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25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.special_tut.special_tut_impl"></a><a class="link" href="special_tut_impl.html" title="Implementation">Implementation</a> | |
28 | </h3></div></div></div> | |
29 | <p> | |
30 | In this section, we'll provide a "recipe" for adding a new special | |
31 | function to this library to make life easier for future authors wishing to | |
32 | contribute. We'll assume the function returns a single floating-point result, | |
33 | and takes two floating-point arguments. For the sake of exposition we'll | |
34 | give the function the name <em class="replaceable"><code>my_special</code></em>. | |
35 | </p> | |
36 | <p> | |
37 | Normally, the implementation of such a function is split into two layers | |
38 | - a public user layer, and an internal implementation layer that does the | |
39 | actual work. The implementation layer is declared inside a <code class="computeroutput"><span class="identifier">detail</span></code> namespace and has a simple signature: | |
40 | </p> | |
41 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">detail</span> <span class="special">{</span> | |
42 | ||
43 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> | |
44 | <span class="identifier">T</span> <span class="identifier">my_special_imp</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span><span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">)</span> | |
45 | <span class="special">{</span> | |
46 | <span class="comment">/* Implementation goes here */</span> | |
47 | <span class="special">}</span> | |
48 | ||
49 | <span class="special">}}}</span> <span class="comment">// namespaces</span> | |
50 | </pre> | |
51 | <p> | |
52 | We'll come back to what can go inside the implementation later, but first | |
53 | lets look at the user layer. This consists of two overloads of the function, | |
54 | with and without a <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument: | |
55 | </p> | |
56 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> | |
57 | ||
58 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
59 | <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span> | |
60 | ||
61 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> | |
62 | <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span> | |
63 | ||
64 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
65 | </pre> | |
66 | <p> | |
67 | Note how each argument has a different template type - this allows for mixed | |
68 | type arguments - the return type is computed from a traits class and is the | |
69 | "common type" of all the arguments after any integer arguments | |
70 | have been promoted to type <code class="computeroutput"><span class="keyword">double</span></code>. | |
71 | </p> | |
72 | <p> | |
73 | The implementation of the non-policy overload is trivial: | |
74 | </p> | |
75 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> | |
76 | ||
77 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span> | |
78 | <span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> | |
79 | <span class="special">{</span> | |
80 | <span class="comment">// Simply forward with a default policy:</span> | |
81 | <span class="keyword">return</span> <span class="identifier">my_special</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><>();</span> | |
82 | <span class="special">}</span> | |
83 | ||
84 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
85 | </pre> | |
86 | <p> | |
87 | The implementation of the other overload is somewhat more complex, as there's | |
88 | some meta-programming to do, but from a runtime perspective is still a one-line | |
89 | forwarding function. Here it is with comments explaining what each line does: | |
90 | </p> | |
91 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> | |
92 | ||
93 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> | |
94 | <span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">my_special</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">)</span> | |
95 | <span class="special">{</span> | |
96 | <span class="comment">//</span> | |
97 | <span class="comment">// We've found some standard library functions to misbehave if any FPU exception flags</span> | |
98 | <span class="comment">// are set prior to their call, this code will clear those flags, then reset them</span> | |
99 | <span class="comment">// on exit:</span> | |
100 | <span class="comment">//</span> | |
101 | <span class="identifier">BOOST_FPU_EXCEPTION_GUARD</span> | |
102 | <span class="comment">//</span> | |
103 | <span class="comment">// The type of the result - the common type of T and U after</span> | |
104 | <span class="comment">// any integer types have been promoted to double:</span> | |
105 | <span class="comment">//</span> | |
106 | <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">promote_args</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">U</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">result_type</span><span class="special">;</span> | |
107 | <span class="comment">//</span> | |
108 | <span class="comment">// The type used for the calculation. This may be a wider type than</span> | |
109 | <span class="comment">// the result in order to ensure full precision:</span> | |
110 | <span class="comment">//</span> | |
111 | <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">evaluation</span><span class="special"><</span><span class="identifier">result_type</span><span class="special">,</span> <span class="identifier">Policy</span><span class="special">>::</span><span class="identifier">type</span> <span class="identifier">value_type</span><span class="special">;</span> | |
112 | <span class="comment">//</span> | |
113 | <span class="comment">// The type of the policy to forward to the actual implementation.</span> | |
114 | <span class="comment">// We disable promotion of float and double as that's [possibly]</span> | |
115 | <span class="comment">// happened already in the line above. Also reset to the default</span> | |
116 | <span class="comment">// any policies we don't use (reduces code bloat if we're called</span> | |
117 | <span class="comment">// multiple times with differing policies we don't actually use).</span> | |
118 | <span class="comment">// Also normalise the type, again to reduce code bloat in case we're</span> | |
119 | <span class="comment">// called multiple times with functionally identical policies that happen</span> | |
120 | <span class="comment">// to be different types.</span> | |
121 | <span class="comment">//</span> | |
122 | <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">normalise</span><span class="special"><</span> | |
123 | <span class="identifier">Policy</span><span class="special">,</span> | |
124 | <span class="identifier">policies</span><span class="special">::</span><span class="identifier">promote_float</span><span class="special"><</span><span class="keyword">false</span><span class="special">>,</span> | |
125 | <span class="identifier">policies</span><span class="special">::</span><span class="identifier">promote_double</span><span class="special"><</span><span class="keyword">false</span><span class="special">>,</span> | |
126 | <span class="identifier">policies</span><span class="special">::</span><span class="identifier">discrete_quantile</span><span class="special"><>,</span> | |
127 | <span class="identifier">policies</span><span class="special">::</span><span class="identifier">assert_undefined</span><span class="special"><></span> <span class="special">>::</span><span class="identifier">type</span> <span class="identifier">forwarding_policy</span><span class="special">;</span> | |
128 | <span class="comment">//</span> | |
129 | <span class="comment">// Whew. Now we can make the actual call to the implementation.</span> | |
130 | <span class="comment">// Arguments are explicitly cast to the evaluation type, and the result</span> | |
131 | <span class="comment">// passed through checked_narrowing_cast which handles things like overflow</span> | |
132 | <span class="comment">// according to the policy passed:</span> | |
133 | <span class="comment">//</span> | |
134 | <span class="keyword">return</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">checked_narrowing_cast</span><span class="special"><</span><span class="identifier">result_type</span><span class="special">,</span> <span class="identifier">forwarding_policy</span><span class="special">>(</span> | |
135 | <span class="identifier">detail</span><span class="special">::</span><span class="identifier">my_special_imp</span><span class="special">(</span> | |
136 | <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">value_type</span><span class="special">>(</span><span class="identifier">a</span><span class="special">),</span> | |
137 | <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">value_type</span><span class="special">>(</span><span class="identifier">x</span><span class="special">),</span> | |
138 | <span class="identifier">forwarding_policy</span><span class="special">()),</span> | |
139 | <span class="string">"boost::math::my_special<%1%>(%1%, %1%)"</span><span class="special">);</span> | |
140 | <span class="special">}</span> | |
141 | ||
142 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
143 | </pre> | |
144 | <p> | |
145 | We're now almost there, we just need to flesh out the details of the implementation | |
146 | layer: | |
147 | </p> | |
148 | <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">detail</span> <span class="special">{</span> | |
149 | ||
150 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span> | |
151 | <span class="identifier">T</span> <span class="identifier">my_special_imp</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span><span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">)</span> | |
152 | <span class="special">{</span> | |
153 | <span class="comment">/* Implementation goes here */</span> | |
154 | <span class="special">}</span> | |
155 | ||
156 | <span class="special">}}}</span> <span class="comment">// namespaces</span> | |
157 | </pre> | |
158 | <p> | |
159 | The following guidelines indicate what (other than basic arithmetic) can | |
160 | go in the implementation: | |
161 | </p> | |
162 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
163 | <li class="listitem"> | |
164 | Error conditions (for example bad arguments) should be handled by calling | |
165 | one of the <a class="link" href="../error_handling.html#math_toolkit.error_handling.finding_more_information">policy | |
166 | based error handlers</a>. | |
167 | </li> | |
168 | <li class="listitem"> | |
169 | Calls to standard library functions should be made unqualified (this | |
170 | allows argument dependent lookup to find standard library functions for | |
171 | user-defined floating point types such as those from <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>). | |
172 | In addition, the macro <code class="computeroutput"><span class="identifier">BOOST_MATH_STD_USING</span></code> | |
173 | should appear at the start of the function (note no semi-colon afterwards!) | |
174 | so that all the math functions in <code class="computeroutput"><span class="keyword">namespace</span> | |
175 | <span class="identifier">std</span></code> are visible in the current | |
176 | scope. | |
177 | </li> | |
178 | <li class="listitem"> | |
179 | Calls to other special functions should be made as fully qualified calls, | |
180 | and include the policy parameter as the last argument, for example <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">pol</span><span class="special">)</span></code>. | |
181 | </li> | |
182 | <li class="listitem"> | |
183 | Where possible, evaluation of series, continued fractions, polynomials, | |
184 | or root finding should use one of the <a class="link" href="../internals_overview.html" title="Overview">boiler-plate | |
185 | functions</a>. In any case, after any iterative method, you should | |
186 | verify that the number of iterations did not exceed the maximum specified | |
187 | in the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> type, and if it did terminate | |
188 | as a result of exceeding the maximum, then the appropriate error handler | |
189 | should be called (see existing code for examples). | |
190 | </li> | |
191 | <li class="listitem"> | |
192 | Numeric constants such as π etc should be obtained via a call to the <a class="link" href="../constants.html" title="The Mathematical Constants">appropriate function</a>, for example: | |
193 | <code class="computeroutput"><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code>. | |
194 | </li> | |
195 | <li class="listitem"> | |
196 | Where tables of coefficients are used (for example for rational approximations), | |
197 | care should be taken to ensure these are initialized at program startup | |
198 | to ensure thread safety when using user-defined number types. See for | |
199 | example the use of <code class="computeroutput"><span class="identifier">erf_initializer</span></code> | |
200 | in <a href="../../../../include/boost/math/special_functions/erf.hpp" target="_top">erf.hpp</a>. | |
201 | </li> | |
202 | </ul></div> | |
203 | <p> | |
204 | Here are some other useful internal functions: | |
205 | </p> | |
206 | <div class="informaltable"><table class="table"> | |
207 | <colgroup> | |
208 | <col> | |
209 | <col> | |
210 | </colgroup> | |
211 | <thead><tr> | |
212 | <th> | |
213 | <p> | |
214 | function | |
215 | </p> | |
216 | </th> | |
217 | <th> | |
218 | <p> | |
219 | Meaning | |
220 | </p> | |
221 | </th> | |
222 | </tr></thead> | |
223 | <tbody> | |
224 | <tr> | |
225 | <td> | |
226 | <p> | |
227 | <code class="computeroutput"><span class="identifier">policies</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> | |
228 | <span class="identifier">Policy</span><span class="special">>()</span></code> | |
229 | </p> | |
230 | </td> | |
231 | <td> | |
232 | <p> | |
233 | Returns number of binary digits in T (possible overridden by the | |
234 | policy). | |
235 | </p> | |
236 | </td> | |
237 | </tr> | |
238 | <tr> | |
239 | <td> | |
240 | <p> | |
241 | <code class="computeroutput"><span class="identifier">policies</span><span class="special">::</span><span class="identifier">get_max_series_iterations</span><span class="special"><</span><span class="identifier">Policy</span><span class="special">>()</span></code> | |
242 | </p> | |
243 | </td> | |
244 | <td> | |
245 | <p> | |
246 | Maximum number of iterations for series evaluation. | |
247 | </p> | |
248 | </td> | |
249 | </tr> | |
250 | <tr> | |
251 | <td> | |
252 | <p> | |
253 | <code class="computeroutput"><span class="identifier">policies</span><span class="special">::</span><span class="identifier">get_max_root_iterations</span><span class="special"><</span><span class="identifier">Policy</span><span class="special">>()</span></code> | |
254 | </p> | |
255 | </td> | |
256 | <td> | |
257 | <p> | |
258 | Maximum number of iterations for root finding. | |
259 | </p> | |
260 | </td> | |
261 | </tr> | |
262 | <tr> | |
263 | <td> | |
264 | <p> | |
265 | <code class="computeroutput"><span class="identifier">polices</span><span class="special">::</span><span class="identifier">get_epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> | |
266 | <span class="identifier">Policy</span><span class="special">>()</span></code> | |
267 | </p> | |
268 | </td> | |
269 | <td> | |
270 | <p> | |
271 | Epsilon for type T, possibly overridden by the Policy. | |
272 | </p> | |
273 | </td> | |
274 | </tr> | |
275 | <tr> | |
276 | <td> | |
277 | <p> | |
278 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
279 | </p> | |
280 | </td> | |
281 | <td> | |
282 | <p> | |
283 | Returns the number of binary digits in T. | |
284 | </p> | |
285 | </td> | |
286 | </tr> | |
287 | <tr> | |
288 | <td> | |
289 | <p> | |
290 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">max_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
291 | </p> | |
292 | </td> | |
293 | <td> | |
294 | <p> | |
295 | Equivalent to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code> | |
296 | </p> | |
297 | </td> | |
298 | </tr> | |
299 | <tr> | |
300 | <td> | |
301 | <p> | |
302 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">min_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
303 | </p> | |
304 | </td> | |
305 | <td> | |
306 | <p> | |
307 | Equivalent to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span></code> | |
308 | </p> | |
309 | </td> | |
310 | </tr> | |
311 | <tr> | |
312 | <td> | |
313 | <p> | |
314 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_max_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
315 | </p> | |
316 | </td> | |
317 | <td> | |
318 | <p> | |
319 | Equivalent to the natural logarithm of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max</span><span class="special">()</span></code> | |
320 | </p> | |
321 | </td> | |
322 | </tr> | |
323 | <tr> | |
324 | <td> | |
325 | <p> | |
326 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_min_value</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
327 | </p> | |
328 | </td> | |
329 | <td> | |
330 | <p> | |
331 | Equivalent to the natural logarithm of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span></code> | |
332 | </p> | |
333 | </td> | |
334 | </tr> | |
335 | <tr> | |
336 | <td> | |
337 | <p> | |
338 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
339 | </p> | |
340 | </td> | |
341 | <td> | |
342 | <p> | |
343 | Equivalent to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>. | |
344 | </p> | |
345 | </td> | |
346 | </tr> | |
347 | <tr> | |
348 | <td> | |
349 | <p> | |
350 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
351 | </p> | |
352 | </td> | |
353 | <td> | |
354 | <p> | |
355 | Equivalent to the square root of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>. | |
356 | </p> | |
357 | </td> | |
358 | </tr> | |
359 | <tr> | |
360 | <td> | |
361 | <p> | |
362 | <code class="computeroutput"><span class="identifier">tools</span><span class="special">::</span><span class="identifier">forth_root_epsilon</span><span class="special"><</span><span class="identifier">T</span><span class="special">>()</span></code> | |
363 | </p> | |
364 | </td> | |
365 | <td> | |
366 | <p> | |
367 | Equivalent to the forth root of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span></code>. | |
368 | </p> | |
369 | </td> | |
370 | </tr> | |
371 | </tbody> | |
372 | </table></div> | |
373 | </div> | |
374 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
375 | <td align="left"></td> | |
376 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
377 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
378 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
379 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
380 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
381 | file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) | |
382 | </p> | |
383 | </div></td> | |
384 | </tr></table> | |
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