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| author | Chanho Park <chanho61.park@samsung.com> | 2014-12-11 18:55:56 +0900 |
|---|---|---|
| committer | Chanho Park <chanho61.park@samsung.com> | 2014-12-11 18:55:56 +0900 |
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diff --git a/libs/multiprecision/doc/html/boost_multiprecision/tut/limits/constants.html b/libs/multiprecision/doc/html/boost_multiprecision/tut/limits/constants.html new file mode 100644 index 0000000000..2a085c9ad8 --- /dev/null +++ b/libs/multiprecision/doc/html/boost_multiprecision/tut/limits/constants.html @@ -0,0 +1,705 @@ +<html> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>std::numeric_limits<> constants</title> +<link rel="stylesheet" href="../../../../../../../doc/src/boostbook.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> +<link rel="home" href="../../../index.html" title="Chapter 1. Boost.Multiprecision"> +<link rel="up" href="../limits.html" title="Numeric Limits"> +<link rel="prev" href="../limits.html" title="Numeric Limits"> +<link rel="next" href="functions.html" title="std::numeric_limits<> functions"> +</head> +<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> +<table cellpadding="2" width="100%"><tr> +<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> +<td align="center"><a href="../../../../../../../index.html">Home</a></td> +<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> +<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> +<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> +<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="../limits.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../limits.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="functions.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +<div class="section"> +<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> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="../limits.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../limits.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="functions.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +</body> +</html> |