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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&lt;T&gt;</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&lt;T&gt;</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 may types which fulfill the concept of Numeric
for which <code>std::is_arithmetic&lt;T&gt; == false</code>.</para>
<para>There are multiple reasons that we cannot use
std::is_arithmetic&lt;T&gt; to identify number-like types for our
purposes:</para>
<para><itemizedlist>
<listitem>
<para>The main purpose of our concept of Numeric&lt;T&gt; is to
indicate that the range of type T is available to be queried. Since
all specializations of Numeric&lt;T&gt; have a member in
std::numeric_limits&lt;T&gt; we have access to this information when
needed through the members std::numeric_limits&lt;T&gt;::min() and
std::numeric_limits&lt;T&gt;::max().</para>
</listitem>
<listitem>
<para>There are types which do not fulfill std::arithmetic&lt;T&gt;
(that is are not built-in numeric types) but still fulfill the
concept of Numeric&lt;T&gt; (number like "things").</para>
</listitem>
<listitem>
<para>The library creates new result types for every expression
which should also fulfill this concept Numeric. But
is_arithmetic&lt;T&gt; cannot be specialized for user types.</para>
</listitem>
</itemizedlist>So is_arithmetic&lt;T&gt; does not correspond to the same
set of types as Numeric&lt;T&gt; does.</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 a Numeric type</entry>
</row>
<row>
<entry><code>t, u</code></entry>
<entry>An object of a type modeling a Numeric type</entry>
</row>
<row>
<entry><code>OS, IS</code></entry>
<entry>A type that is a model of an output or input stream</entry>
</row>
<row>
<entry><code>os, is</code></entry>
<entry>An object of a type modeling output or input stream</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&lt;T&gt;</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. In the safe_numerics library, a type
is considered Numeric if and only if it it has an entry in the
std::numeric_limits table with the following members. Note that this is
different from the the definition of std::is_arithmetic in that the later
is required to support all valid expressions which Numeric does not
require support for all these expressions but only requires that they be
correctly implemented if they are defined. Also is_arithmetic is only
defined for built in numeric types while Numeric applies to any user types
which "look like" a number.<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>Numeric&lt;T&gt;</code></entry>
<entry><code>true_type</code></entry>
<entry/>
</row>
<row>
<entry><code>Numeric&lt;T&gt;()</code></entry>
<entry><code>bool</code></entry>
<entry>true</entry>
</row>
<row>
<entry><code>std::numeric_limits&lt;T&gt;::is_specialized</code></entry>
<entry><code>bool</code></entry>
<entry><code>true</code></entry>
</row>
<row>
<entry><code>std::numeric_limits&lt;T&gt;::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&lt;T&gt;::is_signed</code></entry>
<entry><code>bool</code></entry>
<entry><code>true</code> or <code>false</code></entry>
</row>
</tbody>
</tgroup>
</table></para>
<para>Any or all of the following unary operators MAY be defined. Any such
defined operators shall implement the semantics as described below</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>
<para>Any or all of the following binary operators MAY be defined. Any
defined operators shall implement the semantics as described bellow</para>
<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 &lt; 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 &lt;= 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 &gt; 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 &gt;= 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>
<row>
<entry><code>os &lt;&lt; t</code></entry>
<entry><code><code>OS &amp;</code></code></entry>
<entry>write contents of t to output stream</entry>
</row>
<row>
<entry><code>is &gt;&gt; t</code></entry>
<entry><code><code>IS &amp;</code></code></entry>
<entry>read contents of an input stream into t</entry>
</row>
</tbody>
</tgroup>
</table>
</section>
<section>
<title>Models</title>
<para><code>int, float, safe_signed_integer&lt;int&gt;,
safe_signed_range&lt;int&gt;, checked_result&lt;int&gt;,
etc.</code></para>
</section>
<section>
<title>Header</title>
<para><ulink
url="../../include/boost/safe_numerics/concept/numeric.hpp"><code>#include
&lt;boost/safe_numerics/concepts/numeric.hpp&gt; </code></ulink></para>
</section>
<section>
<title>Note on Usage of <code>std::numeric_limits</code></title>
<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&lt;int&gt;</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&lt;T&gt;" 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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