1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> 4<title>safe<T, PP, EP></title> 5<link rel="stylesheet" href="boostbook.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="index.html" title="Safe Numerics"> 8<link rel="up" href="types.html" title="Types"> 9<link rel="prev" href="types.html" title="Types"> 10<link rel="next" href="safe_range.html" title="safe_signed_range<MIN, MAX, PP, EP> and safe_unsigned_range<MIN, MAX, PP, EP>"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img href="index.html" height="164px" src="pre-boost.jpg" alt="Library Documentation Index"></td> 15<td><h2>Safe Numerics</h2></td> 16</tr></table> 17<div class="spirit-nav"> 18<a accesskey="p" href="types.html"><img src="images/prev.png" alt="Prev"></a><a accesskey="u" href="types.html"><img src="images/up.png" alt="Up"></a><a accesskey="h" href="index.html"><img src="images/home.png" alt="Home"></a><a accesskey="n" href="safe_range.html"><img src="images/next.png" alt="Next"></a> 19</div> 20<div class="section"> 21<div class="titlepage"><div><div><h3 class="title"> 22<a name="safe_numerics.safe"></a>safe<T, PP, EP></h3></div></div></div> 23<div class="toc"><dl class="toc"> 24<dt><span class="section"><a href="safe.html#idm130203562112">Description</a></span></dt> 25<dt><span class="section"><a href="safe.html#idm130203560112">Model of</a></span></dt> 26<dt><span class="section"><a href="safe.html#idm130203556080">Notation</a></span></dt> 27<dt><span class="section"><a href="safe.html#idm130203548784">Associated Types</a></span></dt> 28<dt><span class="section"><a href="safe.html#idm130203541744">Template Parameters</a></span></dt> 29<dt><span class="section"><a href="safe.html#idm130203523280">Valid Expressions</a></span></dt> 30<dt><span class="section"><a href="safe.html#idm130203517392">Examples of use</a></span></dt> 31<dt><span class="section"><a href="safe.html#idm130203247744">Header</a></span></dt> 32</dl></div> 33<div class="section"> 34<div class="titlepage"><div><div><h4 class="title"> 35<a name="idm130203562112"></a>Description</h4></div></div></div> 36<p>A <code class="computeroutput">safe<T, PP , EP></code> can be used anywhere a type T 37 can be used. Any expression which uses this type is guaranteed to return 38 an arithmetically correct value or to trap in some way.</p> 39</div> 40<div class="section"> 41<div class="titlepage"><div><div><h4 class="title"> 42<a name="idm130203560112"></a>Model of</h4></div></div></div> 43<p><a class="link" href="numeric.html" title="Numeric<T>">Integer</a></p> 44<p><a class="link" href="safe_numeric_concept.html" title="SafeNumeric<T>">SafeNumeric</a></p> 45<p>This type inherits all the notation, associated types and template 46 parameters and valid expressions of <a class="link" href="safe_numeric_concept.html" title="SafeNumeric<T>">SafeNumeric</a> types. The 47 following specify additional features of this type.</p> 48</div> 49<div class="section"> 50<div class="titlepage"><div><div><h4 class="title"> 51<a name="idm130203556080"></a>Notation</h4></div></div></div> 52<div class="informaltable"><table class="table"> 53<colgroup> 54<col align="left"> 55<col align="left"> 56</colgroup> 57<thead><tr> 58<th align="left">Symbol</th> 59<th align="left">Description</th> 60</tr></thead> 61<tbody><tr> 62<td align="left"><code class="computeroutput">T</code></td> 63<td align="left">Underlying type from which a safe type is being 64 derived</td> 65</tr></tbody> 66</table></div> 67</div> 68<div class="section"> 69<div class="titlepage"><div><div><h4 class="title"> 70<a name="idm130203548784"></a>Associated Types</h4></div></div></div> 71<div class="informaltable"><table class="table"> 72<colgroup> 73<col align="left"> 74<col align="left"> 75</colgroup> 76<tbody> 77<tr> 78<td align="left"><code class="computeroutput">PP</code></td> 79<td align="left">A type which specifies the result type of an expression 80 using safe types.</td> 81</tr> 82<tr> 83<td align="left"><code class="computeroutput">EP</code></td> 84<td align="left">A type containing members which are called when a correct 85 result cannot be returned</td> 86</tr> 87</tbody> 88</table></div> 89</div> 90<div class="section"> 91<div class="titlepage"><div><div><h4 class="title"> 92<a name="idm130203541744"></a>Template Parameters</h4></div></div></div> 93<div class="informaltable"><table class="table"> 94<colgroup> 95<col> 96<col align="left"> 97<col align="left"> 98</colgroup> 99<thead><tr> 100<th align="left">Parameter</th> 101<th align="left">Type Requirements</th> 102<th align="left">Description</th> 103</tr></thead> 104<tbody> 105<tr> 106<td><code class="computeroutput">T</code></td> 107<td align="left"><a href="http://en.cppreference.com/w/cpp/types/is_integral" target="_top">Integer<T></a></td> 108<td align="left"><p>The underlying type. Currently only integer types are 109 supported</p></td> 110</tr> 111<tr> 112<td><code class="computeroutput">PP</code></td> 113<td align="left"><a class="link" href="numeric.html" title="Numeric<T>">PromotionPolicy<PP></a></td> 114<td align="left"><p>Optional promotion policy. Default value is <a class="link" href="promotion_policies/native.html" title="native"><code class="computeroutput">boost::numeric::native</code></a></p></td> 115</tr> 116<tr> 117<td><code class="computeroutput">EP</code></td> 118<td align="left"><a class="link" href="numeric.html" title="Numeric<T>">Exception 119 Policy<EP></a></td> 120<td align="left"><p>Optional exception policy. Default value is <a class="link" href="exception_policies.html#safe_numerics.exception_policies.default_exception_policy"><code class="computeroutput">boost::numeric::default_exception_policy</code></a></p></td> 121</tr> 122</tbody> 123</table></div> 124<p>See examples below.</p> 125</div> 126<div class="section"> 127<div class="titlepage"><div><div><h4 class="title"> 128<a name="idm130203523280"></a>Valid Expressions</h4></div></div></div> 129<p>Implements all expressions and only those expressions defined by the 130 <a class="link" href="safe_numeric_concept.html" title="SafeNumeric<T>">SafeNumeric</a><T> 131 type requirements. Note that all these expressions are 132 <code class="computeroutput">constexpr</code>. Thus, the result type of such an expression will 133 be another safe type. The actual type of the result of such an expression 134 will depend upon the specific promotion policy template parameter.</p> 135<p>When a binary operand is applied to two instances of safe<T, PP, 136 EP>on of the following must be true:</p> 137<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 138<li class="listitem"><p>The promotion policies of the two operands must be the same or 139 one of them must be void</p></li> 140<li class="listitem"><p>The exception policies of the two operands must be the same or 141 one of them must be void</p></li> 142</ul></div> 143<p>If either of the above is not true, a compile error will 144 result.</p> 145</div> 146<div class="section"> 147<div class="titlepage"><div><div><h4 class="title"> 148<a name="idm130203517392"></a>Examples of use</h4></div></div></div> 149<p>The most common usage would be safe<T> which uses the default 150 promotion and exception policies. This type is meant to be a "drop-in" 151 replacement of the intrinsic integer types. That is, expressions involving 152 these types will be evaluated into result types which reflect the standard 153 rules for evaluation of C++ expressions. Should it occur that such 154 evaluation cannot return a correct result, an exception will be 155 thrown.</p> 156<p>There are two aspects of the operation of this type which can be 157 customized with a policy. The first is the result type of an arithmetic 158 operation. C++ defines the rules which define this result type in terms of 159 the constituent types of the operation. Here we refer to these rules as 160 "type promotion" rules. These rules will sometimes result in a type which 161 cannot hold the actual arithmetic result of the operation. This is the 162 main motivation for making this library in the first place. One way to 163 deal with this problem is to substitute our own type promotion rules for 164 the C++ ones.</p> 165<div class="section"> 166<div class="titlepage"><div><div><h5 class="title"> 167<a name="safe_numerics.drop_in_replacement"></a>As a Drop-in replacement for standard integer types.</h5></div></div></div> 168<p>The following program will throw an exception and emit an error 169 message at runtime if any of several events result in an incorrect 170 arithmetic result. Behavior of this program could vary according to the 171 machine architecture in question.</p> 172<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">exception</span><span class="special">></span> 173<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> 174<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">safe_integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 175 176<span class="keyword">void</span> <span class="identifier">f</span><span class="special">(</span><span class="special">)</span><span class="special">{</span> 177 <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> 178 <span class="identifier">safe</span><span class="special"><</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">j</span><span class="special">;</span> 179 <span class="keyword">try</span> <span class="special">{</span> 180 <span class="identifier">safe</span><span class="special"><</span><span class="keyword">int</span><span class="special">></span> <span class="identifier">i</span><span class="special">;</span> 181 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cin</span> <span class="special">>></span> <span class="identifier">i</span><span class="special">;</span> <span class="comment">// could overflow !</span> 182 <span class="identifier">j</span> <span class="special">=</span> <span class="identifier">i</span> <span class="special">*</span> <span class="identifier">i</span><span class="special">;</span> <span class="comment">// could overflow</span> 183 <span class="special">}</span> 184 <span class="keyword">catch</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span> <span class="special">&</span> <span class="identifier">e</span><span class="special">)</span><span class="special">{</span> 185 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">(</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> 186 <span class="special">}</span> 187 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">j</span><span class="special">;</span> 188<span class="special">}</span></pre> 189<p>The term "drop-in replacement" reveals the aspiration of 190 this library. In most cases, this aspiration is realized. In the 191 following example, the normal implicit conversions function the same for 192 safe integers as they do for built-in integers. </p> 193<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_numerics</span><span class="special">/</span><span class="identifier">safe_integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 194<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">safe_numerics</span><span class="special">;</span> 195 196<span class="keyword">int</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span><span class="special">)</span><span class="special">{</span> 197 <span class="keyword">return</span> <span class="identifier">i</span><span class="special">;</span> 198<span class="special">}</span> 199 200<span class="keyword">using</span> <span class="identifier">safe_t</span> <span class="special">=</span> <span class="identifier">safe</span><span class="special"><</span><span class="keyword">long</span><span class="special">></span><span class="special">;</span> 201 202<span class="keyword">int</span> <span class="identifier">main</span><span class="special">(</span><span class="special">)</span><span class="special">{</span> 203 <span class="keyword">const</span> <span class="keyword">long</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">97</span><span class="special">;</span> 204 <span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span><span class="special">;</span> <span class="comment">// OK - implicit conversion to int</span> 205 <span class="keyword">const</span> <span class="identifier">safe_t</span> <span class="identifier">y</span> <span class="special">=</span> <span class="number">97</span><span class="special">;</span> 206 <span class="identifier">f</span><span class="special">(</span><span class="identifier">y</span><span class="special">)</span><span class="special">;</span> <span class="comment">// Also OK - checked implicit conversion to int</span> 207 <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span> 208<span class="special">}</span></pre> 209<p>When 210 the <code class="computeroutput">safe<long></code> is implicitly converted to an 211 <code class="computeroutput">int</code> when calling <code class="computeroutput">f</code>, the value is checked to be 212 sure that it is within the legal range of an int and will invoke an 213 exception if it cannot. We can easily verify this by altering the 214 exception handling policy in the above example to 215 <code class="computeroutput">loose_trap_policy</code>. This will invoke a compile time error on 216 any conversion might invoke a runtime exception.</p> 217<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_numerics</span><span class="special">/</span><span class="identifier">safe_integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 218<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">safe_numerics</span><span class="special">;</span> 219 220<span class="keyword">int</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span><span class="special">)</span><span class="special">{</span> 221 <span class="keyword">return</span> <span class="identifier">i</span><span class="special">;</span> 222<span class="special">}</span> 223 224<span class="keyword">using</span> <span class="identifier">safe_t</span> <span class="special">=</span> <span class="identifier">safe</span><span class="special"><</span><span class="keyword">long</span><span class="special">,</span> <span class="identifier">native</span><span class="special">,</span> <span class="identifier">loose_trap_policy</span><span class="special">></span><span class="special">;</span> 225 226<span class="keyword">int</span> <span class="identifier">main</span><span class="special">(</span><span class="special">)</span><span class="special">{</span> 227 <span class="keyword">const</span> <span class="keyword">long</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">97</span><span class="special">;</span> 228 <span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span><span class="special">;</span> <span class="comment">// OK - implicit conversion to int</span> 229 <span class="keyword">const</span> <span class="identifier">safe_t</span> <span class="identifier">y</span> <span class="special">=</span> <span class="number">97</span><span class="special">;</span> 230 <span class="identifier">f</span><span class="special">(</span><span class="identifier">y</span><span class="special">)</span><span class="special">;</span> <span class="comment">// Would be OK, but will invoke compile time error</span> 231 <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span> 232<span class="special">}</span></pre> 233<p>But this raises it's own questions. We can see that in this 234 example, the program can never fail:</p> 235<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 236<li class="listitem"><p>The value 97 is assigned to y</p></li> 237<li class="listitem"><p>y is converted to an int</p></li> 238<li class="listitem"><p>and used as an argument to f</p></li> 239</ul></div> 240<p>The conversion can never fail because the value of 97 can always 241 fit into an int. But the library code can't detect this and emits the 242 checking code even though it's not necessary.</p> 243<p>This can be addressed by using a <a class="link" href="safe_literal.html" title="safe_signed_literal<Value, PP , EP> and safe_unsigned_literal<Value, PP, EP>"><code class="computeroutput">safe_literal</code></a>. A 244 safe literal can contain one and only one value. All the functions in 245 this library are marked <code class="computeroutput">constexpr</code>. So it can be determined 246 at compile time that conversion to an <code class="computeroutput">int</code> can never fail 247 and no runtime checking code need be emitted. Making this small change 248 will permit the above example to run with zero runtime overhead while 249 guaranteeing that no error can ever occur.</p> 250<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_numerics</span><span class="special">/</span><span class="identifier">safe_integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 251<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">safe_numerics</span><span class="special">/</span><span class="identifier">safe_integer_literal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 252 253<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">safe_numerics</span><span class="special">;</span> 254 255<span class="keyword">int</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">i</span><span class="special">)</span><span class="special">{</span> 256 <span class="keyword">return</span> <span class="identifier">i</span><span class="special">;</span> 257<span class="special">}</span> 258 259<span class="keyword">template</span><span class="special"><</span><span class="identifier">intmax_t</span> <span class="identifier">N</span><span class="special">></span> 260<span class="keyword">using</span> <span class="identifier">safe_literal</span> <span class="special">=</span> <span class="identifier">safe_signed_literal</span><span class="special"><</span><span class="identifier">N</span><span class="special">,</span> <span class="identifier">native</span><span class="special">,</span> <span class="identifier">loose_trap_policy</span><span class="special">></span><span class="special">;</span> 261 262<span class="keyword">int</span> <span class="identifier">main</span><span class="special">(</span><span class="special">)</span><span class="special">{</span> 263 <span class="keyword">const</span> <span class="keyword">long</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">97</span><span class="special">;</span> 264 <span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span><span class="special">;</span> <span class="comment">// OK - implicit conversion to int</span> 265 <span class="keyword">const</span> <span class="identifier">safe_literal</span><span class="special"><</span><span class="number">97</span><span class="special">></span> <span class="identifier">y</span><span class="special">;</span> 266 <span class="identifier">f</span><span class="special">(</span><span class="identifier">y</span><span class="special">)</span><span class="special">;</span> <span class="comment">// OK - y is a type with min/max = 97;</span> 267 <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span> 268<span class="special">}</span></pre> 269<p>With this trivial example, such efforts would hardly be deemed 270 necessary. But in a more complex case, perhaps including compile time 271 arithmetic expressions, it could be much more difficult to verify that 272 the constant is valid and/or no checking code is needed. And there is 273 also possibility that over the life time of the application, the compile 274 time constants might change, thus rendering any ad hoc analyse obsolete. 275 Using <a class="link" href="safe_literal.html" title="safe_signed_literal<Value, PP , EP> and safe_unsigned_literal<Value, PP, EP>"><code class="computeroutput">safe_literal</code></a> 276 will future-proof your code against well-meaning, but code-breaking 277 updates.</p> 278</div> 279<div class="section"> 280<div class="titlepage"><div><div><h5 class="title"> 281<a name="idm130203320752"></a>Adjust type promotion rules.</h5></div></div></div> 282<p>Another way to avoid arithmetic errors like overflow is to promote 283 types to larger sizes before doing the arithmetic.</p> 284<p>Stepping back, we can see that many of the cases of invalid 285 arithmetic wouldn't exist if the result types were larger. So we can 286 avoid these problems by replacing the C++ type promotion rules for 287 expressions with our own rules. This can be done by specifying a 288 promotion policy <code class="computeroutput"><code class="computeroutput">automatic</code></code>. The policy stores 289 the result of an expression in the smallest size type that can 290 accommodate the largest value that an expression can yield. No checking 291 for exceptions is necessary. The following example illustrates 292 this.</p> 293<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_numerics</span><span class="special">/</span><span class="identifier">safe_integer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 294<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> 295 296<span class="keyword">int</span> <span class="identifier">main</span><span class="special">(</span><span class="keyword">int</span><span class="special">,</span> <span class="keyword">char</span><span class="special">[</span><span class="special">]</span><span class="special">)</span><span class="special">{</span> 297 <span class="keyword">using</span> <span class="identifier">safe_int</span> <span class="special">=</span> <span class="identifier">safe</span><span class="special"><</span> 298 <span class="keyword">int</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">automatic</span><span class="special">,</span> 299 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">default_exception_policy</span> 300 <span class="special">></span><span class="special">;</span> 301 <span class="identifier">safe_int</span> <span class="identifier">i</span><span class="special">;</span> 302 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cin</span> <span class="special">>></span> <span class="identifier">i</span><span class="special">;</span> <span class="comment">// might throw exception</span> 303 <span class="keyword">auto</span> <span class="identifier">j</span> <span class="special">=</span> <span class="identifier">i</span> <span class="special">*</span> <span class="identifier">i</span><span class="special">;</span> <span class="comment">// won't ever trap - result type can hold the maximum value of i * i</span> 304 <span class="identifier">static_assert</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">numeric</span><span class="special">::</span><span class="identifier">is_safe</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">j</span><span class="special">)</span><span class="special">></span><span class="special">::</span><span class="identifier">value</span><span class="special">)</span><span class="special">;</span> <span class="comment">// result is another safe type</span> 305 <span class="identifier">static_assert</span><span class="special">(</span> 306 <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">i</span> <span class="special">*</span> <span class="identifier">i</span><span class="special">)</span><span class="special">></span><span class="special">::</span><span class="identifier">max</span><span class="special">(</span><span class="special">)</span> <span class="special">>=</span> 307 <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">safe_int</span><span class="special">></span><span class="special">::</span><span class="identifier">max</span><span class="special">(</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">safe_int</span><span class="special">></span><span class="special">::</span><span class="identifier">max</span><span class="special">(</span><span class="special">)</span> 308 <span class="special">)</span><span class="special">;</span> <span class="comment">// always true</span> 309 310 <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span> 311<span class="special">}</span></pre> 312</div> 313</div> 314<div class="section"> 315<div class="titlepage"><div><div><h4 class="title"> 316<a name="idm130203247744"></a>Header</h4></div></div></div> 317<p><code class="filename"><a href="../../include/safe_integer.hpp" target="_top">#include 318 <boost/numeric/safe_numerics/safe_integer.hpp></a></code></p> 319</div> 320</div> 321<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 322<td align="left"></td> 323<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 324 Software License</a></p> 325</div></td> 326</tr></table> 327<hr> 328<div class="spirit-nav"> 329<a accesskey="p" href="types.html"><img src="images/prev.png" alt="Prev"></a><a accesskey="u" href="types.html"><img src="images/up.png" alt="Up"></a><a accesskey="h" href="index.html"><img src="images/home.png" alt="Home"></a><a accesskey="n" href="safe_range.html"><img src="images/next.png" alt="Next"></a> 330</div> 331</body> 332</html> 333
