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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
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<a name="boost_multiprecision.tut.limits.constants"></a><a class="link" href="constants.html" title="std::numeric_limits<> constants">std::numeric_limits<>
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constants</a>
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</h4></div></div></div>
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<h5>
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<a name="boost_multiprecision.tut.limits.constants.h0"></a>
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<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>
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</h5>
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<p>
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<code class="computeroutput"><span class="keyword">true</span></code> for all arithmetic types
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(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>
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is specialized.
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</p>
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<p>
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A typical test is
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</p>
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<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>
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<span class="special">{</span>
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<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>
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<span class="comment">// ...</span>
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<span class="special">}</span>
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</pre>
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<p>
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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>
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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
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of <code class="computeroutput"><span class="identifier">numeric_limits</span></code> are indeed
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known at compile time, and don't vary at runtime. For example floating-point
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types with runtime-variable precision such as <code class="computeroutput"><span class="identifier">mpfr_float</span></code>
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have no <code class="computeroutput"><span class="identifier">numeric_limits</span></code>
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specialization as it would be impossible to define all the members at compile
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time. In contrast the precision of a type such as <code class="computeroutput"><span class="identifier">mpfr_float_50</span></code>
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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.
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</p>
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<p>
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Note that not all the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code>
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member constants and functions are meaningful for all user-defined types
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(UDT), such as the decimal and binary multiprecision types provided here.
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More information on this is given in the sections below.
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</p>
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<h5>
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<a name="boost_multiprecision.tut.limits.constants.h1"></a>
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<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>
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</h5>
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<p>
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For floating-point types, ∞ is defined wherever possible, but clearly infinity
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is meaningless for __arbitrary_precision arithmetic backends, and there
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is one floating-point type (GMP's <code class="computeroutput"><span class="identifier">mpf_t</span></code>,
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see <a class="link" href="../floats/gmp_float.html" title="gmp_float">gmp_float</a>)
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which has no notion of infinity or NaN at all.
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</p>
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<p>
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A typical test whether infinity is implemented is
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</p>
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<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>
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<span class="special">{</span>
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<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>
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<span class="special">}</span>
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</pre>
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<p>
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and using tests like this is strongly recommended to improve portability.
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</p>
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<p>
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If the backend is switched to a type that does not support infinity then,
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without checks like this, there will be trouble.
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</p>
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<h5>
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<a name="boost_multiprecision.tut.limits.constants.h2"></a>
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<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>
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</h5>
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<p>
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<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>
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<span class="keyword">true</span></code> if the type <code class="computeroutput"><span class="identifier">T</span></code>
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is signed.
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</p>
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<p>
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For built-in binary types, the sign is held in a single bit, but for other
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types (cpp_dec_float and cpp_bin_float) it may be a separate storage element,
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usually <code class="computeroutput"><span class="keyword">bool</span></code>.
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</p>
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<h5>
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<a name="boost_multiprecision.tut.limits.constants.h3"></a>
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<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>
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</h5>
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<p>
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<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>
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<span class="keyword">true</span></code> if type T uses exact representations.
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</p>
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<p>
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This is defined as <code class="computeroutput"><span class="keyword">true</span></code> for
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all integer types and <code class="computeroutput"><span class="keyword">false</span></code>
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for floating-point types.
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</p>
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<p>
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<a href="http://stackoverflow.com/questions/14203654/stdnumeric-limitsis-exact-what-is-a-usable-definition" target="_top">A
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usable definition</a> has been discussed.
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</p>
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<p>
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ISO/IEC 10967-1, Language independent arithmetic, noted by the C++ Standard
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defines
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</p>
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<pre class="programlisting"><span class="identifier">A</span> <span class="identifier">floating</span><span class="special">-</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>
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</pre>
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<p>
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The important practical distinction is that all integers (up to <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code>)
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can be stored exactly.
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</p>
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<p>
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<a href="http://en.wikipedia.org/wiki/Rational_number" target="_top">Rational</a>
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types using two integer types are also exact.
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</p>
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<p>
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Floating-point types <span class="bold"><strong>cannot store all real values</strong></span>
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(those in the set of ℜ) <span class="bold"><strong>exactly</strong></span>. For example,
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0.5 can be stored exactly in a binary floating-point, but 0.1 cannot. What
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is stored is the nearest representable real value, that is, rounded to
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nearest.
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</p>
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<p>
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Fixed-point types (usually decimal) are also defined as exact, in that
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they only store a <span class="bold"><strong>fixed precision</strong></span>, so
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half cents or pennies (or less) cannot be stored. The results of computations
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are rounded up or down, just like the result of integer division stored
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as an integer result.
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</p>
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<p>
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There are number of proposals to <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3407.html" target="_top">add
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Decimal floating-point Support to C++</a>.
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</p>
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<p>
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<a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2009/n2849.pdf" target="_top">Decimal
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TR</a>.
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</p>
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<p>
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And also <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3352.html" target="_top">C++
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Binary Fixed-Point Arithmetic</a>.
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</p>
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<h5>
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<a name="boost_multiprecision.tut.limits.constants.h4"></a>
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<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>
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</h5>
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<p>
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<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>
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<span class="keyword">true</span></code> if the set of values represented
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by the type <code class="computeroutput"><span class="identifier">T</span></code> is finite.
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</p>
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<p>
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This is <code class="computeroutput"><span class="keyword">true</span></code> for all built-in
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integer, fixed and floating-point types, and most multi-precision types.
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</p>
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<p>
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It is only <code class="computeroutput"><span class="keyword">false</span></code> for a few
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__arbitrary_precision types like <code class="computeroutput"><span class="identifier">cpp_int</span></code>.
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</p>
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<p>
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Rational and fixed-exponent representations are exact but not integer.
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</p>
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<h5>
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<a name="boost_multiprecision.tut.limits.constants.h5"></a>
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<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>
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</h5>
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<p>
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<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
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T can yield a result less than either value.
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">is_modulo</span> <span class="special">==</span>
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<span class="keyword">true</span></code> means that the type does not
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overflow, but, for example, 'wraps around' to zero, when adding one to
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the <code class="computeroutput"><span class="identifier">max</span><span class="special">()</span></code>
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value.
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</p>
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<p>
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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>
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is <code class="computeroutput"><span class="keyword">true</span></code>.
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</p>
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<p>
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<code class="computeroutput"><span class="keyword">bool</span></code> is the only exception.
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</p>
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<p>
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The modulo behaviour is sometimes useful, but also can be unexpected, and
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sometimes undesired, behaviour.
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</p>
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<p>
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Overflow of signed integers can be especially unexpected, possibly causing
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change of sign.
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</p>
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<p>
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Boost.Multiprecision integer type <code class="computeroutput"><span class="identifier">cpp_int</span></code>
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is not modulo because as an __arbitrary_precision types, it expands to
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hold any value that the machine resources permit.
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</p>
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<p>
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However fixed precision <a class="link" href="../ints/cpp_int.html" title="cpp_int">cpp_int</a>'s
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may be modulo if they are unchecked (i.e. they behave just like built in
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integers), but not if they are checked (overflow causes an exception to
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be raised).
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</p>
|
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<p>
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Built-in and multi-precision floating-point types are normally not modulo.
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</p>
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<p>
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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>
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<span class="special">==</span> <span class="keyword">true</span></code>.
|
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</p>
|
|
<h5>
|
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<a name="boost_multiprecision.tut.limits.constants.h6"></a>
|
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<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>
|
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</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
|
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and binary types) or 10 (for decimal types).
|
|
</p>
|
|
<h5>
|
|
<a name="boost_multiprecision.tut.limits.constants.h7"></a>
|
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<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>
|
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</h5>
|
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<p>
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The number of <code class="computeroutput"><span class="identifier">radix</span></code> digits
|
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that be represented without change:
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</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
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<li class="listitem">
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for integer types, the number of <span class="bold"><strong>non-sign bits</strong></span>
|
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in the significand.
|
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</li>
|
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<li class="listitem">
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for floating types, the number of <span class="bold"><strong>radix digits</strong></span>
|
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in the significand.
|
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</li>
|
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</ul></div>
|
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<p>
|
|
The values include any implicit bit, so for example, for the ubiquious
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<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
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binary64 </a>), <code class="computeroutput"><span class="identifier">digits</span></code>
|
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== 53, even though there are only 52 actual bits of the significand stored
|
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in the representation. The value of <code class="computeroutput"><span class="identifier">digits</span></code>
|
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reflects the fact that there is one implicit bit which is always set to
|
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1.
|
|
</p>
|
|
<p>
|
|
The Boost.Multiprecision binary types do not use an implicit bit, so the
|
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<code class="computeroutput"><span class="identifier">digits</span></code> member reflects
|
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exactly how many bits of precision were requested:
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</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>
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<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>
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<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>
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<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>,
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<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,
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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>,
|
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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 the most common <code class="computeroutput"><span class="keyword">double</span></code>
|
|
floating-point type,<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 C++11 <code class="computeroutput"><span class="identifier">max_digits10</span></code> where possible (see <a class="link" href="constants.html#boost_multiprecision.tut.limits.constants.max_digits10">below</a>).
|
|
</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-dependent, 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>
|
|
<span class="emphasis"><em>including a sign and a space</em></span>.
|
|
</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>
|
|
<p>
|
|
The extra two or three least-significant digits are 'noisy' and may be
|
|
junk, but if you want to 'round-trip' - printing a value out as a decimal
|
|
digit string and reading it back in - (most commonly during serialization
|
|
and de-serialization) 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>.
|
|
</p>
|
|
<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 before 2013 and the default floating-point
|
|
format, a small range of double-precision floating-point values with
|
|
a significand of approximately 0.0001 to 0.004 and exponent values of
|
|
1010 to 1014 do not round-trip exactly being off by one least significant
|
|
bit, for probably every third value of the significand.
|
|
</p>
|
|
<p>
|
|
A workaround is using the scientific or exponential format <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">scientific</span></code>.
|
|
</p>
|
|
<p>
|
|
Other compilers also fail to implement round-tripping entirely fault-free,
|
|
for example, see <a href="https://www.exploringbinary.com/incorrectly-rounded-conversions-in-gcc-and-glibc/" target="_top">Incorrectly
|
|
Rounded Conversions in GCC and GLIBC</a>.
|
|
</p>
|
|
<p>
|
|
For more details see <a href="https://www.exploringbinary.com/incorrect-round-trip-conversions-in-visual-c-plus-plus/" target="_top">Incorrect
|
|
Round-Trip Conversions in Visual C++</a>, and references therein
|
|
and <a href="https://arxiv.org/pdf/1310.8121.pdf" target="_top">Easy Accurate Reading
|
|
and Writing of Floating-Point Numbers, Aubrey Jaffer (August 2018)</a>.
|
|
</p>
|
|
<p>
|
|
Microsoft VS2017 and other recent compilers, now use the <a href="https://doi.org/10.1145/3192366.3192369" target="_top">Ryu
|
|
fast float-to-string conversion by Ulf Adams</a> algorithm, claimed
|
|
to be both exact and fast for 32 and 64-bit floating-point numbers.
|
|
</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.
|
|
</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>
|
|
<span class="emphasis"><em>requires cxx11_numeric_limits</em></span> is a suitable test
|
|
for use of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">float</span><span class="special">>::</span><span class="identifier">max_digits10</span></code> to control if a target
|
|
in a jamfile used by a Boost B2/bjam program is built, or not.
|
|
</p></td></tr>
|
|
</table></div>
|
|
<p>
|
|
If <code class="computeroutput"><span class="identifier">max_digits10</span></code> is not
|
|
available, you should use the <a href="http://www.cs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF" target="_top">Kahan
|
|
formula for floating-point type T</a>.
|
|
</p>
|
|
<p>
|
|
In C++, the equations for what Kahan (on page 4) describes as 'at least'
|
|
and 'at most' are:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">static</span> <span class="keyword">long</span> <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">log10Two</span> <span class="special">=</span> <span class="number">0.30102999566398119521373889472449L</span><span class="special">;</span> <span class="comment">// log10(2.)</span>
|
|
|
|
<span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="identifier">floor</span><span class="special">((</span><span class="identifier">significand_digits</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">log10Two</span><span class="special">));</span> <span class="comment">// == digits10 - 'at least' .</span>
|
|
<span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="identifier">ceil</span><span class="special">(</span><span class="number">1</span> <span class="special">+</span> <span class="identifier">significand_digits</span> <span class="special">*</span> <span class="identifier">log10Two</span><span class="special">));</span> <span class="comment">// == max_digits10 - 'at most'.</span>
|
|
</pre>
|
|
<p>
|
|
Unfortunately, these cannot be evaluated (at least by C++03) at <span class="bold"><strong>compile-time</strong></span>. So the following expression is often
|
|
used instead.
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">max_digits10</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="comment">// == 2 + std::numeric_limits<T>::digits for double and 64-bit long double.</span>
|
|
<span class="comment">// == 3 + std::numeric_limits<T>::digits for float, 80-bit long-double and __float128.</span>
|
|
</pre>
|
|
<p>
|
|
often the actual values are computed for the C limits macros:
|
|
</p>
|
|
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">FLT_MAXDIG10</span> <span class="special">(</span><span class="number">2</span><span class="special">+</span><span class="identifier">FLT_MANT_DIG</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="comment">// 9</span>
|
|
<span class="preprocessor">#define</span> <span class="identifier">DBL_MAXDIG10</span> <span class="special">(</span><span class="number">2</span><span class="special">+</span> <span class="special">(</span><span class="identifier">DBL_MANT_DIG</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="comment">// 17</span>
|
|
<span class="preprocessor">#define</span> <span class="identifier">LDBL_MAXDIG10</span> <span class="special">(</span><span class="number">2</span><span class="special">+</span> <span class="special">(</span><span class="identifier">LDBL_MANT_DIG</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="comment">// 17 for MSVC, 18 for others.</span>
|
|
</pre>
|
|
<p>
|
|
The factor 3010U/10000U is <span class="emphasis"><em>log<sub>10</sub>(2) = 0.3010</em></span> that
|
|
can be evaluated at compile-time using only <code class="computeroutput"><span class="keyword">short</span>
|
|
<span class="keyword">unsigned</span> <span class="keyword">int</span></code>s
|
|
to be a desirable <code class="computeroutput"><span class="keyword">const</span></code> or
|
|
<code class="computeroutput"><span class="keyword">constexpr</span></code> (and usually also
|
|
<code class="computeroutput"><span class="keyword">static</span></code>).
|
|
</p>
|
|
<p>
|
|
Boost macros allow this to be done portably, see <a href="http://www.boost.org/doc/libs/1_58_0/libs/config/doc/html/boost_config/boost_macro_reference.html" target="_top">BOOST_CONSTEXPR_OR_CONST
|
|
or BOOST_STATIC_CONSTEXPR</a>.
|
|
</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>
|
|
For example, to be portable (including obselete platforms) for type <code class="computeroutput"><span class="identifier">T</span></code> where <code class="computeroutput"><span class="identifier">T</span></code>
|
|
may be: <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
|
|
<span class="keyword">double</span></code>, <code class="computeroutput"><span class="number">128</span><span class="special">-</span><span class="identifier">bit</span> <span class="identifier">quad</span> <span class="identifier">type</span></code>,
|
|
<code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code> ...
|
|
</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="preprocessor">#if</span> <span class="identifier">defined</span> <span class="identifier">BOOST_NO_CXX11_NUMERIC_LIMITS</span>
|
|
<span class="comment">// No max_digits10 implemented.</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">max_digits10</span><span class="special"><</span><span class="identifier">T</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">// Wrong value for std::numeric_limits<float>::max_digits10.</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">max_digits10</span><span class="special"><</span><span class="identifier">T</span><span class="special">>());</span>
|
|
<span class="preprocessor">#else</span> <span class="comment">// Use the C++11 max_digits10.</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(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">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="comment">// 9</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="identifier">max_digits10</span><span class="special">)</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 © 2002-2019 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>
|
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</div></td>
|
|
</tr></table>
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