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- <?xml version="1.0" encoding="UTF-8"?>
- <!DOCTYPE section PUBLIC "-//Boost//DTD BoostBook XML V1.1//EN"
- "http://www.boost.org/tools/boostbook/dtd/boostbook.dtd">
- <section id="safe_numerics.numeric">
- <title>Numeric<T></title>
- <?dbhtml stop-chunking?>
- <section id="safe_numerics.numeric.description">
- <title>Description</title>
- <para>A type is Numeric if it has the properties of a number.</para>
- <para>More specifically, a type T is Numeric if there exists a
- specialization of <code>std::numeric_limits<T></code>. See the
- documentation for the standard library class <code>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>is_integral</code> and
- <code>is_arithmetic</code>. These latter fulfill the requirement of the
- concept Numeric. But there are types T which fulfill this concept for
- which <code>is_arithmetic<T>::value == false</code>. For example see
- <code>safe_signed_integer<int></code>.</para>
- </section>
- <section>
- <title>Notation</title>
- <informaltable>
- <tgroup cols="2" colsep="1" rowsep="1">
- <colspec align="left"/>
- <colspec align="left" colwidth="3*"/>
- <tbody>
- <row>
- <entry><code>T, U, V</code></entry>
- <entry>A type that is a model of Numeric</entry>
- </row>
- <row>
- <entry><code>t, u</code></entry>
- <entry>An object of a type modeling Numeric</entry>
- </row>
- </tbody>
- </tgroup>
- </informaltable>
- </section>
- <section>
- <title>Associated Types</title>
- <informaltable>
- <tgroup cols="2">
- <colspec align="left"/>
- <colspec align="left" colwidth="3*"/>
- <tbody>
- <row>
- <entry><code>std::numeric_limits<T></code></entry>
- <entry>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.</entry>
- </row>
- </tbody>
- </tgroup>
- </informaltable>
- </section>
- <section>
- <title>Valid Expressions</title>
- <para>In addition to the expressions defined in <ulink
- url="http://www.sgi.com/tech/stl/Assignable.html">Assignable</ulink> the
- following expressions must be valid. Any operations which result in
- integers which cannot be represented as some Numeric type will throw an
- exception.<table>
- <title>General</title>
- <tgroup cols="3">
- <colspec align="left" colwidth="2*"/>
- <colspec align="left" colwidth="1*"/>
- <colspec align="left" colwidth="2*"/>
- <thead>
- <row>
- <entry align="left">Expression</entry>
- <entry>Return Type</entry>
- <entry>Return Value</entry>
- </row>
- </thead>
- <tbody>
- <row>
- <entry><code>std::numeric_limits<T>::is_bounded
- </code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> or <code>false</code></entry>
- </row>
- <row>
- <entry><code>std::numeric_limits<T>::is_integer</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> or <code>false</code></entry>
- </row>
- <row>
- <entry><code>std::numeric_limits<T>::is_signed</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> or <code>false</code></entry>
- </row>
- <row>
- <entry><code>std::numeric_limits<T>::is_specialized
- </code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code></entry>
- </row>
- </tbody>
- </tgroup>
- </table></para>
- <table>
- <title>Unary Operators</title>
- <tgroup cols="3">
- <colspec align="left"/>
- <colspec align="left"/>
- <colspec align="left" colwidth="3*"/>
- <thead>
- <row>
- <entry align="left">Expression</entry>
- <entry>Return Type</entry>
- <entry>Semantics</entry>
- </row>
- </thead>
- <tbody>
- <row>
- <entry><code>-t</code></entry>
- <entry><code>T</code></entry>
- <entry>Invert sign</entry>
- </row>
- <row>
- <entry><code>+t</code></entry>
- <entry><code>T</code></entry>
- <entry>unary plus - a no op</entry>
- </row>
- <row>
- <entry><code>t--</code></entry>
- <entry><code>T</code></entry>
- <entry>post decrement</entry>
- </row>
- <row>
- <entry><code>t++</code></entry>
- <entry><code>T</code></entry>
- <entry>post increment</entry>
- </row>
- <row>
- <entry><code>--t</code></entry>
- <entry><code>T</code></entry>
- <entry>pre decrement</entry>
- </row>
- <row>
- <entry><code>++t</code></entry>
- <entry><code>T</code></entry>
- <entry>pre increment</entry>
- </row>
- </tbody>
- </tgroup>
- </table>
- <table>
- <title>Binary Operators</title>
- <tgroup cols="3">
- <colspec align="left"/>
- <colspec align="left"/>
- <colspec align="left" colwidth="3*"/>
- <thead>
- <row>
- <entry align="left">Expression</entry>
- <entry>Return Type</entry>
- <entry>Semantics</entry>
- </row>
- </thead>
- <tbody>
- <row>
- <entry><code>t - u</code></entry>
- <entry><code>V</code></entry>
- <entry>subtract u from t</entry>
- </row>
- <row>
- <entry><code>t + u</code></entry>
- <entry><code>V</code></entry>
- <entry>add u to t</entry>
- </row>
- <row>
- <entry><code>t * u</code></entry>
- <entry><code>V</code></entry>
- <entry>multiply t by u</entry>
- </row>
- <row>
- <entry><code>t / u</code></entry>
- <entry><code>T</code></entry>
- <entry>divide t by u</entry>
- </row>
- <row>
- <entry><code>t % u</code></entry>
- <entry><code>T</code></entry>
- <entry>t modulus u</entry>
- </row>
- <row>
- <entry><code>t < u</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> if t less than u, <code>false</code>
- otherwise</entry>
- </row>
- <row>
- <entry><code>t <= u</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> if t less than or equal to u,
- <code>false</code> otherwise</entry>
- </row>
- <row>
- <entry><code>t > u</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> if t greater than u, <code>false</code>
- otherwise</entry>
- </row>
- <row>
- <entry><code>t >= u</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> if t greater than or equal to u,
- <code>false</code> otherwise</entry>
- </row>
- <row>
- <entry><code>t == u</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> if t equal to u, <code>false</code>
- otherwise</entry>
- </row>
- <row>
- <entry><code>t != u</code></entry>
- <entry><code>bool</code></entry>
- <entry><code>true</code> if t not equal to u, <code>false</code>
- otherwise</entry>
- </row>
- <row>
- <entry><code>t = u</code></entry>
- <entry><code><code>T</code></code></entry>
- <entry>assign value of u to t</entry>
- </row>
- <row>
- <entry><code>t += u</code></entry>
- <entry><code><code>T</code></code></entry>
- <entry>add u to t and assign to t</entry>
- </row>
- <row>
- <entry><code>t -= u</code></entry>
- <entry><code><code>T</code></code></entry>
- <entry>subtract u from t and assign to t</entry>
- </row>
- <row>
- <entry><code>t *= u</code></entry>
- <entry><code><code>T</code></code></entry>
- <entry>multiply t by u and assign to t</entry>
- </row>
- <row>
- <entry><code>t /= u</code></entry>
- <entry><code><code>T</code></code></entry>
- <entry>divide t by u and assign to t</entry>
- </row>
- </tbody>
- </tgroup>
- </table>
- </section>
- <section>
- <title>Models</title>
- <para><code>int, float, safe_signed_integer<int>,
- safe_signed_range<int>, checked_result<int>,
- etc.</code></para>
- </section>
- <section>
- <title>Header</title>
- <para><ulink
- url="../../include/boost/safe_numerics/concept/numeric.hpp"><code>#include
- <boost/numeric/safe_numerics/concepts/numeric.hpp>
- </code></ulink></para>
- </section>
- <section>
- <title>Note on Usage of <code>std::numeric_limits</code></title>
- <para>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>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>is_arithmetic<T></code> whose name might suggest that it
- should return <code>true</code> for any type which supports the operations
- above. But this is not the case. The standard defines
- <code>is_arithmetic<T></code> as <code>true</code> for any of the
- builtin types <code>int</code>, <code>long</code>, <code>float</code>,
- <code>double</code>, etc and <code>false</code> for any other types. So
- even if a user defined type U were to support the operations above,
- <code>is_arithmetic<U></code> would still return <code>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:</para>
- <para><itemizedlist>
- <listitem>
- <para>Supports the operations above</para>
- </listitem>
- <listitem>
- <para>Specializes the standard type numeric_limits</para>
- </listitem>
- </itemizedlist>while following the C++ standard in using
- <code>is_arithmetic<T></code>, <code>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.</para>
- <para>This in turn raises another question: Is it "legal" to specialize
- <code>std::numeric_limits</code> for one's own types such as
- <code>safe<int></code>. In my view the standard is ambiguous on
- this. See various interpretations: <itemizedlist>
- <listitem>
- <para><ulink
- url="https://stackoverflow.com/questions/16122912/is-it-ok-to-specialize-stdnumeric-limitst-for-user-defined-number-like-class">is-it-ok-to-specialize-stdnumeric-limitst-for-user-defined-number-like-class</ulink></para>
- </listitem>
- <listitem>
- <para><ulink
- url="https://en.cppreference.com/w/cpp/types/numeric_limits">cppreference.com/w/cpp/types/numeric_limits</ulink></para>
- </listitem>
- </itemizedlist></para>
- <para>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.</para>
- <para>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.</para>
- </section>
- </section>
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