programs in the library use the symmetrical form:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../equations/error1.svg"></span>
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</p>
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<p>
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which measures <span class="emphasis"><em>relative difference</em></span> and happens to be less
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error 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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</p>
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<p>
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Special care needs to be taken when one value is zero: we could either take
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the absolute error in this case (but that's cheating as the absolute error
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is likely to be very small), or we could assign a value of either 1 or infinity
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to the relative error in this special case. In the test cases for the special
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functions in this library, everything below a threshold is regarded as "effectively
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zero", otherwise the relative error is assigned the value of 1 if only
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one of the terms is zero. The threshold is currently set at <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><>::</span><span class="identifier">min</span><span class="special">()</span></code>: in other words all denormalised numbers
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are regarded as a zero.
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</p>
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<p>
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All the test programs calculate <span class="emphasis"><em>quantized relative error</em></span>,
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whereas the graphs in this manual are produced with the <span class="emphasis"><em>actual error</em></span>.
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The difference is as follows: in the test programs, the test data is rounded
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to the target real type under test when the program is compiled, so the error
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observed will then be a whole number of <span class="emphasis"><em>units in the last place</em></span>
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either rounded up from the actual error, or rounded down (possibly to zero).
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In contrast the <span class="emphasis"><em>true error</em></span> is obtained by extending the
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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 <span class="emphasis"><em>units in
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the last place</em></span>.
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</p>
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<p>
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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 <span class="emphasis"><em>units
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in the last place</em></span> more accurately reflect the number of contaminated
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digits, and that relative error can <span class="emphasis"><em>"wobble"</em></span>
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by a factor of 2 compared to <span class="emphasis"><em>units in the last place</em></span>.
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In other words: two implementations of the same function, whose maximum relative
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errors differ by a factor of 2, can actually be accurate to the same number
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of binary digits. You have been warned!
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</p>
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<h5>
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<a name="math_toolkit.relative_error.h0"></a>
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<span class="phrase"><a name="math_toolkit.relative_error.zero_error"></a></span><a class="link" href="relative_error.html#math_toolkit.relative_error.zero_error">The
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Impossibility of Zero Error</a>
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</h5>
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<p>
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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
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guard digits. However it should be remembered that if the result is a <span class="emphasis"><em>transcendental
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number</em></span> then as a point of principle we can never be sure that the
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result is accurate to more than 1 ulp. This is an example of what <a href="http://en.wikipedia.org/wiki/William_Kahan" target="_top">http://en.wikipedia.org/wiki/William_Kahan</a>
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called <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma</a>:
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consider what happens if the first guard digit is a one, and the remaining
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guard digits are all zero. Do we have a tie or not? Since the only thing we
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can tell about a transcendental number is that its digits have no particular
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pattern, we can never tell if we have a tie, no matter how many guard digits
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we have. Therefore, we can never be completely sure that the result has been
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rounded in the right direction. Of course, transcendental numbers that just
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happen to be a tie - for however many guard digits we have - are extremely
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rare, and get rarer the more guard digits we have, but even so....
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</p>
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<p>
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Refer to the classic text <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
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Every Computer Scientist Should Know About Floating-Point Arithmetic</a>
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for more information.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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