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 <div class="section">
 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
 <a name="math_toolkit.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error">Relative Error</a>
 </h2></div></div></div>
 <p>
       Given an actual value <span class="emphasis"><em>a</em></span> and a found value <span class="emphasis"><em>v</em></span>
       the relative error can be calculated from:
     </p>
 <p>
       <span class="inlinemediaobject"><img src="../../equations/error2.svg"></span>
     </p>
 <p>
       However the test programs in the library use the symmetrical form:
     </p>
 <p>
       <span class="inlinemediaobject"><img src="../../equations/error1.svg"></span>
     </p>
 <p>
       which measures <span class="emphasis"><em>relative difference</em></span> 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.
     </p>
 <p>
       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 <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;&gt;::</span><span class="identifier">min</span><span class="special">()</span></code>: in other words all denormalised numbers
       are regarded as a zero.
     </p>
 <p>
       All the test programs calculate <span class="emphasis"><em>quantized relative error</em></span>,
       whereas the graphs in this manual are produced with the <span class="emphasis"><em>actual error</em></span>.
       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 <span class="emphasis"><em>units in the last place</em></span>
       either rounded up from the actual error, or rounded down (possibly to zero).
       In contrast the <span class="emphasis"><em>true error</em></span> 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 <span class="emphasis"><em>units in
       the last place</em></span>.
     </p>
 <p>
       Note that throughout this manual and the test programs the relative error is
       usually quoted in units of epsilon. However, remember that <span class="emphasis"><em>units
       in the last place</em></span> more accurately reflect the number of contaminated
       digits, and that relative error can <span class="emphasis"><em>"wobble"</em></span>
       by a factor of 2 compared to <span class="emphasis"><em>units in the last place</em></span>.
       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!
     </p>
 <h5>
 <a name="math_toolkit.relative_error.h0"></a>
       <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
       Impossibility of Zero Error</a>
     </h5>
 <p>
       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 <span class="emphasis"><em>transcendental
       number</em></span> 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 <a href="http://en.wikipedia.org/wiki/William_Kahan" target="_top">http://en.wikipedia.org/wiki/William_Kahan</a>
       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>:
       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....
     </p>
 <p>
       Refer to the classic text <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
       Every Computer Scientist Should Know About Floating-Point Arithmetic</a>
       for more information.
     </p>
 </div>
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 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
       Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
         Distributed under the Boost Software License, Version 1.0. (See accompanying
         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>)
       </p>
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	</div>
	<div class="section">
	<div class="titlepage"><div><div><h2 class="title" style="clear: both">
	<a name="math_toolkit.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error">Relative Error</a>
	</h2></div></div></div>
	<p>
	Given an actual value <span class="emphasis"><em>a</em></span> and a found value <span class="emphasis"><em>v</em></span>
	the relative error can be calculated from:
	</p>
	<p>
	<span class="inlinemediaobject"><img src="../../equations/error2.svg"></span>
	</p>
	<p>
	However the test programs in the library use the symmetrical form:
	</p>
	<p>
	<span class="inlinemediaobject"><img src="../../equations/error1.svg"></span>
	</p>
	<p>
	which measures <span class="emphasis"><em>relative difference</em></span> 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.
	</p>
	<p>
	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 <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
	are regarded as a zero.
	</p>
	<p>
	All the test programs calculate <span class="emphasis"><em>quantized relative error</em></span>,
	whereas the graphs in this manual are produced with the <span class="emphasis"><em>actual error</em></span>.
	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 <span class="emphasis"><em>units in the last place</em></span>
	either rounded up from the actual error, or rounded down (possibly to zero).
	In contrast the <span class="emphasis"><em>true error</em></span> 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 <span class="emphasis"><em>units in
	the last place</em></span>.
	</p>
	<p>
	Note that throughout this manual and the test programs the relative error is
	usually quoted in units of epsilon. However, remember that <span class="emphasis"><em>units
	in the last place</em></span> more accurately reflect the number of contaminated
	digits, and that relative error can <span class="emphasis"><em>"wobble"</em></span>
	by a factor of 2 compared to <span class="emphasis"><em>units in the last place</em></span>.
	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!
	</p>
	<h5>
	<a name="math_toolkit.relative_error.h0"></a>
	<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
	Impossibility of Zero Error</a>
	</h5>
	<p>
	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 <span class="emphasis"><em>transcendental
	number</em></span> 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 <a href="http://en.wikipedia.org/wiki/William_Kahan" target="_top">http://en.wikipedia.org/wiki/William_Kahan</a>
	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>:
	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....
	</p>
	<p>
	Refer to the classic text <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
	Every Computer Scientist Should Know About Floating-Point Arithmetic</a>
	for more information.
	</p>
	</div>
	<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
	<td align="left"></td>
	<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
	Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
	Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
	Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
	Distributed under the Boost Software License, Version 1.0. (See accompanying
	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>)
	</p>
	</div></td>
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