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<td valign="top"><img href="index.html" height="164px" src="pre-boost.jpg" alt="Library Documentation Index"></td>
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<td><h2>Safe Numerics</h2></td>
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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="safe_numerics.introduction"></a>Introduction</h2></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="introduction.html#safe_numerics.introduction.problem">Problem</a></span></dt>
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<dt><span class="section"><a href="introduction.html#safe_numerics.introduction.solution">Solution</a></span></dt>
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<dt><span class="section"><a href="introduction.html#safe_numerics.introduction.implementation">How It Works</a></span></dt>
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<dt><span class="section"><a href="introduction.html#safe_numerics.introduction.additional_features">Additional Features</a></span></dt>
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<dt><span class="section"><a href="introduction.html#safe_numerics.introduction.requirements">Requirements</a></span></dt>
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<dt><span class="section"><a href="introduction.html#safe_numerics.introduction.scope">Scope</a></span></dt>
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</dl></div>
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<p>This library is intended as a drop-in replacement for all built-in
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integer types in any program which must:</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem"><p>be demonstrably and verifiably correct.</p></li>
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<li class="listitem"><p>detect every user error such as input, assignment, etc.</p></li>
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<li class="listitem"><p>be efficient as possible subject to the constraints above.</p></li>
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</ul></div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="safe_numerics.introduction.problem"></a>Problem</h3></div></div></div>
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<p>Arithmetic operations in C/C++ are NOT guaranteed to yield a correct
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mathematical result. This feature is inherited from the early days of C.
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The behavior of <code class="computeroutput">int</code>, <code class="computeroutput">unsigned int</code> and others
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were designed to map closely to the underlying hardware. Computer hardware
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implements these types as a fixed number of bits. When the result of
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arithmetic operations exceeds this number of bits, the result will not be
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arithmetically correct. The following example illustrates just one example
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where this causes problems.</p>
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<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">y</span><span class="special">)</span><span class="special">{</span>
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<span class="comment">// this returns an invalid result for some legal values of x and y !</span>
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<span class="keyword">return</span> <span class="identifier">x</span> <span class="special">+</span> <span class="identifier">y</span><span class="special">;</span>
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<span class="special">}</span>
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</pre>
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<p>It is incumbent upon the C/C++ programmer to guarantee that this
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behavior does not result in incorrect or unexpected operation of the
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program. There are no language facilities which implement such a
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guarantee. A programmer needs to examine each expression individually to
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know that his program will not return an invalid result. There are a
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number of ways to do this. In the above instance,
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[<a class="citation" href="bibliography.html#seacord3"><span class="citation">INT32-C</span></a>] seems to recommend the following
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approach:</p>
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<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">y</span><span class="special">)</span><span class="special">{</span>
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<span class="keyword">if</span> <span class="special">(</span><span class="special">(</span><span class="special">(</span><span class="identifier">y</span> <span class="special">></span> <span class="number">0</span><span class="special">)</span> <span class="special">&&</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">></span> <span class="special">(</span><span class="identifier">INT_MAX</span> <span class="special">-</span> <span class="identifier">y</span><span class="special">)</span><span class="special">)</span><span class="special">)</span>
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<span class="special">||</span> <span class="special">(</span><span class="special">(</span><span class="identifier">y</span> <span class="special"><</span> <span class="number">0</span><span class="special">)</span> <span class="special">&&</span> <span class="special">(</span><span class="identifier">x</span> <span class="special"><</span> <span class="special">(</span><span class="identifier">INT_MIN</span> <span class="special">-</span> <span class="identifier">y</span><span class="special">)</span><span class="special">)</span><span class="special">)</span><span class="special">)</span> <span class="special">{</span>
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<span class="comment">/* Handle error */</span>
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<span class="special">}</span>
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<span class="keyword">return</span> <span class="identifier">x</span> <span class="special">+</span> <span class="identifier">y</span><span class="special">;</span>
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<span class="special">}</span>
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</pre>
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<p>This will indeed trap the error. However, it would be tedious and
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laborious for a programmer to alter his code to do so. Altering code in
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this way for all arithmetic operations would likely render the code
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unreadable and add another source of potential programming errors. This
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approach is clearly not functional when the expression is even a little
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more complex as is shown in the following example.</p>
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<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">y</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">z</span><span class="special">)</span><span class="special">{</span>
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<span class="comment">// this returns an invalid result for some legal values of x and y !</span>
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<span class="keyword">return</span> <span class="identifier">x</span> <span class="special">+</span> <span class="identifier">y</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">;</span>
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<span class="special">}</span>
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</pre>
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<p>This example addresses only the problem of undefined/erroneous
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behavior related to overflow of the addition operation as applied to the
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type <code class="computeroutput">int</code>. Similar problems occur with other built-in integer
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types such as <code class="computeroutput">unsigned</code>, <code class="computeroutput">long</code>, etc. And it also
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applies to other operations such as subtraction, multiplication etc. .
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C/C++ often automatically and silently converts some integer types to
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others in the course of implementing binary operations and similar
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problems occur in this case as well. Since the problems and their solution
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are similar, We'll confine the current discussion to just this one
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example.</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="safe_numerics.introduction.solution"></a>Solution</h3></div></div></div>
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<p>This library implements special versions of <code class="computeroutput">int</code>,
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<code class="computeroutput">unsigned</code>, etc. which behave exactly like the original ones
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<span class="bold"><strong>except</strong></span> that the results of these
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operations are guaranteed to be either arithmetically correct or invoke an
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error. Using this library, the above example would be rendered as:</p>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">safe_numeric</span><span class="special">/</span><span class="identifier">safe_integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">;</span>
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<span class="identifier">safe</span><span class="special"><</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">safe</span><span class="special"><</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">safe</span><span class="special"><</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">y</span><span class="special">)</span><span class="special">{</span>
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<span class="keyword">return</span> <span class="identifier">x</span> <span class="special">+</span> <span class="identifier">y</span><span class="special">;</span> <span class="comment">// throw exception if correct result cannot be returned</span>
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<span class="special">}</span>
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</pre>
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<div class="note"><table border="0" summary="Note">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="images/note.png"></td>
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<th align="left">Note</th>
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</tr>
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<tr><td align="left" valign="top"><p>Library code in this document resides in the name space
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<code class="computeroutput">boost::numeric</code>. This name space has generally been
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eliminated from text, code and examples in order to improve
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readability of the text.</p></td></tr>
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</table></div>
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<p>The addition expression is checked at runtime or (if possible) at
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compile time to trap any possible errors resulting in incorrect arithmetic
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behavior. This will permit one to write arithmetic expressions that cannot
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produce an erroneous result. Instead, one and only one of the following is
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guaranteed to occur.</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem"><p>the expression will yield the correct mathematical
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result</p></li>
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<li class="listitem"><p>the expression will emit a compilation error.</p></li>
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<li class="listitem"><p>the expression will invoke a runtime exception.</p></li>
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</ul></div>
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<p>In other words, the <span class="bold"><strong>library absolutely
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guarantees that no arithmetic expression will yield incorrect
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results</strong></span>.</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="safe_numerics.introduction.implementation"></a>How It Works</h3></div></div></div>
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<p>The library implements special versions of <code class="computeroutput">int</code>,
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<code class="computeroutput">unsigned</code>, etc. named <code class="computeroutput">safe<int></code>,
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<code class="computeroutput">safe<unsigned int></code> etc. These behave exactly like the
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underlying types <span class="bold"><strong>except</strong></span> that expressions
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using these types fulfill the above guarantee. These types are meant to be
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"drop-in" replacements for the built-in types they are meant to replace.
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So things which are legal - such as assignment of a <code class="computeroutput">signed</code> to
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<code class="computeroutput">unsigned</code> value - are not trapped at compile time as they are
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legal C/C++ code. Instead, they are checked at runtime to trap the case
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where this (legal) operation would lead to an arithmetically incorrect
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result.</p>
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<p>Note that the library addresses arithmetical errors generated by
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straightforward C/C++ expressions. Some of these arithmetic errors are
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defined as conforming to the C/C++ standards while others are not. So
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characterizing this library as addressing undefined behavior of C/C++
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numeric expressions would be misleading.</p>
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<p>Facilities particular to C++14 are employed to minimize any runtime
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overhead. In many cases there is no runtime overhead at all. In other
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cases, a program using the library can be slightly altered to achieve the
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above guarantee without any runtime overhead.</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="safe_numerics.introduction.additional_features"></a>Additional Features</h3></div></div></div>
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<p>Operation of safe types is determined by template parameters which
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specify a pair of <a class="link" href="promotion_policies.html" title="Promotion Policies">policy
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classes</a> which specify the behavior for type promotion and error
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handling. In addition to the usage serving as a drop-in replacement for
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standard integer types, users of the library can:</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<p>Select or define an exception policy class to specify handling
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of exceptions.</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
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<li class="listitem"><p>throw exception on runtime, trap at compile time.</p></li>
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<li class="listitem"><p>trap at compiler time all operations which might fail at
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runtime.</p></li>
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<li class="listitem"><p>specify custom functions which should be called at
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runtime</p></li>
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</ul></div>
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</li>
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<li class="listitem">
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<p>Select or define a promotion policy class to alter the C/C++
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type promotion rules. This can be used to </p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
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<li class="listitem"><p>use C/C++ native type promotion rules so that, except
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for throwing/trapping of exceptions on operations resulting in
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incorrect arithmetic behavior, programs will operate
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identically when using/not using safe types.</p></li>
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<li class="listitem"><p>replace C/C++ native promotion rules with ones which are
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arithmetically equivalent but minimize the need for runtime
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checking of arithmetic results.</p></li>
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<li class="listitem"><p>replace C/C++ native promotion rules with ones which
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emulate other machine architectures. This is designed to
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permit the testing of C/C++ code destined to be run on another
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machine on one's development platform. Such a situation often
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occurs while developing code for embedded systems.</p></li>
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</ul></div>
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</li>
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<li class="listitem"><p>Enforce of other program requirements using ranged integer
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types. The library includes the types <code class="computeroutput">safe_range(Min,
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Max)</code> and <code class="computeroutput">safe_literal(N)</code>. These types can be
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used to improve program correctness and performance.</p></li>
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</ul></div>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="safe_numerics.introduction.requirements"></a>Requirements</h3></div></div></div>
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<p>This library is composed entirely of C++ Headers. It requires a
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compiler compatible with the C++14 standard.</p>
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<p>The following Boost Libraries must be installed in order to use this
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library</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem"><p>MPL</p></li>
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<li class="listitem"><p>Integer</p></li>
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<li class="listitem"><p>Config</p></li>
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<li class="listitem"><p>Concept Checking</p></li>
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<li class="listitem"><p>Tribool</p></li>
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<li class="listitem"><p>Enable_if</p></li>
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</ul></div>
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<p>The Safe Numerics library is delivered with an exhaustive
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suite of test programs. Users who choose to run this test suite will also
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need to install the Boost.Preprocessor library.</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="safe_numerics.introduction.scope"></a>Scope</h3></div></div></div>
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<p>This library currently applies only to built-in integer types.
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Analogous issues arise for floating point types but they are not currently
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addressed by this version of the library. User or library defined types
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such as arbitrary precision integers can also have this problem. Extension
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of this library to these other types is not currently under development
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but may be addressed in the future. This is one reason why the library
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name is "safe numeric" rather than "safe integer" library.</p>
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</div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2012 Robert Ramey<p><a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">Subject to Boost
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Software License</a></p>
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</div></td>
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</tr></table>
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