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26 | <div class="titlepage"><div><div><h2 class="title" style="clear: both"> | |
27 | <a name="math_toolkit.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error">Relative Error</a> | |
28 | </h2></div></div></div> | |
29 | <p> | |
30 | Given an actual value <span class="emphasis"><em>a</em></span> and a found value <span class="emphasis"><em>v</em></span> | |
31 | the relative error can be calculated from: | |
32 | </p> | |
33 | <p> | |
34 | <span class="inlinemediaobject"><img src="../../equations/error2.svg"></span> | |
35 | </p> | |
36 | <p> | |
37 | However the test programs in the library use the symmetrical form: | |
38 | </p> | |
39 | <p> | |
40 | <span class="inlinemediaobject"><img src="../../equations/error1.svg"></span> | |
41 | </p> | |
42 | <p> | |
43 | which measures <span class="emphasis"><em>relative difference</em></span> and happens to be less | |
44 | error prone in use since we don't have to worry which value is the "true" | |
45 | result, and which is the experimental one. It guarantees to return a value | |
46 | at least as large as the relative error. | |
47 | </p> | |
48 | <p> | |
49 | Special care needs to be taken when one value is zero: we could either take | |
50 | the absolute error in this case (but that's cheating as the absolute error | |
51 | is likely to be very small), or we could assign a value of either 1 or infinity | |
52 | to the relative error in this special case. In the test cases for the special | |
53 | functions in this library, everything below a threshold is regarded as "effectively | |
54 | zero", otherwise the relative error is assigned the value of 1 if only | |
55 | 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 | |
56 | are regarded as a zero. | |
57 | </p> | |
58 | <p> | |
59 | All the test programs calculate <span class="emphasis"><em>quantized relative error</em></span>, | |
60 | whereas the graphs in this manual are produced with the <span class="emphasis"><em>actual error</em></span>. | |
61 | The difference is as follows: in the test programs, the test data is rounded | |
62 | to the target real type under test when the program is compiled, so the error | |
63 | observed will then be a whole number of <span class="emphasis"><em>units in the last place</em></span> | |
64 | either rounded up from the actual error, or rounded down (possibly to zero). | |
65 | In contrast the <span class="emphasis"><em>true error</em></span> is obtained by extending the | |
66 | precision of the calculated value, and then comparing to the actual value: | |
67 | in this case the calculated error may be some fraction of <span class="emphasis"><em>units in | |
68 | the last place</em></span>. | |
69 | </p> | |
70 | <p> | |
71 | Note that throughout this manual and the test programs the relative error is | |
72 | usually quoted in units of epsilon. However, remember that <span class="emphasis"><em>units | |
73 | in the last place</em></span> more accurately reflect the number of contaminated | |
74 | digits, and that relative error can <span class="emphasis"><em>"wobble"</em></span> | |
75 | by a factor of 2 compared to <span class="emphasis"><em>units in the last place</em></span>. | |
76 | In other words: two implementations of the same function, whose maximum relative | |
77 | errors differ by a factor of 2, can actually be accurate to the same number | |
78 | of binary digits. You have been warned! | |
79 | </p> | |
80 | <h5> | |
81 | <a name="math_toolkit.relative_error.h0"></a> | |
82 | <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 | |
83 | Impossibility of Zero Error</a> | |
84 | </h5> | |
85 | <p> | |
86 | For many of the functions in this library, it is assumed that the error is | |
87 | "effectively zero" if the computation can be done with a number of | |
88 | guard digits. However it should be remembered that if the result is a <span class="emphasis"><em>transcendental | |
89 | number</em></span> then as a point of principle we can never be sure that the | |
90 | 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> | |
91 | 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>: | |
92 | consider what happens if the first guard digit is a one, and the remaining | |
93 | guard digits are all zero. Do we have a tie or not? Since the only thing we | |
94 | can tell about a transcendental number is that its digits have no particular | |
95 | pattern, we can never tell if we have a tie, no matter how many guard digits | |
96 | we have. Therefore, we can never be completely sure that the result has been | |
97 | rounded in the right direction. Of course, transcendental numbers that just | |
98 | happen to be a tie - for however many guard digits we have - are extremely | |
99 | rare, and get rarer the more guard digits we have, but even so.... | |
100 | </p> | |
101 | <p> | |
102 | Refer to the classic text <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What | |
103 | Every Computer Scientist Should Know About Floating-Point Arithmetic</a> | |
104 | for more information. | |
105 | </p> | |
106 | </div> | |
107 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
108 | <td align="left"></td> | |
109 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
110 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
111 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
112 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
113 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
114 | 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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