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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="boost_numericconversion.definitions"></a><a class="link" href="definitions.html" title="Definitions">Definitions</a>
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</h2></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.introduction">Introduction</a></span></dt>
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.types_and_values">Types
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and Values</a></span></dt>
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.c___arithmetic_types">C++
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Arithmetic Types</a></span></dt>
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.numeric_types">Numeric
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Types</a></span></dt>
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.range_and_precision">Range
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and Precision</a></span></dt>
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.exact__correctly_rounded_and_out_of_range_representations">Exact,
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Correctly Rounded and Out-Of-Range Representations</a></span></dt>
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.standard__numeric__conversions">Standard
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(numeric) Conversions</a></span></dt>
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<dt><span class="section"><a href="definitions.html#boost_numericconversion.definitions.subranged_conversion_direction__subtype_and_supertype">Subranged
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Conversion Direction, Subtype and Supertype</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_numericconversion.definitions.introduction"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.introduction" title="Introduction">Introduction</a>
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</h3></div></div></div>
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<p>
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This section provides definitions of terms used in the Numeric Conversion
|
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library.
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</p>
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<div class="sidebar">
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<div class="titlepage"></div>
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<p>
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<span class="bold"><strong>Notation</strong></span> <span class="underline">underlined
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text</span> denotes terms defined in the C++ standard.
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</p>
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<p>
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<span class="bold"><strong>bold face</strong></span> denotes terms defined here but
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not in the standard.
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</p>
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</div>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_numericconversion.definitions.types_and_values"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.types_and_values" title="Types and Values">Types
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and Values</a>
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</h3></div></div></div>
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<p>
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As defined by the <span class="underline">C++ Object Model</span>
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(§1.7) the <span class="underline">storage</span> or memory on which
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a C++ program runs is a contiguous sequence of <span class="underline">bytes</span>
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where each byte is a contiguous sequence of bits.
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</p>
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<p>
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An <span class="underline">object</span> is a region of storage (§1.8)
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and has a type (§3.9).
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</p>
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<p>
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A <span class="underline">type</span> is a discrete set of values.
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</p>
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<p>
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An object of type <code class="computeroutput"><span class="identifier">T</span></code> has an
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<span class="underline">object representation</span> which is the
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sequence of bytes stored in the object (§3.9/4)
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</p>
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<p>
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An object of type <code class="computeroutput"><span class="identifier">T</span></code> has a
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<span class="underline">value representation</span> which is the set
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of bits that determine the <span class="emphasis"><em>value</em></span> of an object of that
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type (§3.9/4). For <span class="underline">POD</span> types (§3.9/10),
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this bitset is given by the object representation, but not all the bits in
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the storage need to participate in the value representation (except for character
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types): for example, some bits might be used for padding or there may be
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trap-bits.
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</p>
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<p>
|
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<span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
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</p>
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<p>
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The <span class="bold"><strong>typed value</strong></span> that is held by an object
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is the value which is determined by its value representation.
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</p>
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<p>
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An <span class="bold"><strong>abstract value</strong></span> (untyped) is the conceptual
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information that is represented in a type (i.e. the number π).
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</p>
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<p>
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The <span class="bold"><strong>intrinsic value</strong></span> of an object is the
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binary value of the sequence of unsigned characters which form its object
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representation.
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
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</p>
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<p>
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<span class="emphasis"><em>Abstract</em></span> values can be <span class="bold"><strong>represented</strong></span>
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in a given type.
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</p>
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<p>
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To <span class="bold"><strong>represent</strong></span> an abstract value <code class="computeroutput"><span class="identifier">V</span></code> in a type <code class="computeroutput"><span class="identifier">T</span></code>
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is to obtain a typed value <code class="computeroutput"><span class="identifier">v</span></code>
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which corresponds to the abstract value <code class="computeroutput"><span class="identifier">V</span></code>.
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</p>
|
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<p>
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The operation is denoted using the <code class="computeroutput"><span class="identifier">rep</span><span class="special">()</span></code> operator, as in: <code class="computeroutput"><span class="identifier">v</span><span class="special">=</span><span class="identifier">rep</span><span class="special">(</span><span class="identifier">V</span><span class="special">)</span></code>. <code class="computeroutput"><span class="identifier">v</span></code> is the <span class="bold"><strong>representation</strong></span>
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of <code class="computeroutput"><span class="identifier">V</span></code> in the type <code class="computeroutput"><span class="identifier">T</span></code>.
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</p>
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<p>
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For example, the abstract value π can be represented in the type <code class="computeroutput"><span class="keyword">double</span></code> as the <code class="computeroutput"><span class="keyword">double</span>
|
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<span class="identifier">value</span> <span class="identifier">M_PI</span></code>
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and in the type <code class="computeroutput"><span class="keyword">int</span></code> as the
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<code class="computeroutput"><span class="keyword">int</span> <span class="identifier">value</span>
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|
<span class="number">3</span></code>
|
|
</p>
|
|
<p>
|
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<span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
|
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</p>
|
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<p>
|
|
Conversely, <span class="emphasis"><em>typed values</em></span> can be <span class="bold"><strong>abstracted</strong></span>.
|
|
</p>
|
|
<p>
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To <span class="bold"><strong>abstract</strong></span> a typed value <code class="computeroutput"><span class="identifier">v</span></code> of type <code class="computeroutput"><span class="identifier">T</span></code>
|
|
is to obtain the abstract value <code class="computeroutput"><span class="identifier">V</span></code>
|
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whose representation in <code class="computeroutput"><span class="identifier">T</span></code>
|
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is <code class="computeroutput"><span class="identifier">v</span></code>.
|
|
</p>
|
|
<p>
|
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The operation is denoted using the <code class="computeroutput"><span class="identifier">abt</span><span class="special">()</span></code> operator, as in: <code class="computeroutput"><span class="identifier">V</span><span class="special">=</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">v</span><span class="special">)</span></code>.
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</p>
|
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<p>
|
|
<code class="computeroutput"><span class="identifier">V</span></code> is the <span class="bold"><strong>abstraction</strong></span>
|
|
of <code class="computeroutput"><span class="identifier">v</span></code> of type <code class="computeroutput"><span class="identifier">T</span></code>.
|
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</p>
|
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<p>
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Abstraction is just an abstract operation (you can't do it); but it is defined
|
|
nevertheless because it will be used to give the definitions in the rest
|
|
of this document.
|
|
</p>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h3 class="title">
|
|
<a name="boost_numericconversion.definitions.c___arithmetic_types"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.c___arithmetic_types" title="C++ Arithmetic Types">C++
|
|
Arithmetic Types</a>
|
|
</h3></div></div></div>
|
|
<p>
|
|
The C++ language defines <span class="underline">fundamental types</span>
|
|
(§3.9.1). The following subsets of the fundamental types are intended to
|
|
represent <span class="emphasis"><em>numbers</em></span>:
|
|
</p>
|
|
<div class="variablelist">
|
|
<p class="title"><b></b></p>
|
|
<dl class="variablelist">
|
|
<dt><span class="term"><span class="underline">signed integer types</span> (§3.9.1/2):</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="special">{</span><span class="keyword">signed</span>
|
|
<span class="keyword">char</span><span class="special">,</span>
|
|
<span class="keyword">signed</span> <span class="keyword">short</span>
|
|
<span class="keyword">int</span><span class="special">,</span>
|
|
<span class="keyword">signed</span> <span class="keyword">int</span><span class="special">,</span> <span class="keyword">signed</span> <span class="keyword">long</span> <span class="keyword">int</span><span class="special">}</span></code> Can be used to represent general integer
|
|
numbers (both negative and positive).
|
|
</p></dd>
|
|
<dt><span class="term"><span class="underline">unsigned integer types</span> (§3.9.1/3):</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="special">{</span><span class="keyword">unsigned</span>
|
|
<span class="keyword">char</span><span class="special">,</span>
|
|
<span class="keyword">unsigned</span> <span class="keyword">short</span>
|
|
<span class="keyword">int</span><span class="special">,</span>
|
|
<span class="keyword">unsigned</span> <span class="keyword">int</span><span class="special">,</span> <span class="keyword">unsigned</span>
|
|
<span class="keyword">long</span> <span class="keyword">int</span><span class="special">}</span></code> Can be used to represent positive
|
|
integer numbers with modulo-arithmetic.
|
|
</p></dd>
|
|
<dt><span class="term"><span class="underline">floating-point types</span> (§3.9.1/8):</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="special">{</span><span class="keyword">float</span><span class="special">,</span><span class="keyword">double</span><span class="special">,</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">}</span></code>
|
|
Can be used to represent real numbers.
|
|
</p></dd>
|
|
<dt><span class="term"><span class="underline">integral or integer types</span> (§3.9.1/7):</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="special">{{</span><span class="keyword">signed</span>
|
|
<span class="identifier">integers</span><span class="special">},{</span><span class="keyword">unsigned</span> <span class="identifier">integers</span><span class="special">},</span> <span class="keyword">bool</span><span class="special">,</span> <span class="keyword">char</span> <span class="keyword">and</span> <span class="keyword">wchar_t</span><span class="special">}</span></code>
|
|
</p></dd>
|
|
<dt><span class="term"><span class="underline">arithmetic types</span> (§3.9.1/8):</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="special">{{</span><span class="identifier">integer</span>
|
|
<span class="identifier">types</span><span class="special">},{</span><span class="identifier">floating</span> <span class="identifier">types</span><span class="special">}}</span></code>
|
|
</p></dd>
|
|
</dl>
|
|
</div>
|
|
<p>
|
|
The integer types are required to have a <span class="emphasis"><em>binary</em></span> value
|
|
representation.
|
|
</p>
|
|
<p>
|
|
Additionally, the signed/unsigned integer types of the same base type (<code class="computeroutput"><span class="keyword">short</span></code>, <code class="computeroutput"><span class="keyword">int</span></code>
|
|
or <code class="computeroutput"><span class="keyword">long</span></code>) are required to have
|
|
the same value representation, that is:
|
|
</p>
|
|
<pre class="programlisting"> <span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="special">-</span><span class="number">3</span> <span class="special">;</span> <span class="comment">// suppose value representation is: 10011 (sign bit + 4 magnitude bits)</span>
|
|
<span class="keyword">unsigned</span> <span class="keyword">int</span> <span class="identifier">u</span> <span class="special">=</span> <span class="identifier">i</span> <span class="special">;</span> <span class="comment">// u is required to have the same 10011 as its value representation.</span>
|
|
</pre>
|
|
<p>
|
|
In other words, the integer types signed/unsigned X use the same value representation
|
|
but a different <span class="emphasis"><em>interpretation</em></span> of it; that is, their
|
|
<span class="emphasis"><em>typed values</em></span> might differ.
|
|
</p>
|
|
<p>
|
|
Another consequence of this is that the range for signed X is always a smaller
|
|
subset of the range of unsigned X, as required by §3.9.1/3.
|
|
</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>
|
|
Always remember that unsigned types, unlike signed types, have modulo-arithmetic;
|
|
that is, they do not overflow. This means that:
|
|
</p>
|
|
<p>
|
|
<span class="bold"><strong>-</strong></span> Always be extra careful when mixing
|
|
signed/unsigned types
|
|
</p>
|
|
<p>
|
|
<span class="bold"><strong>-</strong></span> Use unsigned types only when you need
|
|
modulo arithmetic or very very large numbers. Don't use unsigned types
|
|
just because you intend to deal with positive values only (you can do this
|
|
with signed types as well).
|
|
</p>
|
|
</td></tr>
|
|
</table></div>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h3 class="title">
|
|
<a name="boost_numericconversion.definitions.numeric_types"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.numeric_types" title="Numeric Types">Numeric
|
|
Types</a>
|
|
</h3></div></div></div>
|
|
<p>
|
|
This section introduces the following definitions intended to integrate arithmetic
|
|
types with user-defined types which behave like numbers. Some definitions
|
|
are purposely broad in order to include a vast variety of user-defined number
|
|
types.
|
|
</p>
|
|
<p>
|
|
Within this library, the term <span class="emphasis"><em>number</em></span> refers to an abstract
|
|
numeric value.
|
|
</p>
|
|
<p>
|
|
A type is <span class="bold"><strong>numeric</strong></span> if:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
It is an arithmetic type, or,
|
|
</li>
|
|
<li class="listitem">
|
|
It is a user-defined type which
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
|
|
<li class="listitem">
|
|
Represents numeric abstract values (i.e. numbers).
|
|
</li>
|
|
<li class="listitem">
|
|
Can be converted (either implicitly or explicitly) to/from at least
|
|
one arithmetic type.
|
|
</li>
|
|
<li class="listitem">
|
|
Has <a class="link" href="definitions.html#boost_numericconversion.definitions.range_and_precision" title="Range and Precision">range</a>
|
|
(possibly unbounded) and <a class="link" href="definitions.html#boost_numericconversion.definitions.range_and_precision" title="Range and Precision">precision</a>
|
|
(possibly dynamic or unlimited).
|
|
</li>
|
|
<li class="listitem">
|
|
Provides an specialization of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code>.
|
|
</li>
|
|
</ul></div>
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
A numeric type is <span class="bold"><strong>signed</strong></span> if the abstract
|
|
values it represent include negative numbers.
|
|
</p>
|
|
<p>
|
|
A numeric type is <span class="bold"><strong>unsigned</strong></span> if the abstract
|
|
values it represent exclude negative numbers.
|
|
</p>
|
|
<p>
|
|
A numeric type is <span class="bold"><strong>modulo</strong></span> if it has modulo-arithmetic
|
|
(does not overflow).
|
|
</p>
|
|
<p>
|
|
A numeric type is <span class="bold"><strong>integer</strong></span> if the abstract
|
|
values it represent are whole numbers.
|
|
</p>
|
|
<p>
|
|
A numeric type is <span class="bold"><strong>floating</strong></span> if the abstract
|
|
values it represent are real numbers.
|
|
</p>
|
|
<p>
|
|
An <span class="bold"><strong>arithmetic value</strong></span> is the typed value of
|
|
an arithmetic type
|
|
</p>
|
|
<p>
|
|
A <span class="bold"><strong>numeric value</strong></span> is the typed value of a
|
|
numeric type
|
|
</p>
|
|
<p>
|
|
These definitions simply generalize the standard notions of arithmetic types
|
|
and values by introducing a superset called <span class="underline">numeric</span>.
|
|
All arithmetic types and values are numeric types and values, but not vice
|
|
versa, since user-defined numeric types are not arithmetic types.
|
|
</p>
|
|
<p>
|
|
The following examples clarify the differences between arithmetic and numeric
|
|
types (and values):
|
|
</p>
|
|
<pre class="programlisting"><span class="comment">// A numeric type which is not an arithmetic type (is user-defined)</span>
|
|
<span class="comment">// and which is intended to represent integer numbers (i.e., an 'integer' numeric type)</span>
|
|
<span class="keyword">class</span> <span class="identifier">MyInt</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">MyInt</span> <span class="special">(</span> <span class="keyword">long</span> <span class="keyword">long</span> <span class="identifier">v</span> <span class="special">)</span> <span class="special">;</span>
|
|
<span class="keyword">long</span> <span class="keyword">long</span> <span class="identifier">to_builtin</span><span class="special">();</span>
|
|
<span class="special">}</span> <span class="special">;</span>
|
|
<span class="keyword">namespace</span> <span class="identifier">std</span> <span class="special">{</span>
|
|
<span class="keyword">template</span><span class="special"><></span> <span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">MyInt</span><span class="special">></span> <span class="special">{</span> <span class="special">...</span> <span class="special">}</span> <span class="special">;</span>
|
|
<span class="special">}</span>
|
|
|
|
<span class="comment">// A 'floating' numeric type (double) which is also an arithmetic type (built-in),</span>
|
|
<span class="comment">// with a float numeric value.</span>
|
|
<span class="keyword">double</span> <span class="identifier">pi</span> <span class="special">=</span> <span class="identifier">M_PI</span> <span class="special">;</span>
|
|
|
|
<span class="comment">// A 'floating' numeric type with a whole numeric value.</span>
|
|
<span class="comment">// NOTE: numeric values are typed valued, hence, they are, for instance,</span>
|
|
<span class="comment">// integer or floating, despite the value itself being whole or including</span>
|
|
<span class="comment">// a fractional part.</span>
|
|
<span class="keyword">double</span> <span class="identifier">two</span> <span class="special">=</span> <span class="number">2.0</span> <span class="special">;</span>
|
|
|
|
<span class="comment">// An integer numeric type with an integer numeric value.</span>
|
|
<span class="identifier">MyInt</span> <span class="identifier">i</span><span class="special">(</span><span class="number">1234</span><span class="special">);</span>
|
|
</pre>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h3 class="title">
|
|
<a name="boost_numericconversion.definitions.range_and_precision"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.range_and_precision" title="Range and Precision">Range
|
|
and Precision</a>
|
|
</h3></div></div></div>
|
|
<p>
|
|
Given a number set <code class="computeroutput"><span class="identifier">N</span></code>, some
|
|
of its elements are representable in a numeric type <code class="computeroutput"><span class="identifier">T</span></code>.
|
|
</p>
|
|
<p>
|
|
The set of representable values of type <code class="computeroutput"><span class="identifier">T</span></code>,
|
|
or numeric set of <code class="computeroutput"><span class="identifier">T</span></code>, is a
|
|
set of numeric values whose elements are the representation of some subset
|
|
of <code class="computeroutput"><span class="identifier">N</span></code>.
|
|
</p>
|
|
<p>
|
|
For example, the interval of <code class="computeroutput"><span class="keyword">int</span></code>
|
|
values <code class="computeroutput"><span class="special">[</span><span class="identifier">INT_MIN</span><span class="special">,</span><span class="identifier">INT_MAX</span><span class="special">]</span></code> is the set of representable values of type
|
|
<code class="computeroutput"><span class="keyword">int</span></code>, i.e. the <code class="computeroutput"><span class="keyword">int</span></code> numeric set, and corresponds to the representation
|
|
of the elements of the interval of abstract values <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">INT_MIN</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">INT_MAX</span><span class="special">)]</span></code>
|
|
from the integer numbers.
|
|
</p>
|
|
<p>
|
|
Similarly, the interval of <code class="computeroutput"><span class="keyword">double</span></code>
|
|
values <code class="computeroutput"><span class="special">[-</span><span class="identifier">DBL_MAX</span><span class="special">,</span><span class="identifier">DBL_MAX</span><span class="special">]</span></code> is the <code class="computeroutput"><span class="keyword">double</span></code>
|
|
numeric set, which corresponds to the subset of the real numbers from <code class="computeroutput"><span class="identifier">abt</span><span class="special">(-</span><span class="identifier">DBL_MAX</span><span class="special">)</span></code> to <code class="computeroutput"><span class="identifier">abt</span><span class="special">(</span><span class="identifier">DBL_MAX</span><span class="special">)</span></code>.
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
|
|
</p>
|
|
<p>
|
|
Let <span class="bold"><strong><code class="computeroutput"><span class="identifier">next</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code></strong></span>
|
|
denote the lowest numeric value greater than x.
|
|
</p>
|
|
<p>
|
|
Let <span class="bold"><strong><code class="computeroutput"><span class="identifier">prev</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span></code></strong></span>
|
|
denote the highest numeric value lower then x.
|
|
</p>
|
|
<p>
|
|
Let <span class="bold"><strong><code class="computeroutput"><span class="identifier">v</span><span class="special">=</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">V</span><span class="special">))</span></code></strong></span> and <span class="bold"><strong><code class="computeroutput"><span class="identifier">v</span><span class="special">=</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">V</span><span class="special">))</span></code></strong></span>
|
|
be identities that relate a numeric typed value <code class="computeroutput"><span class="identifier">v</span></code>
|
|
with a number <code class="computeroutput"><span class="identifier">V</span></code>.
|
|
</p>
|
|
<p>
|
|
An ordered pair of numeric values <code class="computeroutput"><span class="identifier">x</span></code>,<code class="computeroutput"><span class="identifier">y</span></code> s.t. <code class="computeroutput"><span class="identifier">x</span><span class="special"><</span><span class="identifier">y</span></code> are
|
|
<span class="bold"><strong>consecutive</strong></span> iff <code class="computeroutput"><span class="identifier">next</span><span class="special">(</span><span class="identifier">x</span><span class="special">)==</span><span class="identifier">y</span></code>.
|
|
</p>
|
|
<p>
|
|
The abstract distance between consecutive numeric values is usually referred
|
|
to as a <span class="underline">Unit in the Last Place</span>, or
|
|
<span class="bold"><strong>ulp</strong></span> for short. A ulp is a quantity whose
|
|
abstract magnitude is relative to the numeric values it corresponds to: If
|
|
the numeric set is not evenly distributed, that is, if the abstract distance
|
|
between consecutive numeric values varies along the set -as is the case with
|
|
the floating-point types-, the magnitude of 1ulp after the numeric value
|
|
<code class="computeroutput"><span class="identifier">x</span></code> might be (usually is) different
|
|
from the magnitude of a 1ulp after the numeric value y for <code class="computeroutput"><span class="identifier">x</span><span class="special">!=</span><span class="identifier">y</span></code>.
|
|
</p>
|
|
<p>
|
|
Since numbers are inherently ordered, a <span class="bold"><strong>numeric set</strong></span>
|
|
of type <code class="computeroutput"><span class="identifier">T</span></code> is an ordered sequence
|
|
of numeric values (of type <code class="computeroutput"><span class="identifier">T</span></code>)
|
|
of the form:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">REP</span><span class="special">(</span><span class="identifier">T</span><span class="special">)={</span><span class="identifier">l</span><span class="special">,</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">l</span><span class="special">),</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">next</span><span class="special">(</span><span class="identifier">l</span><span class="special">)),...,</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">h</span><span class="special">)),</span><span class="identifier">prev</span><span class="special">(</span><span class="identifier">h</span><span class="special">),</span><span class="identifier">h</span><span class="special">}</span>
|
|
</pre>
|
|
<p>
|
|
where <code class="computeroutput"><span class="identifier">l</span></code> and <code class="computeroutput"><span class="identifier">h</span></code> are respectively the lowest and highest
|
|
values of type <code class="computeroutput"><span class="identifier">T</span></code>, called
|
|
the boundary values of type <code class="computeroutput"><span class="identifier">T</span></code>.
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
|
|
</p>
|
|
<p>
|
|
A numeric set is discrete. It has a <span class="bold"><strong>size</strong></span>
|
|
which is the number of numeric values in the set, a <span class="bold"><strong>width</strong></span>
|
|
which is the abstract difference between the highest and lowest boundary
|
|
values: <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)-</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)]</span></code>, and a <span class="bold"><strong>density</strong></span>
|
|
which is the relation between its size and width: <code class="computeroutput"><span class="identifier">density</span><span class="special">=</span><span class="identifier">size</span><span class="special">/</span><span class="identifier">width</span></code>.
|
|
</p>
|
|
<p>
|
|
The integer types have density 1, which means that there are no unrepresentable
|
|
integer numbers between <code class="computeroutput"><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code>
|
|
and <code class="computeroutput"><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code> (i.e.
|
|
there are no gaps). On the other hand, floating types have density much smaller
|
|
than 1, which means that there are real numbers unrepresented between consecutive
|
|
floating values (i.e. there are gaps).
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
|
|
</p>
|
|
<p>
|
|
The interval of <span class="underline">abstract values</span> <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)]</span></code>
|
|
is the range of the type <code class="computeroutput"><span class="identifier">T</span></code>,
|
|
denoted <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code>.
|
|
</p>
|
|
<p>
|
|
A range is a set of abstract values and not a set of numeric values. In other
|
|
documents, such as the C++ standard, the word <code class="computeroutput"><span class="identifier">range</span></code>
|
|
is <span class="emphasis"><em>sometimes</em></span> used as synonym for <code class="computeroutput"><span class="identifier">numeric</span>
|
|
<span class="identifier">set</span></code>, that is, as the ordered sequence
|
|
of numeric values from <code class="computeroutput"><span class="identifier">l</span></code>
|
|
to <code class="computeroutput"><span class="identifier">h</span></code>. In this document, however,
|
|
a range is an abstract interval which subtends the numeric set.
|
|
</p>
|
|
<p>
|
|
For example, the sequence <code class="computeroutput"><span class="special">[-</span><span class="identifier">DBL_MAX</span><span class="special">,</span><span class="identifier">DBL_MAX</span><span class="special">]</span></code>
|
|
is the numeric set of the type <code class="computeroutput"><span class="keyword">double</span></code>,
|
|
and the real interval <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(-</span><span class="identifier">DBL_MAX</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">DBL_MAX</span><span class="special">)]</span></code>
|
|
is its range.
|
|
</p>
|
|
<p>
|
|
Notice, for instance, that the range of a floating-point type is <span class="emphasis"><em>continuous</em></span>
|
|
unlike its numeric set.
|
|
</p>
|
|
<p>
|
|
This definition was chosen because:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
<span class="bold"><strong>(a)</strong></span> The discrete set of numeric values
|
|
is already given by the numeric set.
|
|
</li>
|
|
<li class="listitem">
|
|
<span class="bold"><strong>(b)</strong></span> Abstract intervals are easier to
|
|
compare and overlap since only boundary values need to be considered.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
This definition allows for a concise definition of <code class="computeroutput"><span class="identifier">subranged</span></code>
|
|
as given in the last section.
|
|
</p>
|
|
<p>
|
|
The width of a numeric set, as defined, is exactly equivalent to the width
|
|
of a range.
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../images/space.png" alt="space"></span>
|
|
</p>
|
|
<p>
|
|
The <span class="bold"><strong>precision</strong></span> of a type is given by the
|
|
width or density of the numeric set.
|
|
</p>
|
|
<p>
|
|
For integer types, which have density 1, the precision is conceptually equivalent
|
|
to the range and is determined by the number of bits used in the value representation:
|
|
The higher the number of bits the bigger the size of the numeric set, the
|
|
wider the range, and the higher the precision.
|
|
</p>
|
|
<p>
|
|
For floating types, which have density <<1, the precision is given
|
|
not by the width of the range but by the density. In a typical implementation,
|
|
the range is determined by the number of bits used in the exponent, and the
|
|
precision by the number of bits used in the mantissa (giving the maximum
|
|
number of significant digits that can be exactly represented). The higher
|
|
the number of exponent bits the wider the range, while the higher the number
|
|
of mantissa bits, the higher the precision.
|
|
</p>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h3 class="title">
|
|
<a name="boost_numericconversion.definitions.exact__correctly_rounded_and_out_of_range_representations"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.exact__correctly_rounded_and_out_of_range_representations" title="Exact, Correctly Rounded and Out-Of-Range Representations">Exact,
|
|
Correctly Rounded and Out-Of-Range Representations</a>
|
|
</h3></div></div></div>
|
|
<p>
|
|
Given an abstract value <code class="computeroutput"><span class="identifier">V</span></code>
|
|
and a type <code class="computeroutput"><span class="identifier">T</span></code> with its corresponding
|
|
range <code class="computeroutput"><span class="special">[</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">),</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)]</span></code>:
|
|
</p>
|
|
<p>
|
|
If <code class="computeroutput"><span class="identifier">V</span> <span class="special"><</span>
|
|
<span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code> or
|
|
<code class="computeroutput"><span class="identifier">V</span> <span class="special">></span>
|
|
<span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code>, <code class="computeroutput"><span class="identifier">V</span></code> is <span class="bold"><strong>not representable</strong></span>
|
|
(cannot be represented) in the type <code class="computeroutput"><span class="identifier">T</span></code>,
|
|
or, equivalently, it's representation in the type <code class="computeroutput"><span class="identifier">T</span></code>
|
|
is <span class="bold"><strong>out of range</strong></span>, or <span class="bold"><strong>overflows</strong></span>.
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">V</span> <span class="special"><</span>
|
|
<span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code>,
|
|
the <span class="bold"><strong>overflow is negative</strong></span>.
|
|
</li>
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">V</span> <span class="special">></span>
|
|
<span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code>,
|
|
the <span class="bold"><strong>overflow is positive</strong></span>.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
If <code class="computeroutput"><span class="identifier">V</span> <span class="special">>=</span>
|
|
<span class="identifier">abt</span><span class="special">(</span><span class="identifier">l</span><span class="special">)</span></code> and
|
|
<code class="computeroutput"><span class="identifier">V</span> <span class="special"><=</span>
|
|
<span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span></code>, <code class="computeroutput"><span class="identifier">V</span></code> is <span class="bold"><strong>representable</strong></span>
|
|
(can be represented) in the type <code class="computeroutput"><span class="identifier">T</span></code>,
|
|
or, equivalently, its representation in the type <code class="computeroutput"><span class="identifier">T</span></code>
|
|
is <span class="bold"><strong>in range</strong></span>, or <span class="bold"><strong>does
|
|
not overflow</strong></span>.
|
|
</p>
|
|
<p>
|
|
Notice that a numeric type, such as a C++ unsigned type, can define that
|
|
any <code class="computeroutput"><span class="identifier">V</span></code> does not overflow by
|
|
always representing not <code class="computeroutput"><span class="identifier">V</span></code>
|
|
itself but the abstract value <code class="computeroutput"><span class="identifier">U</span>
|
|
<span class="special">=</span> <span class="special">[</span> <span class="identifier">V</span> <span class="special">%</span> <span class="special">(</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)+</span><span class="number">1</span><span class="special">)</span>
|
|
<span class="special">]</span></code>, which is always in range.
|
|
</p>
|
|
<p>
|
|
Given an abstract value <code class="computeroutput"><span class="identifier">V</span></code>
|
|
represented in the type <code class="computeroutput"><span class="identifier">T</span></code>
|
|
as <code class="computeroutput"><span class="identifier">v</span></code>, the <span class="bold"><strong>roundoff</strong></span>
|
|
error of the representation is the abstract difference: <code class="computeroutput"><span class="special">(</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">v</span><span class="special">)-</span><span class="identifier">V</span><span class="special">)</span></code>.
|
|
</p>
|
|
<p>
|
|
Notice that a representation is an <span class="emphasis"><em>operation</em></span>, hence,
|
|
the roundoff error corresponds to the representation operation and not to
|
|
the numeric value itself (i.e. numeric values do not have any error themselves)
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
If the roundoff is 0, the representation is <span class="bold"><strong>exact</strong></span>,
|
|
and <code class="computeroutput"><span class="identifier">V</span></code> is exactly representable
|
|
in the type <code class="computeroutput"><span class="identifier">T</span></code>.
|
|
</li>
|
|
<li class="listitem">
|
|
If the roundoff is not 0, the representation is <span class="bold"><strong>inexact</strong></span>,
|
|
and <code class="computeroutput"><span class="identifier">V</span></code> is inexactly representable
|
|
in the type <code class="computeroutput"><span class="identifier">T</span></code>.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
If a representation <code class="computeroutput"><span class="identifier">v</span></code> in
|
|
a type <code class="computeroutput"><span class="identifier">T</span></code> -either exact or
|
|
inexact-, is any of the adjacents of <code class="computeroutput"><span class="identifier">V</span></code>
|
|
in that type, that is, if <code class="computeroutput"><span class="identifier">v</span><span class="special">==</span><span class="identifier">prev</span></code>
|
|
or <code class="computeroutput"><span class="identifier">v</span><span class="special">==</span><span class="identifier">next</span></code>, the representation is faithfully
|
|
rounded. If the choice between <code class="computeroutput"><span class="identifier">prev</span></code>
|
|
and <code class="computeroutput"><span class="identifier">next</span></code> matches a given
|
|
<span class="bold"><strong>rounding direction</strong></span>, it is <span class="bold"><strong>correctly
|
|
rounded</strong></span>.
|
|
</p>
|
|
<p>
|
|
All exact representations are correctly rounded, but not all inexact representations
|
|
are. In particular, C++ requires numeric conversions (described below) and
|
|
the result of arithmetic operations (not covered by this document) to be
|
|
correctly rounded, but batch operations propagate roundoff, thus final results
|
|
are usually incorrectly rounded, that is, the numeric value <code class="computeroutput"><span class="identifier">r</span></code> which is the computed result is neither
|
|
of the adjacents of the abstract value <code class="computeroutput"><span class="identifier">R</span></code>
|
|
which is the theoretical result.
|
|
</p>
|
|
<p>
|
|
Because a correctly rounded representation is always one of adjacents of
|
|
the abstract value being represented, the roundoff is guaranteed to be at
|
|
most 1ulp.
|
|
</p>
|
|
<p>
|
|
The following examples summarize the given definitions. Consider:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
A numeric type <code class="computeroutput"><span class="identifier">Int</span></code> representing
|
|
integer numbers with a <span class="emphasis"><em>numeric set</em></span>: <code class="computeroutput"><span class="special">{-</span><span class="number">2</span><span class="special">,-</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">}</span></code> and
|
|
<span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[-</span><span class="number">2</span><span class="special">,</span><span class="number">2</span><span class="special">]</span></code>
|
|
</li>
|
|
<li class="listitem">
|
|
A numeric type <code class="computeroutput"><span class="identifier">Cardinal</span></code>
|
|
representing integer numbers with a <span class="emphasis"><em>numeric set</em></span>:
|
|
<code class="computeroutput"><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</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="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">,</span><span class="number">8</span><span class="special">,</span><span class="number">9</span><span class="special">}</span></code> and <span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[</span><span class="number">0</span><span class="special">,</span><span class="number">9</span><span class="special">]</span></code> (no
|
|
modulo-arithmetic here)
|
|
</li>
|
|
<li class="listitem">
|
|
A numeric type <code class="computeroutput"><span class="identifier">Real</span></code> representing
|
|
real numbers with a <span class="emphasis"><em>numeric set</em></span>: <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,-</span><span class="number">0.5</span><span class="special">,-</span><span class="number">0.0</span><span class="special">,+</span><span class="number">0.0</span><span class="special">,+</span><span class="number">0.5</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">1.5</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and
|
|
<span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
|
|
</li>
|
|
<li class="listitem">
|
|
A numeric type <code class="computeroutput"><span class="identifier">Whole</span></code>
|
|
representing real numbers with a <span class="emphasis"><em>numeric set</em></span>: <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and
|
|
<span class="emphasis"><em>range</em></span>: <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
First, notice that the types <code class="computeroutput"><span class="identifier">Real</span></code>
|
|
and <code class="computeroutput"><span class="identifier">Whole</span></code> both represent
|
|
real numbers, have the same range, but different precision.
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
The integer number <code class="computeroutput"><span class="number">1</span></code> (an
|
|
abstract value) can be exactly represented in any of these types.
|
|
</li>
|
|
<li class="listitem">
|
|
The integer number <code class="computeroutput"><span class="special">-</span><span class="number">1</span></code>
|
|
can be exactly represented in <code class="computeroutput"><span class="identifier">Int</span></code>,
|
|
<code class="computeroutput"><span class="identifier">Real</span></code> and <code class="computeroutput"><span class="identifier">Whole</span></code>, but cannot be represented in
|
|
<code class="computeroutput"><span class="identifier">Cardinal</span></code>, yielding negative
|
|
overflow.
|
|
</li>
|
|
<li class="listitem">
|
|
The real number <code class="computeroutput"><span class="number">1.5</span></code> can be
|
|
exactly represented in <code class="computeroutput"><span class="identifier">Real</span></code>,
|
|
and inexactly represented in the other types.
|
|
</li>
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="number">1.5</span></code> is represented as
|
|
either <code class="computeroutput"><span class="number">1</span></code> or <code class="computeroutput"><span class="number">2</span></code> in any of the types (except <code class="computeroutput"><span class="identifier">Real</span></code>), the representation is correctly
|
|
rounded.
|
|
</li>
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="number">0.5</span></code> is represented as
|
|
<code class="computeroutput"><span class="special">+</span><span class="number">1.5</span></code>
|
|
in the type <code class="computeroutput"><span class="identifier">Real</span></code>, it
|
|
is incorrectly rounded.
|
|
</li>
|
|
<li class="listitem">
|
|
<code class="computeroutput"><span class="special">(-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">)</span></code>
|
|
are the <code class="computeroutput"><span class="identifier">Real</span></code> adjacents
|
|
of any real number in the interval <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">]</span></code>, yet there are no <code class="computeroutput"><span class="identifier">Real</span></code>
|
|
adjacents for <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span>
|
|
<span class="special">-</span><span class="number">2.0</span></code>,
|
|
nor for <code class="computeroutput"><span class="identifier">x</span> <span class="special">></span>
|
|
<span class="special">+</span><span class="number">2.0</span></code>.
|
|
</li>
|
|
</ul></div>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h3 class="title">
|
|
<a name="boost_numericconversion.definitions.standard__numeric__conversions"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.standard__numeric__conversions" title="Standard (numeric) Conversions">Standard
|
|
(numeric) Conversions</a>
|
|
</h3></div></div></div>
|
|
<p>
|
|
The C++ language defines <span class="underline">Standard Conversions</span>
|
|
(§4) some of which are conversions between arithmetic types.
|
|
</p>
|
|
<p>
|
|
These are <span class="underline">Integral promotions</span> (§4.5),
|
|
<span class="underline">Integral conversions</span> (§4.7), <span class="underline">Floating point promotions</span> (§4.6), <span class="underline">Floating point conversions</span> (§4.8) and <span class="underline">Floating-integral conversions</span> (§4.9).
|
|
</p>
|
|
<p>
|
|
In the sequel, integral and floating point promotions are called <span class="bold"><strong>arithmetic promotions</strong></span>, and these plus integral, floating-point
|
|
and floating-integral conversions are called <span class="bold"><strong>arithmetic
|
|
conversions</strong></span> (i.e, promotions are conversions).
|
|
</p>
|
|
<p>
|
|
Promotions, both Integral and Floating point, are <span class="emphasis"><em>value-preserving</em></span>,
|
|
which means that the typed value is not changed with the conversion.
|
|
</p>
|
|
<p>
|
|
In the sequel, consider a source typed value <code class="computeroutput"><span class="identifier">s</span></code>
|
|
of type <code class="computeroutput"><span class="identifier">S</span></code>, the source abstract
|
|
value <code class="computeroutput"><span class="identifier">N</span><span class="special">=</span><span class="identifier">abt</span><span class="special">(</span><span class="identifier">s</span><span class="special">)</span></code>, a destination type <code class="computeroutput"><span class="identifier">T</span></code>;
|
|
and whenever possible, a result typed value <code class="computeroutput"><span class="identifier">t</span></code>
|
|
of type <code class="computeroutput"><span class="identifier">T</span></code>.
|
|
</p>
|
|
<p>
|
|
Integer to integer conversions are always defined:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">T</span></code> is unsigned, the
|
|
abstract value which is effectively represented is not <code class="computeroutput"><span class="identifier">N</span></code> but <code class="computeroutput"><span class="identifier">M</span><span class="special">=[</span> <span class="identifier">N</span> <span class="special">%</span> <span class="special">(</span> <span class="identifier">abt</span><span class="special">(</span><span class="identifier">h</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span>
|
|
<span class="special">)</span> <span class="special">]</span></code>,
|
|
where <code class="computeroutput"><span class="identifier">h</span></code> is the highest
|
|
unsigned typed value of type <code class="computeroutput"><span class="identifier">T</span></code>.
|
|
</li>
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">T</span></code> is signed and <code class="computeroutput"><span class="identifier">N</span></code> is not directly representable, the
|
|
result <code class="computeroutput"><span class="identifier">t</span></code> is <span class="underline">implementation-defined</span>, which means that
|
|
the C++ implementation is required to produce a value <code class="computeroutput"><span class="identifier">t</span></code>
|
|
even if it is totally unrelated to <code class="computeroutput"><span class="identifier">s</span></code>.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
Floating to Floating conversions are defined only if <code class="computeroutput"><span class="identifier">N</span></code>
|
|
is representable; if it is not, the conversion has <span class="underline">undefined
|
|
behavior</span>.
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">N</span></code> is exactly representable,
|
|
<code class="computeroutput"><span class="identifier">t</span></code> is required to be the
|
|
exact representation.
|
|
</li>
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">N</span></code> is inexactly representable,
|
|
<code class="computeroutput"><span class="identifier">t</span></code> is required to be one
|
|
of the two adjacents, with an implementation-defined choice of rounding
|
|
direction; that is, the conversion is required to be correctly rounded.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
Floating to Integer conversions represent not <code class="computeroutput"><span class="identifier">N</span></code>
|
|
but <code class="computeroutput"><span class="identifier">M</span><span class="special">=</span><span class="identifier">trunc</span><span class="special">(</span><span class="identifier">N</span><span class="special">)</span></code>, were
|
|
<code class="computeroutput"><span class="identifier">trunc</span><span class="special">()</span></code>
|
|
is to truncate: i.e. to remove the fractional part, if any.
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">M</span></code> is not representable
|
|
in <code class="computeroutput"><span class="identifier">T</span></code>, the conversion
|
|
has <span class="underline">undefined behavior</span> (unless
|
|
<code class="computeroutput"><span class="identifier">T</span></code> is <code class="computeroutput"><span class="keyword">bool</span></code>,
|
|
see §4.12).
|
|
</li></ul></div>
|
|
<p>
|
|
Integer to Floating conversions are always defined.
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">N</span></code> is exactly representable,
|
|
<code class="computeroutput"><span class="identifier">t</span></code> is required to be the
|
|
exact representation.
|
|
</li>
|
|
<li class="listitem">
|
|
If <code class="computeroutput"><span class="identifier">N</span></code> is inexactly representable,
|
|
<code class="computeroutput"><span class="identifier">t</span></code> is required to be one
|
|
of the two adjacents, with an implementation-defined choice of rounding
|
|
direction; that is, the conversion is required to be correctly rounded.
|
|
</li>
|
|
</ul></div>
|
|
</div>
|
|
<div class="section">
|
|
<div class="titlepage"><div><div><h3 class="title">
|
|
<a name="boost_numericconversion.definitions.subranged_conversion_direction__subtype_and_supertype"></a><a class="link" href="definitions.html#boost_numericconversion.definitions.subranged_conversion_direction__subtype_and_supertype" title="Subranged Conversion Direction, Subtype and Supertype">Subranged
|
|
Conversion Direction, Subtype and Supertype</a>
|
|
</h3></div></div></div>
|
|
<p>
|
|
Given a source type <code class="computeroutput"><span class="identifier">S</span></code> and
|
|
a destination type <code class="computeroutput"><span class="identifier">T</span></code>, there
|
|
is a <span class="bold"><strong>conversion direction</strong></span> denoted: <code class="computeroutput"><span class="identifier">S</span><span class="special">-></span><span class="identifier">T</span></code>.
|
|
</p>
|
|
<p>
|
|
For any two ranges the following <span class="emphasis"><em>range relation</em></span> can
|
|
be defined: A range <code class="computeroutput"><span class="identifier">X</span></code> can
|
|
be <span class="emphasis"><em>entirely contained</em></span> in a range <code class="computeroutput"><span class="identifier">Y</span></code>,
|
|
in which case it is said that <code class="computeroutput"><span class="identifier">X</span></code>
|
|
is enclosed by <code class="computeroutput"><span class="identifier">Y</span></code>.
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="bold"><strong>Formally:</strong></span> <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>
|
|
is enclosed by <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code> iif
|
|
<code class="computeroutput"><span class="special">(</span><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span>
|
|
<span class="identifier">intersection</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">))</span>
|
|
<span class="special">==</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>.
|
|
</p></blockquote></div>
|
|
<p>
|
|
If the source type range, <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>,
|
|
is not enclosed in the target type range, <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code>;
|
|
that is, if <code class="computeroutput"><span class="special">(</span><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span>
|
|
<span class="special">&</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">))</span>
|
|
<span class="special">!=</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>,
|
|
the conversion direction is said to be <span class="bold"><strong>subranged</strong></span>,
|
|
which means that <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code> is not
|
|
entirely contained in <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code> and
|
|
therefore there is some portion of the source range which falls outside the
|
|
target range. In other words, if a conversion direction <code class="computeroutput"><span class="identifier">S</span><span class="special">-></span><span class="identifier">T</span></code>
|
|
is subranged, there are values in <code class="computeroutput"><span class="identifier">S</span></code>
|
|
which cannot be represented in <code class="computeroutput"><span class="identifier">T</span></code>
|
|
because they are out of range. Notice that for <code class="computeroutput"><span class="identifier">S</span><span class="special">-></span><span class="identifier">T</span></code>,
|
|
the adjective subranged applies to <code class="computeroutput"><span class="identifier">T</span></code>.
|
|
</p>
|
|
<p>
|
|
Examples:
|
|
</p>
|
|
<p>
|
|
Given the following numeric types all representing real numbers:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
<code class="computeroutput"><span class="identifier">X</span></code> with numeric set <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and
|
|
range <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
|
|
</li>
|
|
<li class="listitem">
|
|
<code class="computeroutput"><span class="identifier">Y</span></code> with numeric set <code class="computeroutput"><span class="special">{-</span><span class="number">2.0</span><span class="special">,-</span><span class="number">1.5</span><span class="special">,-</span><span class="number">1.0</span><span class="special">,-</span><span class="number">0.5</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">0.5</span><span class="special">,+</span><span class="number">1.0</span><span class="special">,+</span><span class="number">1.5</span><span class="special">,+</span><span class="number">2.0</span><span class="special">}</span></code> and range <code class="computeroutput"><span class="special">[-</span><span class="number">2.0</span><span class="special">,+</span><span class="number">2.0</span><span class="special">]</span></code>
|
|
</li>
|
|
<li class="listitem">
|
|
<code class="computeroutput"><span class="identifier">Z</span></code> with numeric set <code class="computeroutput"><span class="special">{-</span><span class="number">1.0</span><span class="special">,</span><span class="number">0.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">}</span></code> and range <code class="computeroutput"><span class="special">[-</span><span class="number">1.0</span><span class="special">,+</span><span class="number">1.0</span><span class="special">]</span></code>
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
For:
|
|
</p>
|
|
<div class="variablelist">
|
|
<p class="title"><b></b></p>
|
|
<dl class="variablelist">
|
|
<dt><span class="term">(a) X->Y:</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span> <span class="special">&</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">Y</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span></code>,
|
|
then <code class="computeroutput"><span class="identifier">X</span><span class="special">-></span><span class="identifier">Y</span></code> is not subranged. Thus, all values
|
|
of type <code class="computeroutput"><span class="identifier">X</span></code> are representable
|
|
in the type <code class="computeroutput"><span class="identifier">Y</span></code>.
|
|
</p></dd>
|
|
<dt><span class="term">(b) Y->X:</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">Y</span><span class="special">)</span> <span class="special">&</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">Y</span><span class="special">)</span></code>,
|
|
then <code class="computeroutput"><span class="identifier">Y</span><span class="special">-></span><span class="identifier">X</span></code> is not subranged. Thus, all values
|
|
of type <code class="computeroutput"><span class="identifier">Y</span></code> are representable
|
|
in the type <code class="computeroutput"><span class="identifier">X</span></code>, but
|
|
in this case, some values are <span class="emphasis"><em>inexactly</em></span> representable
|
|
(all the halves). (note: it is to permit this case that a range is
|
|
an interval of abstract values and not an interval of typed values)
|
|
</p></dd>
|
|
<dt><span class="term">(b) X->Z:</span></dt>
|
|
<dd><p>
|
|
<code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span> <span class="special">&</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">Z</span><span class="special">)</span> <span class="special">!=</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span></code>,
|
|
then <code class="computeroutput"><span class="identifier">X</span><span class="special">-></span><span class="identifier">Z</span></code> is subranged. Thus, some values
|
|
of type <code class="computeroutput"><span class="identifier">X</span></code> are not representable
|
|
in the type <code class="computeroutput"><span class="identifier">Z</span></code>, they
|
|
fall out of range <code class="computeroutput"><span class="special">(-</span><span class="number">2.0</span>
|
|
<span class="keyword">and</span> <span class="special">+</span><span class="number">2.0</span><span class="special">)</span></code>.
|
|
</p></dd>
|
|
</dl>
|
|
</div>
|
|
<p>
|
|
It is possible that <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code> is not
|
|
enclosed by <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code>, while
|
|
neither is <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">T</span><span class="special">)</span></code> enclosed
|
|
by <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="identifier">S</span><span class="special">)</span></code>; for
|
|
example, <code class="computeroutput"><span class="identifier">UNSIG</span><span class="special">=[</span><span class="number">0</span><span class="special">,</span><span class="number">255</span><span class="special">]</span></code> is not enclosed by <code class="computeroutput"><span class="identifier">SIG</span><span class="special">=[-</span><span class="number">128</span><span class="special">,</span><span class="number">127</span><span class="special">]</span></code>; neither
|
|
is <code class="computeroutput"><span class="identifier">SIG</span></code> enclosed by <code class="computeroutput"><span class="identifier">UNSIG</span></code>. This implies that is possible that
|
|
a conversion direction is subranged both ways. This occurs when a mixture
|
|
of signed/unsigned types are involved and indicates that in both directions
|
|
there are values which can fall out of range.
|
|
</p>
|
|
<p>
|
|
Given the range relation (subranged or not) of a conversion direction <code class="computeroutput"><span class="identifier">S</span><span class="special">-></span><span class="identifier">T</span></code>, it is possible to classify <code class="computeroutput"><span class="identifier">S</span></code> and <code class="computeroutput"><span class="identifier">T</span></code>
|
|
as <span class="bold"><strong>supertype</strong></span> and <span class="bold"><strong>subtype</strong></span>:
|
|
If the conversion is subranged, which means that <code class="computeroutput"><span class="identifier">T</span></code>
|
|
cannot represent all possible values of type <code class="computeroutput"><span class="identifier">S</span></code>,
|
|
<code class="computeroutput"><span class="identifier">S</span></code> is the supertype and <code class="computeroutput"><span class="identifier">T</span></code> the subtype; otherwise, <code class="computeroutput"><span class="identifier">T</span></code> is the supertype and <code class="computeroutput"><span class="identifier">S</span></code>
|
|
the subtype.
|
|
</p>
|
|
<p>
|
|
For example:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">float</span><span class="special">)=[-</span><span class="identifier">FLT_MAX</span><span class="special">,</span><span class="identifier">FLT_MAX</span><span class="special">]</span></code>
|
|
and <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">double</span><span class="special">)=[-</span><span class="identifier">DBL_MAX</span><span class="special">,</span><span class="identifier">DBL_MAX</span><span class="special">]</span></code>
|
|
</p></blockquote></div>
|
|
<p>
|
|
If <code class="computeroutput"><span class="identifier">FLT_MAX</span> <span class="special"><</span>
|
|
<span class="identifier">DBL_MAX</span></code>:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
<code class="computeroutput"><span class="keyword">double</span><span class="special">-></span><span class="keyword">float</span></code> is subranged and <code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">double</span></code>,
|
|
<code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">float</span></code>.
|
|
</li>
|
|
<li class="listitem">
|
|
<code class="computeroutput"><span class="keyword">float</span><span class="special">-></span><span class="keyword">double</span></code> is not subranged and <code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">double</span></code>, <code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">float</span></code>.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
Notice that while <code class="computeroutput"><span class="keyword">double</span><span class="special">-></span><span class="keyword">float</span></code> is subranged, <code class="computeroutput"><span class="keyword">float</span><span class="special">-></span><span class="keyword">double</span></code>
|
|
is not, which yields the same supertype,subtype for both directions.
|
|
</p>
|
|
<p>
|
|
Now consider:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">int</span><span class="special">)=[</span><span class="identifier">INT_MIN</span><span class="special">,</span><span class="identifier">INT_MAX</span><span class="special">]</span></code> and <code class="computeroutput"><span class="identifier">R</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="keyword">int</span><span class="special">)=[</span><span class="number">0</span><span class="special">,</span><span class="identifier">UINT_MAX</span><span class="special">]</span></code>
|
|
</p></blockquote></div>
|
|
<p>
|
|
A C++ implementation is required to have <code class="computeroutput"><span class="identifier">UINT_MAX</span>
|
|
<span class="special">></span> <span class="identifier">INT_MAX</span></code>
|
|
(§3.9/3), so:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
'int->unsigned' is subranged (negative values fall out of range) and
|
|
<code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">int</span></code>, <code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">unsigned</span></code>.
|
|
</li>
|
|
<li class="listitem">
|
|
'unsigned->int' is <span class="emphasis"><em>also</em></span> subranged (high positive
|
|
values fall out of range) and <code class="computeroutput"><span class="identifier">supertype</span><span class="special">=</span><span class="keyword">unsigned</span></code>,
|
|
<code class="computeroutput"><span class="identifier">subtype</span><span class="special">=</span><span class="keyword">int</span></code>.
|
|
</li>
|
|
</ul></div>
|
|
<p>
|
|
In this case, the conversion is subranged in both directions and the supertype,subtype
|
|
pairs are not invariant (under inversion of direction). This indicates that
|
|
none of the types can represent all the values of the other.
|
|
</p>
|
|
<p>
|
|
When the supertype is the same for both <code class="computeroutput"><span class="identifier">S</span><span class="special">-></span><span class="identifier">T</span></code>
|
|
and <code class="computeroutput"><span class="identifier">T</span><span class="special">-></span><span class="identifier">S</span></code>, it is effectively indicating a type
|
|
which can represent all the values of the subtype. Consequently, if a conversion
|
|
<code class="computeroutput"><span class="identifier">X</span><span class="special">-></span><span class="identifier">Y</span></code> is not subranged, but the opposite <code class="computeroutput"><span class="special">(</span><span class="identifier">Y</span><span class="special">-></span><span class="identifier">X</span><span class="special">)</span></code> is,
|
|
so that the supertype is always <code class="computeroutput"><span class="identifier">Y</span></code>,
|
|
it is said that the direction <code class="computeroutput"><span class="identifier">X</span><span class="special">-></span><span class="identifier">Y</span></code>
|
|
is <span class="bold"><strong>correctly rounded value preserving</strong></span>, meaning
|
|
that all such conversions are guaranteed to produce results in range and
|
|
correctly rounded (even if inexact). For example, all integer to floating
|
|
conversions are correctly rounded value preserving.
|
|
</p>
|
|
</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 © 2004-2007 Fernando
|
|
Luis Cacciola Carballal<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
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