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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="boost_multiprecision.tut.limits.constants"></a><a class="link" href="constants.html" title="std::numeric_limits<> constants">std::numeric_limits<> |
| constants</a> |
| </h4></div></div></div> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h0"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_specialized"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_specialized">is_specialized</a> |
| </h5> |
| <p> |
| <code class="computeroutput"><span class="keyword">true</span></code> for all arithmetic types |
| (integer, floating and fixed-point) 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">numeric_limits</span></code> |
| is specialized. |
| </p> |
| <p> |
| A typical test is |
| </p> |
| <pre class="programlisting"><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">T</span><span class="special">>::</span><span class="identifier">is_specialized</span> <span class="special">==</span> <span class="keyword">false</span><span class="special">)</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">T</span><span class="special">).</span><span class="identifier">name</span><span class="special">()</span> <span class="special"><<</span> <span class="string">" is not specialized for std::numeric_limits!"</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">// ...</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| Typically <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">is_specialized</span></code> |
| is <code class="computeroutput"><span class="keyword">true</span></code> for all <code class="computeroutput"><span class="identifier">T</span></code> where the compile-time constant members |
| of <code class="computeroutput"><span class="identifier">numeric_limits</span></code> are indeed |
| known at compile time, and don't vary at runtime. For example floating |
| point types with runtime-variable precision such as <code class="computeroutput"><span class="identifier">mpfr_float</span></code> |
| have no <code class="computeroutput"><span class="identifier">numeric_limits</span></code> |
| specialization as it would be impossible to define all the members at compile |
| time. In contrast the precision of a type such as <code class="computeroutput"><span class="identifier">mpfr_float_50</span></code> |
| is known at compile time, and so it <span class="emphasis"><em>does</em></span> have a <code class="computeroutput"><span class="identifier">numeric_limits</span></code> specialization. |
| </p> |
| <p> |
| Note that not all the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code> |
| member constants and functions are meaningful for all user-defined types |
| (UDT), such as the decimal and binary multiprecision types provided here. |
| More information on this is given in the sections below. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h1"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.infinity"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.infinity">infinity</a> |
| </h5> |
| <p> |
| For floating-point types, ∞ is defined wherever possible, but clearly infinity |
| is meaningless for __arbitrary_precision arithmetic backends, and there |
| is one floating point type (GMP's <code class="computeroutput"><span class="identifier">mpf_t</span></code>, |
| see <a class="link" href="../floats/gmp_float.html" title="gmp_float">gmp_float</a>) |
| which has no notion of infinity or NaN at all. |
| </p> |
| <p> |
| A typical test whether infinity is implemented is |
| </p> |
| <pre class="programlisting"><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">T</span><span class="special">>::</span><span class="identifier">has_infinity</span><span class="special">)</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">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="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> |
| and using tests like this is strongly recommended to improve portability. |
| </p> |
| <p> |
| If the backend is switched to a type that does not support infinity then, |
| without checks like this, there will be trouble. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h2"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_signed"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_signed">is_signed</a> |
| </h5> |
| <p> |
| <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_signed</span> <span class="special">==</span> |
| <span class="keyword">true</span></code> if the type <code class="computeroutput"><span class="identifier">T</span></code> |
| is signed. |
| </p> |
| <p> |
| For built-in binary types, the sign is held in a single bit, but for other |
| types (cpp_dec_float and cpp_bin_float) it may be a separate storage element, |
| usually <code class="computeroutput"><span class="keyword">bool</span></code>. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h3"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_exact"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_exact">is_exact</a> |
| </h5> |
| <p> |
| <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_exact</span> <span class="special">==</span> |
| <span class="keyword">true</span></code> if type T uses exact representations. |
| </p> |
| <p> |
| This is defined as <code class="computeroutput"><span class="keyword">true</span></code> for |
| all integer types and <code class="computeroutput"><span class="keyword">false</span></code> |
| for floating-point types. |
| </p> |
| <p> |
| <a href="http://stackoverflow.com/questions/14203654/stdnumeric-limitsis-exact-what-is-a-usable-definition" target="_top">A |
| usable definition</a> has been discussed. |
| </p> |
| <p> |
| ISO/IEC 10967-1, Language independent arithmetic, noted by the C++ Standard |
| defines |
| </p> |
| <pre class="programlisting"><span class="identifier">A</span> <span class="identifier">floating</span> <span class="identifier">point</span> <span class="identifier">type</span> <span class="identifier">F</span> <span class="identifier">shall</span> <span class="identifier">be</span> <span class="identifier">a</span> <span class="identifier">finite</span> <span class="identifier">subset</span> <span class="identifier">of</span> <span class="special">[</span><span class="identifier">real</span><span class="special">].</span> |
| </pre> |
| <p> |
| The important practical distinction is that all integers (up to <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code>) |
| can be stored exactly. |
| </p> |
| <p> |
| <a href="http://en.wikipedia.org/wiki/Rational_number" target="_top">Rational</a> |
| types using two integer types are also exact. |
| </p> |
| <p> |
| Floating-point types <span class="bold"><strong>cannot store all real values</strong></span> |
| (those in the set of ℜ) <span class="bold"><strong>exactly</strong></span>. For example, |
| 0.5 can be stored exactly in a binary floating-point, but 0.1 cannot. What |
| is stored is the nearest representable real value, that is, rounded to |
| nearest. |
| </p> |
| <p> |
| Fixed-point types (usually decimal) are also defined as exact, in that |
| they only store a <span class="bold"><strong>fixed precision</strong></span>, so |
| half cents or pennies (or less) cannot be stored. The results of computations |
| are rounded up or down, just like the result of integer division stored |
| as an integer result. |
| </p> |
| <p> |
| There are number of proposals to <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3407.html" target="_top">add |
| Decimal Floating Point Support to C++</a>. |
| </p> |
| <p> |
| <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2009/n2849.pdf" target="_top">Decimal |
| TR</a>. |
| </p> |
| <p> |
| And also <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3352.html" target="_top">C++ |
| Binary Fixed-Point Arithmetic</a>. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h4"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_bounded"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_bounded">is_bounded</a> |
| </h5> |
| <p> |
| <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_bounded</span> <span class="special">==</span> |
| <span class="keyword">true</span></code> if the set of values represented |
| by the type <code class="computeroutput"><span class="identifier">T</span></code> is finite. |
| </p> |
| <p> |
| This is <code class="computeroutput"><span class="keyword">true</span></code> for all built-in |
| integer, fixed and floating-point types, and most multi-precision types. |
| </p> |
| <p> |
| It is only <code class="computeroutput"><span class="keyword">false</span></code> for a few |
| __arbitrary_precision types like <code class="computeroutput"><span class="identifier">cpp_int</span></code>. |
| </p> |
| <p> |
| Rational and fixed-exponent representations are exact but not integer. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h5"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.is_modulo"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.is_modulo">is_modulo</a> |
| </h5> |
| <p> |
| <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_modulo</span></code> is defined as <code class="computeroutput"><span class="keyword">true</span></code> if adding two positive values of type |
| T can yield a result less than either value. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="identifier">is_modulo</span> <span class="special">==</span> |
| <span class="keyword">true</span></code> means that the type does not |
| overflow, but, for example, 'wraps around' to zero, when adding one to |
| the <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code> |
| value. |
| </p> |
| <p> |
| For most built-in integer types, <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">is_modulo</span></code> |
| is <code class="computeroutput"><span class="keyword">true</span></code>. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="keyword">bool</span></code> is the only exception. |
| </p> |
| <p> |
| The modulo behaviour is sometimes useful, but also can be unexpected, and |
| sometimes undesired, behaviour. |
| </p> |
| <p> |
| Overflow of signed integers can be especially unexpected, possibly causing |
| change of sign. |
| </p> |
| <p> |
| Boost.Multiprecision integer type <code class="computeroutput"><span class="identifier">cpp_int</span></code> |
| is not modulo because as an __arbitrary_precision types, it expands to |
| hold any value that the machine resources permit. |
| </p> |
| <p> |
| However fixed precision <a class="link" href="../ints/cpp_int.html" title="cpp_int">cpp_int</a>'s |
| may be modulo if they are unchecked (i.e. they behave just like built in |
| integers), but not if they are checked (overflow causes an exception to |
| be raised). |
| </p> |
| <p> |
| Built-in and multi-precision floating-point types are normally not modulo. |
| </p> |
| <p> |
| Where possible, overflow is 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">infinity</span><span class="special">()</span></code>, provided <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">has_infinity</span> |
| <span class="special">==</span> <span class="keyword">true</span></code>. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h6"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.radix"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.radix">radix</a> |
| </h5> |
| <p> |
| Constant <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">radix</span></code> returns either 2 (for built-in |
| and binary types) or 10 (for decimal types). |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h7"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.digits"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.digits">digits</a> |
| </h5> |
| <p> |
| The number of <code class="computeroutput"><span class="identifier">radix</span></code> digits |
| that be represented without change: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| for integer types, the number of <span class="bold"><strong>non-sign bits</strong></span> |
| in the significand. |
| </li> |
| <li class="listitem"> |
| for floating types, the number of <span class="bold"><strong>radix digits</strong></span> |
| in the significand. |
| </li> |
| </ul></div> |
| <p> |
| The values include any implicit bit, so for example, for the ubiquious |
| <code class="computeroutput"><span class="keyword">double</span></code> using 64 bits (<a href="http://en.wikipedia.org/wiki/Double_precision_floating-point_format" target="_top">IEEE |
| binary64 </a>), <code class="computeroutput"><span class="identifier">digits</span></code> |
| == 53, even though there are only 52 actual bits of the significand stored |
| in the representation. The value of <code class="computeroutput"><span class="identifier">digits</span></code> |
| reflects the fact that there is one implicit bit which is always set to |
| 1. |
| </p> |
| <p> |
| The Boost.Multiprecision binary types do not use an implicit bit, so the |
| <code class="computeroutput"><span class="identifier">digits</span></code> member reflects |
| exactly how many bits of precision were requested: |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">53</span><span class="special">,</span> <span class="identifier">digit_base_2</span><span class="special">></span> <span class="special">></span> <span class="identifier">float64</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">113</span><span class="special">,</span> <span class="identifier">digit_base_2</span><span class="special">></span> <span class="special">></span> <span class="identifier">float128</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">float64</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">==</span> <span class="number">53.</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">float128</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">==</span> <span class="number">113.</span> |
| </pre> |
| <p> |
| For the most common case of <code class="computeroutput"><span class="identifier">radix</span> |
| <span class="special">==</span> <span class="number">2</span></code>, |
| <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">digits</span></code> is the number of bits in the representation, |
| not counting any sign bit. |
| </p> |
| <p> |
| For a decimal integer type, when <code class="computeroutput"><span class="identifier">radix</span> |
| <span class="special">==</span> <span class="number">10</span></code>, |
| it is the number of decimal digits. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h8"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.digits10"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.digits10">digits10</a> |
| </h5> |
| <p> |
| Constant <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">digits10</span></code> returns the number of decimal |
| digits that can be represented without change or loss. |
| </p> |
| <p> |
| For example, <code class="computeroutput"><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">unsigned</span> <span class="keyword">char</span><span class="special">>::</span><span class="identifier">digits10</span></code> is 2. |
| </p> |
| <p> |
| This somewhat inscrutable definition means that an <code class="computeroutput"><span class="keyword">unsigned</span> |
| <span class="keyword">char</span></code> can hold decimal values <code class="computeroutput"><span class="number">0.</span><span class="special">.</span><span class="number">99</span></code> |
| without loss of precision or accuracy, usually from truncation. |
| </p> |
| <p> |
| Had the definition been 3 then that would imply it could hold 0..999, but |
| as we all know, an 8-bit <code class="computeroutput"><span class="keyword">unsigned</span> |
| <span class="keyword">char</span></code> can only hold 0..255, and an |
| attempt to store 256 or more will involve loss or change. |
| </p> |
| <p> |
| For bounded integers, it is thus <span class="bold"><strong>one less</strong></span> |
| than number of decimal digits you need to display the biggest integer |
| <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>. |
| This value can be used to predict the layout width required for |
| </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">std</span><span class="special">::</span><span class="identifier">setw</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">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="comment">// digits10+1, and +1 for sign.</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpos</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">short</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)()</span> <span class="comment">// +32767</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">setw</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">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</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="keyword">short</span><span class="special">>::</span><span class="identifier">min</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="comment">// -32767</span> |
| </pre> |
| <p> |
| For example, <code class="computeroutput"><span class="keyword">unsigned</span> <span class="keyword">short</span></code> |
| is often stored in 16 bits, so the maximum value is 0xFFFF or 65535. |
| </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">std</span><span class="special">::</span><span class="identifier">setw</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">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="comment">// digits10+1, and +1 for sign.</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showpos</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">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)()</span> <span class="comment">// 65535</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">setw</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">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="special">+</span><span class="number">1</span> <span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="comment">// digits10+1, and +1 for sign.</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">unsigned</span> <span class="keyword">short</span><span class="special">>::</span><span class="identifier">min</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="comment">// 0</span> |
| </pre> |
| <p> |
| For bounded floating-point types, if we create a <code class="computeroutput"><span class="keyword">double</span></code> |
| with a value with <code class="computeroutput"><span class="identifier">digits10</span></code> |
| (usually 15) decimal digits, <code class="computeroutput"><span class="number">1e15</span></code> |
| or <code class="computeroutput"><span class="number">1000000000000000</span></code> : |
| </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">1e15</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">dp1</span> <span class="special">=</span> <span class="identifier">d</span><span class="special">+</span><span class="number">1</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="string">"\n"</span> <span class="special"><<</span> <span class="identifier">dp1</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">// 1000000000000000</span> |
| <span class="comment">// 1000000000000001</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dp1</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">// 1</span> |
| </pre> |
| <p> |
| and we can increment this value to <code class="computeroutput"><span class="number">1000000000000001</span></code> |
| as expected and show the difference too. |
| </p> |
| <p> |
| But if we try to repeat this with more than <code class="computeroutput"><span class="identifier">digits10</span></code> |
| 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">1e16</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">dp1</span> <span class="special">=</span> <span class="identifier">d</span><span class="special">+</span><span class="number">1</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="string">"\n"</span> <span class="special"><<</span> <span class="identifier">dp1</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">// 10000000000000000</span> |
| <span class="comment">// 10000000000000000</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">dp1</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">// 0 !!!</span> |
| </pre> |
| <p> |
| then we find that when we add one it has no effect, and display show that |
| there is loss of precision. See <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">Loss |
| of significance or cancellation error</a>. |
| </p> |
| <p> |
| So <code class="computeroutput"><span class="identifier">digits10</span></code> is the number |
| of decimal digits <span class="bold"><strong>guaranteed</strong></span> to be correct. |
| </p> |
| <p> |
| For example, 'round-tripping' for <code class="computeroutput"><span class="keyword">double</span></code>: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| If a decimal string with at most <code class="computeroutput"><span class="identifier">digits10</span></code>( |
| == 15) significant decimal digits is converted to <code class="computeroutput"><span class="keyword">double</span></code> |
| and then converted back to the same number of significant decimal digits, |
| then the final string will match the original 15 decimal digit string. |
| </li> |
| <li class="listitem"> |
| If a <code class="computeroutput"><span class="keyword">double</span></code> floating-point |
| number is converted to a decimal string with at least 17 decimal digits |
| and then converted back to <code class="computeroutput"><span class="keyword">double</span></code>, |
| then the result will be binary identical to the original <code class="computeroutput"><span class="keyword">double</span></code> value. |
| </li> |
| </ul></div> |
| <p> |
| For most purposes, you will much more likely want <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">max_digits10</span></code>, |
| the number of decimal digits that ensure that a change of one least significant |
| bit (ULP) produces a different decimal digits string. |
| </p> |
| <p> |
| For nearly all floating-point types, <code class="computeroutput"><span class="identifier">max_digits10</span></code> |
| is <code class="computeroutput"><span class="identifier">digits10</span><span class="special">+</span><span class="number">2</span></code>, but you should use <code class="computeroutput"><span class="identifier">max_digits10</span></code> |
| where possible. |
| </p> |
| <p> |
| If <code class="computeroutput"><span class="identifier">max_digits10</span></code> is not |
| available, you should using the <a href="http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF" target="_top">Kahan |
| formula for floating-point type T</a> |
| </p> |
| <pre class="programlisting"><span class="identifier">max_digits10</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">digits</span> <span class="special">*</span> <span class="number">3010U</span><span class="special">/</span><span class="number">10000U</span><span class="special">;</span> |
| </pre> |
| <p> |
| The factor is log<sub>10</sub>(2) = 0.3010 but must be evaluated at compile time using |
| only integers. |
| </p> |
| <p> |
| (See also <a href="http://www.loria.fr/~zimmerma/mca/mca-cup-0.5.9.pdf" target="_top">Richard |
| P. Brent and Paul Zimmerman, Modern Computer Arithmetic</a> Equation |
| 3.8 on page 116.). |
| </p> |
| <p> |
| The extra two (or 3) least significant digits are 'noisy' and may be junk, |
| but if you want to 'round-trip' - printing a value out and reading it back |
| in - you must use <code class="computeroutput"><span class="identifier">os</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="identifier">T</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">)</span></code>. |
| For at least one popular compiler, you must also use <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span></code> |
| format. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h9"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.max_digits10"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.max_digits10">max_digits10</a> |
| </h5> |
| <p> |
| <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_digits10</span></code> was added for floating-point |
| because <code class="computeroutput"><span class="identifier">digits10</span></code> decimal |
| digits are insufficient to show a least significant bit (ULP) change giving |
| puzzling displays like |
| </p> |
| <pre class="programlisting"><span class="number">0.666666666666667</span> <span class="special">!=</span> <span class="number">0.666666666666667</span> |
| </pre> |
| <p> |
| from failure to 'round-trip', for example: |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">write</span> <span class="special">=</span> <span class="number">2.</span><span class="special">/</span><span class="number">3</span><span class="special">;</span> <span class="comment">// Any arbitrary value that cannot be represented exactly.</span> |
| <span class="keyword">double</span> <span class="identifier">read</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">stringstream</span> <span class="identifier">s</span><span class="special">;</span> |
| <span class="identifier">s</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">digits10</span><span class="special">);</span> <span class="comment">// or `float64_t` for 64-bit IEE754 double.</span> |
| <span class="identifier">s</span> <span class="special"><<</span> <span class="identifier">write</span><span class="special">;</span> |
| <span class="identifier">s</span> <span class="special">>></span> <span class="identifier">read</span><span class="special">;</span> |
| <span class="keyword">if</span><span class="special">(</span><span class="identifier">read</span> <span class="special">!=</span> <span class="identifier">write</span><span class="special">)</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">setprecision</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">digits10</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="identifier">read</span> <span class="special"><<</span> <span class="string">" != "</span> <span class="special"><<</span> <span class="identifier">write</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> |
| If you wish to ensure that a change of one least significant bit (ULP) |
| produces a different decimal digits string, then <code class="computeroutput"><span class="identifier">max_digits10</span></code> |
| is the precision to use. |
| </p> |
| <p> |
| For example: |
| </p> |
| <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">pi</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">double_constants</span><span class="special">::</span><span class="identifier">pi</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">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="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pi</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">// 3.1415926535897931</span> |
| </pre> |
| <p> |
| will display π to the maximum possible precision using a <code class="computeroutput"><span class="keyword">double</span></code>. |
| </p> |
| <p> |
| and similarly for a much higher precision type: |
| </p> |
| <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</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="special">></span> <span class="identifier">cpp_dec_float_50</span><span class="special">;</span> <span class="comment">// 50 decimal digits.</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_50</span><span class="special">;</span> |
| |
| <span class="identifier">cpp_dec_float_50</span> <span class="identifier">pi</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">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="identifier">cpp_dec_float_50</span><span class="special">>::</span><span class="identifier">max_digits10</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">pi</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">// 3.141592653589793238462643383279502884197169399375105820974944592307816406</span> |
| </pre> |
| <p> |
| For integer types, <code class="computeroutput"><span class="identifier">max_digits10</span></code> |
| is implementation-dependant, but is usually <code class="computeroutput"><span class="identifier">digits10</span> |
| <span class="special">+</span> <span class="number">2</span></code>. |
| This is the output field width required for the maximum value of the type |
| T <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> |
| including a sign and a space. |
| </p> |
| <p> |
| So this will produce neat columns. |
| </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">std</span><span class="special">::</span><span class="identifier">setw</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">int</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">)</span> <span class="special">...</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> |
| For Microsoft Visual Studio 2010, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> |
| is wrongly defined as 8. It should be 9. |
| </p></td></tr> |
| </table></div> |
| <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> |
| For Microsoft Visual Studio, and default float format, a small range |
| of values approximately 0.0001 to 0.004, with exponent values of 3f2 |
| to 3f6, are wrongly input by one least significant bit, probably every |
| third value of significand. |
| </p> |
| <p> |
| A workaround is using scientific or exponential format <code class="computeroutput"><span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span></code>. |
| </p> |
| </td></tr> |
| </table></div> |
| <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> |
| BOOST_NO_CXX11_NUMERIC_LIMITS is a suitable feature-test macro to determine |
| if <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> is implemented on any |
| platform. If <code class="computeroutput"><span class="identifier">max_digits10</span></code> |
| is not available, you should using the <a href="http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF" target="_top">Kahan |
| formula for floating-point type T</a>. See above. |
| </p></td></tr> |
| </table></div> |
| <p> |
| For example, to be portable, including older platforms: |
| </p> |
| <pre class="programlisting"> <span class="keyword">typedef</span> <span class="keyword">float</span> <span class="identifier">T</span><span class="special">;</span> <span class="comment">// Any type: `double`, cpp_dec_float_50, bin_128bit_double_type ...</span> |
| |
| <span class="preprocessor">#if</span> <span class="identifier">defined</span><span class="special">(</span><span class="identifier">BOOST_NO_CXX11_NUMERIC_LIMITS</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">precision</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">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">*</span> <span class="number">3010U</span><span class="special">/</span><span class="number">10000U</span><span class="special">);</span> |
| <span class="preprocessor">#else</span> |
| <span class="preprocessor"># if</span> <span class="special">(</span><span class="identifier">_MSC_VER</span> <span class="special"><=</span> <span class="number">1600</span><span class="special">)</span> <span class="comment">// Correct wrong value for float.</span> |
| <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="number">2</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">digits</span> <span class="special">*</span> <span class="number">3010U</span><span class="special">/</span><span class="number">10000U</span><span class="special">);</span> |
| <span class="preprocessor"># else</span> |
| <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="identifier">T</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span> |
| <span class="preprocessor"># endif</span> |
| <span class="preprocessor">#endif</span> |
| |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"std::cout.precision = "</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">precision</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="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">1.2345678901234567889</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">"x = "</span> <span class="special"><<</span> <span class="identifier">x</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">//</span> |
| </pre> |
| <p> |
| which should output: |
| </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="number">9</span> |
| <span class="identifier">x</span> <span class="special">=</span> <span class="number">1.23456789</span> |
| </pre> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h10"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.round_style"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.round_style">round_style</a> |
| </h5> |
| <p> |
| The rounding style determines how the result of floating-point operations |
| is treated when the result cannot be <span class="bold"><strong>exactly represented</strong></span> |
| in the significand. Various rounding modes may be provided: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| round to nearest up or down (default for floating-point types). |
| </li> |
| <li class="listitem"> |
| round up (toward positive infinity). |
| </li> |
| <li class="listitem"> |
| round down (toward negative infinity). |
| </li> |
| <li class="listitem"> |
| round toward zero (integer types). |
| </li> |
| <li class="listitem"> |
| no rounding (if decimal radix). |
| </li> |
| <li class="listitem"> |
| rounding mode is not determinable. |
| </li> |
| </ul></div> |
| <p> |
| For integer types, <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_style</span></code> |
| is always towards zero, so |
| </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">round_style</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round_to_zero</span><span class="special">;</span> |
| </pre> |
| <p> |
| A decimal type, <code class="computeroutput"><span class="identifier">cpp_dec_float</span></code> |
| rounds in no particular direction, which is to say it doesn't round at |
| all. And since there are several guard digits, it's not really the same |
| as truncation (round toward zero) either. |
| </p> |
| <p> |
| For floating-point types, it is normal to round to nearest. |
| </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">round_style</span> <span class="special">==</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">round_to_nearest</span><span class="special">;</span> |
| </pre> |
| <p> |
| See 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></code> for the maximum error (in |
| ULP) that rounding can cause. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h11"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.has_denorm_loss"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.has_denorm_loss">has_denorm_loss</a> |
| </h5> |
| <p> |
| <code class="computeroutput"><span class="keyword">true</span></code> if a loss of precision |
| is detected as a <a href="http://en.wikipedia.org/wiki/Denormalization" target="_top">denormalization</a> |
| loss, rather than an inexact result. |
| </p> |
| <p> |
| Always <code class="computeroutput"><span class="keyword">false</span></code> for integer types. |
| </p> |
| <p> |
| <code class="computeroutput"><span class="keyword">false</span></code> for all types which |
| do not have <code class="computeroutput"><span class="identifier">has_denorm</span></code> |
| == <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span></code>. |
| </p> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h12"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.denorm_style"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.denorm_style">denorm_style</a> |
| </h5> |
| <p> |
| <a href="http://en.wikipedia.org/wiki/Denormal_number" target="_top">Denormalized |
| values</a> are representations with a variable number of exponent bits |
| that can permit gradual underflow, so that, if type T is <code class="computeroutput"><span class="keyword">double</span></code>. |
| </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">denorm_min</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">T</span><span class="special">>::</span><span class="identifier">min</span><span class="special">()</span> |
| </pre> |
| <p> |
| A type may have any of the following <code class="computeroutput"><span class="keyword">enum</span> |
| <span class="identifier">float_denorm_style</span></code> values: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_absent</span></code>, if it does not allow |
| denormalized values. (Always used for all integer and exact types). |
| </li> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_present</span></code>, if the floating-point |
| type allows denormalized values. |
| </li> |
| <li class="listitem"> |
| <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">denorm_indeterminate</span></code>, if indeterminate |
| at compile time. |
| </li> |
| </ul></div> |
| <h5> |
| <a name="boost_multiprecision.tut.limits.constants.h13"></a> |
| <span class="phrase"><a name="boost_multiprecision.tut.limits.constants.tinyness_before_rounding"></a></span><a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.tinyness_before_rounding">Tinyness |
| before rounding</a> |
| </h5> |
| <p> |
| <code class="computeroutput"><span class="keyword">bool</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">tinyness_before</span></code> |
| </p> |
| <p> |
| <code class="computeroutput"><span class="keyword">true</span></code> if a type can determine |
| that a value is too small to be represent as a normalized value before |
| rounding it. |
| </p> |
| <p> |
| Generally true for <code class="computeroutput"><span class="identifier">is_iec559</span></code> |
| floating-point built-in types, but false for integer types. |
| </p> |
| <p> |
| Standard-compliant IEEE 754 floating-point implementations may detect the |
| floating-point underflow at three predefined moments: |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| After computation of a result with absolute value smaller than <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>, |
| such implementation detects <span class="emphasis"><em>tinyness before rounding</em></span> |
| (e.g. UltraSparc). |
| </li> |
| <li class="listitem"> |
| After rounding of the result 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">digits</span></code> |
| bits, if the result is tiny, such implementation detects <span class="emphasis"><em>tinyness |
| after rounding</em></span> (e.g. SuperSparc). |
| </li> |
| <li class="listitem"> |
| If the conversion of the rounded tiny result to subnormal form resulted |
| in the loss of precision, such implementation detects <span class="emphasis"><em>denorm |
| loss</em></span>. |
| </li> |
| </ol></div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <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>) |
| </p> |
| </div></td> |
| </tr></table> |
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