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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="boost_multiprecision.tut.limits.functions"></a><a class="link" href="functions.html" title="std::numeric_limits<> functions">std::numeric_limits<> |
| functions</a> |
| </h4></div></div></div> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h0"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.max_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.max_function">max function</a> |
| </h5> |
| <p> |
| Function <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> |
| returns the largest finite value that can be represented by the type T. |
| If there is no such value (and <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">bounded</span></code> |
| is <code class="computeroutput"><span class="keyword">false</span></code>) then returns <code class="computeroutput"><span class="identifier">T</span><span class="special">()</span></code>. |
| </p> |
| <p> |
| For built-in types there is usually a corresponding MACRO value TYPE_MAX, |
| where TYPE is CHAR, INT, FLOAT etc. |
| </p> |
| <p> |
| Other types, including those provided by a typedef, for example <code class="computeroutput"><span class="identifier">INT64_T_MAX</span></code> for <code class="computeroutput"><span class="identifier">int64_t</span></code>, |
| may provide a macro definition. |
| </p> |
| <p> |
| To cater for situations where no <code class="computeroutput"><span class="identifier">numeric_limits</span></code> |
| specialization is available (for example because the precision of the type |
| varies at runtime), packaged versions of this (and other functions) are |
| provided using |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">precision</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| |
| <span class="identifier">T</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><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> |
| </pre> |
| <p> |
| Of course, these simply use <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> if available, but otherwise 'do something |
| sensible'. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h1"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.lowest_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.lowest_function">lowest |
| function</a> |
| </h5> |
| <p> |
| Since C++11: <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">lowest</span><span class="special">()</span></code> |
| is |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| For integral types, the same as function <code class="computeroutput"><span class="identifier">min</span><span class="special">()</span></code>. |
| </li> |
| <li class="listitem"> |
| For floating-point types, generally the negative of <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code> (but implementation-dependent). |
| </li> |
| </ul></div> |
| <pre class="programlisting"><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="keyword">double</span><span class="special">>::</span><span class="identifier">max</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="keyword">double</span><span class="special">>::</span><span class="identifier">lowest</span><span class="special">();</span> |
| </pre> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h2"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.min_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.min_function">min function</a> |
| </h5> |
| <p> |
| Function <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> |
| returns the minimum finite value that can be represented by the type T. |
| </p> |
| <p> |
| For built-in types there is usually a corresponding MACRO value TYPE_MIN, |
| where TYPE is CHAR, INT, FLOAT etc. |
| </p> |
| <p> |
| Other types, including those provided by a typedef, for example <code class="computeroutput"><span class="identifier">INT64_T_MIN</span></code> for <code class="computeroutput"><span class="identifier">int64_t</span></code>, |
| may provide a macro definition. |
| </p> |
| <p> |
| For floating-point types, it is more fully defined as the <span class="emphasis"><em>minimum |
| positive normalized value</em></span>. |
| </p> |
| <p> |
| See <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">denorm_min</span><span class="special">()</span></code> |
| for the smallest denormalized value, provided |
| </p> |
| <pre class="programlisting"><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">has_denorm</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span> |
| </pre> |
| <p> |
| To cater for situations where no <code class="computeroutput"><span class="identifier">numeric_limits</span></code> |
| specialization is available (for example because the precision of the type |
| varies at runtime), packaged versions of this (and other functions) are |
| provided using |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">precision</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| |
| <span class="identifier">T</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><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> |
| </pre> |
| <p> |
| Of course, these simply use <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> if available. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h3"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.denorm_min_function"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.denorm_min_function">denorm_min |
| function</a> |
| </h5> |
| <p> |
| Function <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">denorm_min</span><span class="special">()</span></code> |
| returns the smallest <a href="http://en.wikipedia.org/wiki/Denormal_number" target="_top">denormalized |
| value</a>, provided |
| </p> |
| <pre class="programlisting"><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">has_denorm</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span> |
| </pre> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</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="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> |
| <span class="keyword">if</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="keyword">double</span><span class="special">>::</span><span class="identifier">has_denorm</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="keyword">double</span> <span class="identifier">d</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="keyword">double</span><span class="special">>::</span><span class="identifier">denorm_min</span><span class="special">();</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">d</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 4.9406564584124654e-324</span> |
| |
| <span class="keyword">int</span> <span class="identifier">exponent</span><span class="special">;</span> |
| |
| <span class="keyword">double</span> <span class="identifier">significand</span> <span class="special">=</span> <span class="identifier">frexp</span><span class="special">(</span><span class="identifier">d</span><span class="special">,</span> <span class="special">&</span><span class="identifier">exponent</span><span class="special">);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"exponent = "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">exponent</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// fffffbcf</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"significand = "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">significand</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.50000000000000000</span> |
| <span class="special">}</span> |
| <span class="keyword">else</span> |
| <span class="special">{</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"No denormalization. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| The exponent is effectively reduced from -308 to -324 (though it remains |
| encoded as zero and leading zeros appear in the significand, thereby losing |
| precision until the significand reaches zero). |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h4"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.round_error"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.round_error">round_error</a> |
| </h5> |
| <p> |
| Function <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">round_error</span><span class="special">()</span></code> |
| returns the maximum error (in units of <a href="http://en.wikipedia.org/wiki/Unit_in_the_last_place" target="_top">ULP</a>) |
| that can be caused by any basic arithmetic operation. |
| </p> |
| <pre class="programlisting"><span class="identifier">round_style</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round_indeterminate</span><span class="special">;</span> |
| </pre> |
| <p> |
| The rounding style is indeterminable at compile time. |
| </p> |
| <p> |
| For floating-point types, when rounding is to nearest, only half a bit |
| is lost by rounding, and <code class="computeroutput"><span class="identifier">round_error</span> |
| <span class="special">==</span> <span class="number">0.5</span></code>. |
| In contrast when rounding is towards zero, or plus/minus infinity, we can |
| loose up to one bit from rounding, and <code class="computeroutput"><span class="identifier">round_error</span> |
| <span class="special">==</span> <span class="number">1</span></code>. |
| </p> |
| <p> |
| For integer types, rounding always to zero, so at worst almost one bit |
| can be rounded, so <code class="computeroutput"><span class="identifier">round_error</span> |
| <span class="special">==</span> <span class="number">1</span></code>. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">round_error</span><span class="special">()</span></code> |
| can be used with <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> |
| to estimate the maximum potential error caused by rounding. For typical |
| floating-point types, <code class="computeroutput"><span class="identifier">round_error</span><span class="special">()</span> <span class="special">=</span> <span class="number">1</span><span class="special">/</span><span class="number">2</span></code>, so half |
| epsilon is the maximum potential error. |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">round_err</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="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span> <span class="comment">// 2.2204460492503131e-016</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="keyword">double</span><span class="special">>::</span><span class="identifier">round_error</span><span class="special">();</span> <span class="comment">// 1/2</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">round_err</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1.1102230246251565e-016</span> |
| </pre> |
| <p> |
| There are, of course, many occasions when much bigger loss of precision |
| occurs, for example, caused by <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">Loss |
| of significance or cancellation error</a> or very many iterations. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h5"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.epsilon"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.epsilon">epsilon</a> |
| </h5> |
| <p> |
| Function <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> |
| is meaningful only for non-integral types. |
| </p> |
| <p> |
| It returns the difference between <code class="computeroutput"><span class="number">1.0</span></code> |
| and the next value representable by the floating-point type T. So it is |
| a one least-significant-bit change in this floating-point value. |
| </p> |
| <p> |
| For <code class="computeroutput"><span class="keyword">double</span></code> (<code class="computeroutput"><span class="identifier">float_64t</span></code>) it is <code class="computeroutput"><span class="number">2.2204460492503131e-016</span></code> |
| showing all possibly significant 17 decimal digits. |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</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="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> |
| <span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">eps</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="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> |
| <span class="keyword">double</span> <span class="identifier">dpeps</span> <span class="special">=</span> <span class="identifier">d</span><span class="special">+</span><span class="identifier">eps</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span> <span class="comment">// Ensure all trailing zeros are shown.</span> |
| <span class="special"><<</span> <span class="identifier">d</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="comment">// 1.0000000000000000</span> |
| <span class="special"><<</span> <span class="identifier">dpeps</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 2.2204460492503131e-016</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dpeps</span> <span class="special">-</span> <span class="identifier">d</span> <span class="comment">// 1.0000000000000002</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| We can explicitly increment by one bit using the function <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">float_next</span><span class="special">()</span></code> |
| and the result is the same as adding <code class="computeroutput"><span class="identifier">epsilon</span></code>. |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">one</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">nad</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">float_next</span><span class="special">(</span><span class="identifier">one</span><span class="special">);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">nad</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="comment">// 1.0000000000000002</span> |
| <span class="special"><<</span> <span class="identifier">nad</span> <span class="special">-</span> <span class="identifier">one</span> <span class="comment">// 2.2204460492503131e-016</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| Adding any smaller value, like half <code class="computeroutput"><span class="identifier">epsilon</span></code>, |
| will have no effect on this value. |
| </p> |
| <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</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="keyword">double</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> |
| <span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">eps</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="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> |
| <span class="keyword">double</span> <span class="identifier">dpeps</span> <span class="special">=</span> <span class="identifier">d</span> <span class="special">+</span> <span class="identifier">eps</span><span class="special">/</span><span class="number">2</span><span class="special">;</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpoint</span> <span class="comment">// Ensure all trailing zeros are shown.</span> |
| <span class="special"><<</span> <span class="identifier">dpeps</span> <span class="special"><<</span> <span class="string">"\n"</span> <span class="comment">// 1.0000000000000000</span> |
| <span class="special"><<</span> <span class="identifier">eps</span><span class="special">/</span><span class="number">2</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 1.1102230246251565e-016</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dpeps</span> <span class="special">-</span> <span class="identifier">d</span> <span class="comment">// 0.00000000000000000</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| So this cancellation error leaves the values equal, despite adding half |
| <code class="computeroutput"><span class="identifier">epsilon</span></code>. |
| </p> |
| <p> |
| To achieve greater portability over platform and floating-point type, Boost.Math |
| and Boost.Multiprecion provide a package of functions that 'do something |
| sensible' if the standard <code class="computeroutput"><span class="identifier">numeric_limits</span></code> |
| is not available. To use these <code class="computeroutput"><span class="preprocessor">#include</span> |
| <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">precision</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code>. |
| </p> |
| <p> |
| A tolerance might be defined using this version of epsilon thus: |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">tolerance</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span> <span class="special">*</span> <span class="number">2</span><span class="special">;</span> |
| </pre> |
| <h6> |
| <a name="boost_multiprecision.tut.limits.functions.h6"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.tolerance_for_floating_point_com"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.tolerance_for_floating_point_com">Tolerance |
| for Floating-point Comparisons</a> |
| </h6> |
| <p> |
| <code class="computeroutput"><span class="identifier">epsilon</span></code> is very useful |
| to compute a tolerance when comparing floating-point values, a much more |
| difficult task than is commonly imagined. |
| </p> |
| <p> |
| For more information you probably want (but still need) see <a href="http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html" target="_top">What |
| Every Computer Scientist Should Know About Floating-Point Arithmetic</a> |
| </p> |
| <p> |
| The naive test comparing the absolute difference between two values and |
| a tolerance does not give useful results if the values are too large or |
| too small. |
| </p> |
| <p> |
| So Boost.Test uses an algorithm first devised by Knuth for reliably checking |
| if floating-point values are close enough. |
| </p> |
| <p> |
| See Donald. E. Knuth. The art of computer programming (vol II). Copyright |
| 1998 Addison-Wesley Longman, Inc., 0-201-89684-2. Addison-Wesley Professional; |
| 3rd edition. |
| </p> |
| <p> |
| See also: |
| </p> |
| <p> |
| <a href="http://adtmag.com/articles/2000/03/15/comparing-floats-how-to-determine-if-floating-quantities-are-close-enough-once-a-tolerance-has-been.aspx" target="_top">Alberto |
| Squassia, Comparing floats</a> |
| </p> |
| <p> |
| <a href="http://adtmag.com/articles/2000/03/16/comparing-floats-how-to-determine-if-floating-quantities-are-close-enough-once-a-tolerance-has-been.aspx" target="_top">Alberto |
| Squassia, Comparing floats code</a> |
| </p> |
| <p> |
| <a href="../../../../../../../libs/test/doc/html/utf/testing-tools/floating_point_comparison.html" target="_top">floating-point |
| comparison</a>. |
| </p> |
| <p> |
| For example, if we want a tolerance that might suit about 9 arithmetical |
| operations, say sqrt(9) = 3, we could define: |
| </p> |
| <pre class="programlisting"><span class="identifier">T</span> <span class="identifier">tolerance</span> <span class="special">=</span> <span class="number">3</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">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> |
| </pre> |
| <p> |
| This is very widely used in Boost.Math testing with Boost.Test's macro |
| <code class="computeroutput"><span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span></code> |
| </p> |
| <pre class="programlisting"><span class="identifier">T</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="number">1.0</span><span class="special">;</span> |
| <span class="identifier">T</span> <span class="identifier">calculated</span> <span class="special">=</span> <span class="number">1.0</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">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> |
| |
| <span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span> |
| </pre> |
| <p> |
| used thus: |
| </p> |
| <pre class="programlisting"><span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span> |
| </pre> |
| <p> |
| (There is also a version using tolerance as a percentage rather than a |
| fraction). |
| </p> |
| <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special">;</span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float</span><span class="special">;</span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">et_off</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">cpp_dec_float</span><span class="special"><</span><span class="number">50</span><span class="special">>,</span> <span class="identifier">et_off</span> <span class="special">></span> <span class="identifier">cpp_dec_float_50</span><span class="special">;</span> <span class="comment">// 50 decimal digits.</span> |
| </pre> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| that Boost.Test does not yet allow floating-point comparisons with expression |
| templates on, so the default expression template parameter has been replaced |
| by <code class="computeroutput"><span class="identifier">et_off</span></code>. |
| </p></td></tr> |
| </table></div> |
| <pre class="programlisting"><span class="identifier">cpp_dec_float_50</span> <span class="identifier">tolerance</span> <span class="special">=</span> <span class="number">3</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">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> |
| <span class="identifier">cpp_dec_float_50</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">two_pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>();</span> |
| <span class="identifier">cpp_dec_float_50</span> <span class="identifier">calculated</span> <span class="special">=</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>();</span> |
| |
| <span class="identifier">BOOST_CHECK_CLOSE_FRACTION</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span> |
| </pre> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h7"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.infinity_positive_and_negative"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.infinity_positive_and_negative">Infinity |
| - positive and negative</a> |
| </h5> |
| <p> |
| For floating-point types only, for which <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">has_infinity</span> |
| <span class="special">==</span> <span class="keyword">true</span></code>, |
| function <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">infinity</span><span class="special">()</span></code> |
| provides an implementation-defined representation for ∞. |
| </p> |
| <p> |
| The 'representation' is a particular bit pattern reserved for infinity. |
| For IEEE754 system (for which <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">is_iec559</span> |
| <span class="special">==</span> <span class="keyword">true</span></code>) |
| <a href="http://en.wikipedia.org/wiki/IEEE_754-1985#Positive_and_negative_infinity" target="_top">positive |
| and negative infinity</a> are assigned bit patterns for all defined |
| floating-point types. |
| </p> |
| <p> |
| Confusingly, the string resulting from outputting this representation, |
| is also implementation-defined. And the string that can be input to generate |
| the representation is also implementation-defined. |
| </p> |
| <p> |
| For example, the output is <code class="computeroutput"><span class="number">1.</span><span class="special">#</span><span class="identifier">INF</span></code> |
| on Microsoft systems, but <code class="computeroutput"><span class="identifier">inf</span></code> |
| on most *nix platforms. |
| </p> |
| <p> |
| This implementation-defined-ness has hampered use of infinity (and NaNs) |
| but Boost.Math and Boost.Multiprecision work hard to provide a sensible |
| representation for <span class="bold"><strong>all</strong></span> floating-point |
| types, not just the built-in types, which with the use of suitable facets |
| to define the input and output strings, makes it possible to use these |
| useful features portably and including Boost.Serialization. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.functions.h8"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.not_a_number_nan"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.not_a_number_nan">Not-A-Number |
| NaN</a> |
| </h5> |
| <h6> |
| <a name="boost_multiprecision.tut.limits.functions.h9"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.quiet_nan"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.quiet_nan">Quiet_NaN</a> |
| </h6> |
| <p> |
| For floating-point types only, for which <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">has_quiet_NaN</span> |
| <span class="special">==</span> <span class="keyword">true</span></code>, |
| function <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">quiet_NaN</span><span class="special">()</span></code> |
| provides an implementation-defined representation for NaN. |
| </p> |
| <p> |
| <a href="http://en.wikipedia.org/wiki/NaN" target="_top">NaNs</a> are values to |
| indicate that the result of an assignment or computation is meaningless. |
| A typical example is <code class="computeroutput"><span class="number">0</span><span class="special">/</span><span class="number">0</span></code> but there are many others. |
| </p> |
| <p> |
| NaNs may also be used, to represent missing values: for example, these |
| could, by convention, be ignored in calculations of statistics like means. |
| </p> |
| <p> |
| Many of the problems with a representation for <a href="http://en.wikipedia.org/wiki/NaN" target="_top">Not-A-Number</a> |
| has hampered portable use, similar to those with infinity. |
| </p> |
| <p> |
| NaN can be used with binary multiprecision types like <code class="computeroutput"><span class="identifier">cpp_bin_float_quad</span></code>: |
| </p> |
| <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_quad</span><span class="special">;</span> |
| |
| <span class="keyword">if</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">cpp_bin_float_quad</span><span class="special">>::</span><span class="identifier">has_quiet_NaN</span> <span class="special">==</span> <span class="keyword">true</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">tolerance</span> <span class="special">=</span> <span class="number">3</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">cpp_bin_float_quad</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">();</span> |
| |
| <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">NaN</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">cpp_bin_float_quad</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">();</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"cpp_bin_float_quad NaN is "</span> <span class="special"><<</span> <span class="identifier">NaN</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// cpp_bin_float_quad NaN is nan</span> |
| |
| <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">expected</span> <span class="special">=</span> <span class="identifier">NaN</span><span class="special">;</span> |
| <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">calculated</span> <span class="special">=</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">NaN</span><span class="special">;</span> |
| <span class="comment">// Comparisons of NaN's always fail:</span> |
| <span class="keyword">bool</span> <span class="identifier">b</span> <span class="special">=</span> <span class="identifier">expected</span> <span class="special">==</span> <span class="identifier">calculated</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">b</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="identifier">BOOST_CHECK_NE</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">expected</span><span class="special">);</span> |
| <span class="identifier">BOOST_CHECK_NE</span><span class="special">(</span><span class="identifier">expected</span><span class="special">,</span> <span class="identifier">calculated</span><span class="special">);</span> |
| <span class="special">}</span> |
| <span class="keyword">else</span> |
| <span class="special">{</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Type "</span> <span class="special"><<</span> <span class="keyword">typeid</span><span class="special">(</span><span class="identifier">cpp_bin_float_quad</span><span class="special">).</span><span class="identifier">name</span><span class="special">()</span> <span class="special"><<</span> <span class="string">" does not have NaNs!"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| But using Boost.Math and suitable facets can permit portable use of both |
| NaNs and positive and negative infinity. |
| </p> |
| <p> |
| See <a href="../../../../../../../libs/math/example/nonfinite_facet_sstream.cpp" target="_top">boost:/libs/math/example/nonfinite_facet_sstream.cpp</a> |
| and we also need |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">nonfinite_num_facets</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <p> |
| Then we can equally well use a multiprecision type cpp_bin_float_quad: |
| </p> |
| <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_quad</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">cpp_bin_float_quad</span> <span class="identifier">T</span><span class="special">;</span> |
| |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">nonfinite_num_put</span><span class="special">;</span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">nonfinite_num_get</span><span class="special">;</span> |
| <span class="special">{</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">old_locale</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">tmp_locale</span><span class="special">(</span><span class="identifier">old_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_put</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">new_locale</span><span class="special">(</span><span class="identifier">tmp_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_get</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span> |
| <span class="identifier">ss</span><span class="special">.</span><span class="identifier">imbue</span><span class="special">(</span><span class="identifier">new_locale</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">inf</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">T</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">();</span> |
| <span class="identifier">ss</span> <span class="special"><<</span> <span class="identifier">inf</span><span class="special">;</span> <span class="comment">// Write out.</span> |
| <span class="identifier">assert</span><span class="special">(</span><span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special">==</span> <span class="string">"inf"</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">r</span><span class="special">;</span> |
| <span class="identifier">ss</span> <span class="special">>></span> <span class="identifier">r</span><span class="special">;</span> <span class="comment">// Read back in.</span> |
| <span class="identifier">assert</span><span class="special">(</span><span class="identifier">inf</span> <span class="special">==</span> <span class="identifier">r</span><span class="special">);</span> <span class="comment">// Confirms that the floating-point values really are identical.</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity output was "</span> <span class="special"><<</span> <span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</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> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"infinity input was "</span> <span class="special"><<</span> <span class="identifier">r</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <pre class="programlisting"><span class="identifier">infinity</span> <span class="identifier">output</span> <span class="identifier">was</span> <span class="identifier">inf</span> |
| <span class="identifier">infinity</span> <span class="identifier">input</span> <span class="identifier">was</span> <span class="identifier">inf</span> |
| </pre> |
| <p> |
| Similarly we can do the same with NaN (except that we cannot use <code class="computeroutput"><span class="identifier">assert</span></code>) |
| </p> |
| <pre class="programlisting"><span class="special">{</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">old_locale</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">tmp_locale</span><span class="special">(</span><span class="identifier">old_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_put</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">locale</span> <span class="identifier">new_locale</span><span class="special">(</span><span class="identifier">tmp_locale</span><span class="special">,</span> <span class="keyword">new</span> <span class="identifier">nonfinite_num_get</span><span class="special"><</span><span class="keyword">char</span><span class="special">>);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">ss</span><span class="special">;</span> |
| <span class="identifier">ss</span><span class="special">.</span><span class="identifier">imbue</span><span class="special">(</span><span class="identifier">new_locale</span><span class="special">);</span> |
| <span class="identifier">T</span> <span class="identifier">n</span><span class="special">;</span> |
| <span class="identifier">T</span> <span class="identifier">NaN</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">T</span><span class="special">>::</span><span class="identifier">quiet_NaN</span><span class="special">();</span> |
| <span class="identifier">ss</span> <span class="special"><<</span> <span class="identifier">NaN</span><span class="special">;</span> <span class="comment">// Write out.</span> |
| <span class="identifier">assert</span><span class="special">(</span><span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</span><span class="special">()</span> <span class="special">==</span> <span class="string">"nan"</span><span class="special">);</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"NaN output was "</span> <span class="special"><<</span> <span class="identifier">ss</span><span class="special">.</span><span class="identifier">str</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> |
| <span class="identifier">ss</span> <span class="special">>></span> <span class="identifier">n</span><span class="special">;</span> <span class="comment">// Read back in.</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"NaN input was "</span> <span class="special"><<</span> <span class="identifier">n</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| NaN output was nan NaN input was nan |
| </p> |
| <h6> |
| <a name="boost_multiprecision.tut.limits.functions.h10"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.functions.signaling_nan"></a></span><a class="link" href="functions.html#boost_multiprecision.tut.limits.functions.signaling_nan">Signaling |
| NaN</a> |
| </h6> |
| <p> |
| For floating-point types only, for which <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">has_signaling_NaN</span> |
| <span class="special">==</span> <span class="keyword">true</span></code>, |
| function <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">signaling_NaN</span><span class="special">()</span></code> |
| provides an implementation-defined representation for NaN that causes a |
| hardware trap. It should be noted however, that at least one implementation |
| of this function causes a hardware trap to be triggered simply by calling |
| <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">signaling_NaN</span><span class="special">()</span></code>, |
| and not only by using the value returned. |
| </p> |
| </div> |
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| <td align="right"><div class="copyright-footer">Copyright © 2002-2013 John Maddock and Christopher Kormanyos<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
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