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- <title>Numeric<T></title>
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- <td><h2>Safe Numerics</h2></td>
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- <div class="section">
- <div class="titlepage"><div><div><h3 class="title">
- <a name="safe_numerics.numeric"></a>Numeric<T></h3></div></div></div>
- <div class="toc"><dl class="toc">
- <dt><span class="section"><a href="numeric.html#safe_numerics.numeric.description">Description</a></span></dt>
- <dt><span class="section"><a href="numeric.html#idm130204085440">Notation</a></span></dt>
- <dt><span class="section"><a href="numeric.html#idm130204078192">Associated Types</a></span></dt>
- <dt><span class="section"><a href="numeric.html#idm130204072896">Valid Expressions</a></span></dt>
- <dt><span class="section"><a href="numeric.html#idm130203973088">Models</a></span></dt>
- <dt><span class="section"><a href="numeric.html#idm130203971280">Header</a></span></dt>
- <dt><span class="section"><a href="numeric.html#idm130203969088">Note on Usage of <code class="computeroutput">std::numeric_limits</code></a></span></dt>
- </dl></div>
- <div class="section">
- <div class="titlepage"><div><div><h4 class="title">
- <a name="safe_numerics.numeric.description"></a>Description</h4></div></div></div>
- <p>A type is Numeric if it has the properties of a number.</p>
- <p>More specifically, a type T is Numeric if there exists a
- specialization of <code class="computeroutput">std::numeric_limits<T></code>. See the
- documentation for the standard library class <code class="computeroutput">numeric_limits</code>.
- The standard library includes such specializations for all the built-in
- numeric types. Note that this concept is distinct from the C++ standard
- library type traits <code class="computeroutput">is_integral</code> and
- <code class="computeroutput">is_arithmetic</code>. These latter fulfill the requirement of the
- concept Numeric. But there are types T which fulfill this concept for
- which <code class="computeroutput">is_arithmetic<T>::value == false</code>. For example see
- <code class="computeroutput">safe_signed_integer<int></code>.</p>
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h4 class="title">
- <a name="idm130204085440"></a>Notation</h4></div></div></div>
- <div class="informaltable"><table class="table">
- <colgroup>
- <col align="left">
- <col align="left">
- </colgroup>
- <tbody>
- <tr>
- <td align="left"><code class="computeroutput">T, U, V</code></td>
- <td align="left">A type that is a model of Numeric</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t, u</code></td>
- <td align="left">An object of a type modeling Numeric</td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h4 class="title">
- <a name="idm130204078192"></a>Associated Types</h4></div></div></div>
- <div class="informaltable"><table class="table">
- <colgroup>
- <col align="left">
- <col align="left">
- </colgroup>
- <tbody><tr>
- <td align="left"><code class="computeroutput">std::numeric_limits<T></code></td>
- <td align="left">The numeric_limits class template provides a C++ program
- with information about various properties of the implementation's
- representation of the arithmetic types. See C++ standard
- 18.3.2.2.</td>
- </tr></tbody>
- </table></div>
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h4 class="title">
- <a name="idm130204072896"></a>Valid Expressions</h4></div></div></div>
- <p>In addition to the expressions defined in <a href="http://www.sgi.com/tech/stl/Assignable.html" target="_top">Assignable</a> the
- following expressions must be valid. Any operations which result in
- integers which cannot be represented as some Numeric type will throw an
- exception.</p>
- <div class="table">
- <a name="idm130204071072"></a><p class="title"><b>Table 1. General</b></p>
- <div class="table-contents"><table class="table" summary="General">
- <colgroup>
- <col align="left">
- <col align="left">
- <col align="left">
- </colgroup>
- <thead><tr>
- <th align="left">Expression</th>
- <th align="left">Return Type</th>
- <th align="left">Return Value</th>
- </tr></thead>
- <tbody>
- <tr>
- <td align="left"><code class="computeroutput">std::numeric_limits<T>::is_bounded
- </code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> or <code class="computeroutput">false</code>
- </td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">std::numeric_limits<T>::is_integer</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> or <code class="computeroutput">false</code>
- </td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">std::numeric_limits<T>::is_signed</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> or <code class="computeroutput">false</code>
- </td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">std::numeric_limits<T>::is_specialized
- </code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left"><code class="computeroutput">true</code></td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <p><br class="table-break"></p>
- <div class="table">
- <a name="idm130204050048"></a><p class="title"><b>Table 2. Unary Operators</b></p>
- <div class="table-contents"><table class="table" summary="Unary Operators">
- <colgroup>
- <col align="left">
- <col align="left">
- <col align="left">
- </colgroup>
- <thead><tr>
- <th align="left">Expression</th>
- <th align="left">Return Type</th>
- <th align="left">Semantics</th>
- </tr></thead>
- <tbody>
- <tr>
- <td align="left"><code class="computeroutput">-t</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">Invert sign</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">+t</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">unary plus - a no op</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t--</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">post decrement</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t++</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">post increment</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">--t</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">pre decrement</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">++t</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">pre increment</td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <br class="table-break"><div class="table">
- <a name="idm130204029296"></a><p class="title"><b>Table 3. Binary Operators</b></p>
- <div class="table-contents"><table class="table" summary="Binary Operators">
- <colgroup>
- <col align="left">
- <col align="left">
- <col align="left">
- </colgroup>
- <thead><tr>
- <th align="left">Expression</th>
- <th align="left">Return Type</th>
- <th align="left">Semantics</th>
- </tr></thead>
- <tbody>
- <tr>
- <td align="left"><code class="computeroutput">t - u</code></td>
- <td align="left"><code class="computeroutput">V</code></td>
- <td align="left">subtract u from t</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t + u</code></td>
- <td align="left"><code class="computeroutput">V</code></td>
- <td align="left">add u to t</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t * u</code></td>
- <td align="left"><code class="computeroutput">V</code></td>
- <td align="left">multiply t by u</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t / u</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">divide t by u</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t % u</code></td>
- <td align="left"><code class="computeroutput">T</code></td>
- <td align="left">t modulus u</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t < u</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> if t less than u, <code class="computeroutput">false</code>
- otherwise</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t <= u</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> if t less than or equal to u,
- <code class="computeroutput">false</code> otherwise</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t > u</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> if t greater than u, <code class="computeroutput">false</code>
- otherwise</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t >= u</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> if t greater than or equal to u,
- <code class="computeroutput">false</code> otherwise</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t == u</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> if t equal to u, <code class="computeroutput">false</code>
- otherwise</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t != u</code></td>
- <td align="left"><code class="computeroutput">bool</code></td>
- <td align="left">
- <code class="computeroutput">true</code> if t not equal to u, <code class="computeroutput">false</code>
- otherwise</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t = u</code></td>
- <td align="left"><code class="computeroutput"><code class="computeroutput">T</code></code></td>
- <td align="left">assign value of u to t</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t += u</code></td>
- <td align="left"><code class="computeroutput"><code class="computeroutput">T</code></code></td>
- <td align="left">add u to t and assign to t</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t -= u</code></td>
- <td align="left"><code class="computeroutput"><code class="computeroutput">T</code></code></td>
- <td align="left">subtract u from t and assign to t</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t *= u</code></td>
- <td align="left"><code class="computeroutput"><code class="computeroutput">T</code></code></td>
- <td align="left">multiply t by u and assign to t</td>
- </tr>
- <tr>
- <td align="left"><code class="computeroutput">t /= u</code></td>
- <td align="left"><code class="computeroutput"><code class="computeroutput">T</code></code></td>
- <td align="left">divide t by u and assign to t</td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <br class="table-break">
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h4 class="title">
- <a name="idm130203973088"></a>Models</h4></div></div></div>
- <p><code class="computeroutput">int, float, safe_signed_integer<int>,
- safe_signed_range<int>, checked_result<int>,
- etc.</code></p>
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h4 class="title">
- <a name="idm130203971280"></a>Header</h4></div></div></div>
- <p><a href="../../include/concept/numeric.hpp" target="_top"><code class="computeroutput">#include
- <boost/numeric/safe_numerics/concepts/numeric.hpp>
- </code></a></p>
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h4 class="title">
- <a name="idm130203969088"></a>Note on Usage of <code class="computeroutput">std::numeric_limits</code>
- </h4></div></div></div>
- <p>We define the word "Numeric" in terms of the operations which are
- supported by "Numeric" types. This is in line with the current and
- historical usage of the word "concept" in the context of C++. It is also
- common to define compile time predicates such as
- "<code class="computeroutput">is_numeric<T></code>" to permit one to include expressions in
- his code which will generated a compile time error if the specified type
- (T) does not support the operations required. But this is not always true.
- In the C++ standard library there is a predicate
- <code class="computeroutput">is_arithmetic<T></code> whose name might suggest that it
- should return <code class="computeroutput">true</code> for any type which supports the operations
- above. But this is not the case. The standard defines
- <code class="computeroutput">is_arithmetic<T></code> as <code class="computeroutput">true</code> for any of the
- builtin types <code class="computeroutput">int</code>, <code class="computeroutput">long</code>, <code class="computeroutput">float</code>,
- <code class="computeroutput">double</code>, etc and <code class="computeroutput">false</code> for any other types. So
- even if a user defined type U were to support the operations above,
- <code class="computeroutput">is_arithmetic<U></code> would still return <code class="computeroutput">false</code>.
- This is quite unintuitive and not a good match for our purposes. Hence we
- define our own term "Numeric" to designate any type T which:</p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem"><p>Supports the operations above</p></li>
- <li class="listitem"><p>Specializes the standard type numeric_limits</p></li>
- </ul></div>
- <p>while following the C++ standard in using
- <code class="computeroutput">is_arithmetic<T></code>, <code class="computeroutput">is_integral<T></code> to
- detect specific types only. The standard types are useful in various
- aspects of the implementation - which of course is done in terms of the
- standard types.</p>
- <p>This in turn raises another question: Is it "legal" to specialize
- <code class="computeroutput">std::numeric_limits</code> for one's own types such as
- <code class="computeroutput">safe<int></code>. In my view the standard is ambiguous on
- this. See various interpretations: </p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem"><p><a href="https://stackoverflow.com/questions/16122912/is-it-ok-to-specialize-stdnumeric-limitst-for-user-defined-number-like-class" target="_top">is-it-ok-to-specialize-stdnumeric-limitst-for-user-defined-number-like-class</a></p></li>
- <li class="listitem"><p><a href="https://en.cppreference.com/w/cpp/types/numeric_limits" target="_top">cppreference.com/w/cpp/types/numeric_limits</a></p></li>
- </ul></div>
- <p>In any case, it seems pretty clear that no harm will come of it. In
- spite of the consideration given to this issue, it turns out that the
- found no real need to implement these predicates. For example, there is no
- "is_numeric<T>" implemented as part of the safe numerics library.
- This may change in the future though. Even if not used, defining and
- maintaining these type requirements in this document has been very
- valuable in keeping the concepts and code more unified and
- understandable.</p>
- <p>Remember that above considerations apply to other numeric types used
- in this library even though we don't explicitly repeat this information
- for every case.</p>
- </div>
- </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 © 2012-2018 Robert Ramey<p><a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">Subject to Boost
- Software License</a></p>
- </div></td>
- </tr></table>
- <hr>
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