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- <div class="section">
- <div class="titlepage"><div><div><h3 class="title">
- <a name="boost_multiprecision.ref.cpp_bin_float_ref"></a><a class="link" href="cpp_bin_float_ref.html" title="cpp_bin_float">cpp_bin_float</a>
- </h3></div></div></div>
- <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">multiprecision</span><span class="special">{</span>
- <span class="keyword">enum</span> <span class="identifier">digit_base_type</span>
- <span class="special">{</span>
- <span class="identifier">digit_base_2</span> <span class="special">=</span> <span class="number">2</span><span class="special">,</span>
- <span class="identifier">digit_base_10</span> <span class="special">=</span> <span class="number">10</span>
- <span class="special">};</span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">unsigned</span> <span class="identifier">Digits</span><span class="special">,</span> <span class="identifier">digit_base_type</span> <span class="identifier">base</span> <span class="special">=</span> <span class="identifier">digit_base_10</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Allocator</span> <span class="special">=</span> <span class="keyword">void</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Exponent</span> <span class="special">=</span> <span class="keyword">int</span><span class="special">,</span> <span class="identifier">ExponentMin</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">ExponentMax</span> <span class="special">=</span> <span class="number">0</span><span class="special">></span>
- <span class="keyword">class</span> <span class="identifier">cpp_bin_float</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">50</span><span class="special">></span> <span class="special">></span> <span class="identifier">cpp_bin_float_50</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">100</span><span class="special">></span> <span class="special">></span> <span class="identifier">cpp_bin_float_100</span><span class="special">;</span>
- <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">backends</span><span class="special">::</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">24</span><span class="special">,</span> <span class="identifier">backends</span><span class="special">::</span><span class="identifier">digit_base_2</span><span class="special">,</span> <span class="keyword">void</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">int16_t</span><span class="special">,</span> <span class="special">-</span><span class="number">126</span><span class="special">,</span> <span class="number">127</span><span class="special">>,</span> <span class="identifier">et_off</span><span class="special">></span> <span class="identifier">cpp_bin_float_single</span><span class="special">;</span>
- <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">backends</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">backends</span><span class="special">::</span><span class="identifier">digit_base_2</span><span class="special">,</span> <span class="keyword">void</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">int16_t</span><span class="special">,</span> <span class="special">-</span><span class="number">1022</span><span class="special">,</span> <span class="number">1023</span><span class="special">>,</span> <span class="identifier">et_off</span><span class="special">></span> <span class="identifier">cpp_bin_float_double</span><span class="special">;</span>
- <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">backends</span><span class="special">::</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">64</span><span class="special">,</span> <span class="identifier">backends</span><span class="special">::</span><span class="identifier">digit_base_2</span><span class="special">,</span> <span class="keyword">void</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">int16_t</span><span class="special">,</span> <span class="special">-</span><span class="number">16382</span><span class="special">,</span> <span class="number">16383</span><span class="special">>,</span> <span class="identifier">et_off</span><span class="special">></span> <span class="identifier">cpp_bin_float_double_extended</span><span class="special">;</span>
- <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">backends</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">backends</span><span class="special">::</span><span class="identifier">digit_base_2</span><span class="special">,</span> <span class="keyword">void</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">int16_t</span><span class="special">,</span> <span class="special">-</span><span class="number">16382</span><span class="special">,</span> <span class="number">16383</span><span class="special">>,</span> <span class="identifier">et_off</span><span class="special">></span> <span class="identifier">cpp_bin_float_quad</span><span class="special">;</span>
- <span class="keyword">typedef</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">backends</span><span class="special">::</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">237</span><span class="special">,</span> <span class="identifier">backends</span><span class="special">::</span><span class="identifier">digit_base_2</span><span class="special">,</span> <span class="keyword">void</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">int32_t</span><span class="special">,</span> <span class="special">-</span><span class="number">262142</span><span class="special">,</span> <span class="number">262143</span><span class="special">>,</span> <span class="identifier">et_off</span><span class="special">></span> <span class="identifier">cpp_bin_float_oct</span><span class="special">;</span>
- <span class="special">}}</span> <span class="comment">// namespaces</span>
- </pre>
- <p>
- Class template <code class="computeroutput"><span class="identifier">cpp_bin_float</span></code>
- fulfils all of the requirements for a <a class="link" href="backendconc.html" title="Backend Requirements">Backend</a>
- type. Its members and non-member functions are deliberately not documented:
- these are considered implementation details that are subject to change.
- </p>
- <p>
- The class takes six template parameters:
- </p>
- <div class="variablelist">
- <p class="title"><b></b></p>
- <dl class="variablelist">
- <dt><span class="term">Digits</span></dt>
- <dd><p>
- The number of digits precision the type should support. This is normally
- expressed as base-10 digits, but that can be changed via the second
- template parameter.
- </p></dd>
- <dt><span class="term">base</span></dt>
- <dd><p>
- An enumerated value (either <code class="computeroutput"><span class="identifier">digit_base_10</span></code>
- or <code class="computeroutput"><span class="identifier">digit_base_2</span></code>) that
- indicates whether <code class="computeroutput"><span class="identifier">Digits</span></code>
- is base-10 or base-2
- </p></dd>
- <dt><span class="term">Allocator</span></dt>
- <dd><p>
- The allocator used: defaults to type <code class="computeroutput"><span class="keyword">void</span></code>,
- meaning all storage is within the class, and no dynamic allocation
- is performed, but can be set to a standard library allocator if dynamic
- allocation makes more sense.
- </p></dd>
- <dt><span class="term">Exponent</span></dt>
- <dd><p>
- A signed integer type to use as the type of the exponent - defaults
- to <code class="computeroutput"><span class="keyword">int</span></code>.
- </p></dd>
- <dt><span class="term">ExponentMin</span></dt>
- <dd><p>
- The smallest (most negative) permitted exponent, defaults to zero,
- meaning "define as small as possible given the limitations of
- the type and our internal requirements".
- </p></dd>
- <dt><span class="term">ExponentMax</span></dt>
- <dd><p>
- The largest (most positive) permitted exponent, defaults to zero, meaning
- "define as large as possible given the limitations of the type
- and our internal requirements".
- </p></dd>
- </dl>
- </div>
- <p>
- The type of <code class="computeroutput"><span class="identifier">number_category</span><span class="special"><</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="identifier">Args</span><span class="special">...></span> <span class="special">>::</span><span class="identifier">type</span></code> is <code class="computeroutput"><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">int_</span><span class="special"><</span><span class="identifier">number_kind_floating_point</span><span class="special">></span></code>.
- </p>
- <p>
- More information on this type can be found in the <a class="link" href="../tut/floats/cpp_bin_float.html" title="cpp_bin_float">tutorial</a>.
- </p>
- <h5>
- <a name="boost_multiprecision.ref.cpp_bin_float_ref.h0"></a>
- <span class="phrase"><a name="boost_multiprecision.ref.cpp_bin_float_ref.implementation_notes"></a></span><a class="link" href="cpp_bin_float_ref.html#boost_multiprecision.ref.cpp_bin_float_ref.implementation_notes">Implementation
- Notes</a>
- </h5>
- <p>
- Internally, an N-bit <code class="computeroutput"><span class="identifier">cpp_bin_float</span></code>
- is represented as an N-bit unsigned integer along with an exponent and a
- sign. The integer part is normalized so that it's most significant bit is
- always 1. The decimal point is assumed to be directly after the most significant
- bit of the integer part. The special values zero, infinity and NaN all have
- the integer part set to zero, and the exponent to one of 3 special values
- above the maximum permitted exponent.
- </p>
- <p>
- Multiplication is trivial: multiply the two N-bit integer mantissa's to obtain
- a 2N-bit number, then round and adjust the sign and exponent.
- </p>
- <p>
- Addition and subtraction proceed similarly - if the exponents are such that
- there is overlap between the two values, then left shift the larger value
- to produce a number with between N and 2N bits, then perform integer addition
- or subtraction, round, and adjust the exponent.
- </p>
- <p>
- Division proceeds as follows: first scale the numerator by some power of
- 2 so that integer division will produce either an N-bit or N+1 bit result
- plus a remainder. If we get an N bit result then the size of twice the remainder
- compared to the denominator gives us the rounding direction. Otherwise we
- have one extra bit in the result which we can use to determine rounding (in
- this case ties occur only if the remainder is zero and the extra bit is a
- 1).
- </p>
- <p>
- Square root uses integer square root in a manner analogous to division.
- </p>
- <p>
- Decimal string to binary conversion proceeds as follows: first parse the
- digits to produce an integer multiplied by a decimal exponent. Note that
- we stop parsing digits once we have parsed as many as can possibly effect
- the result - this stops the integer part growing too large when there are
- a very large number of input digits provided. At this stage if the decimal
- exponent is positive then the result is an integer and we can in principle
- simply multiply by 10^N to get an exact integer result. In practice however,
- that could produce some very large integers. We also need to be able to divide
- by 10^N in the event that the exponent is negative. Therefore calculation
- of the 10^N values plus the multiplication or division are performed using
- limited precision integer arithmetic, plus an exponent, and a track of the
- accumulated error. At the end of the calculation we will either be able to
- round unambiguously, or the error will be such that we can't tell which way
- to round. In the latter case we simply up the precision and try again until
- we have an unambiguously rounded result.
- </p>
- <p>
- Binary to decimal conversion proceeds very similarly to the above, our aim
- is to calculate <code class="computeroutput"><span class="identifier">mantissa</span> <span class="special">*</span> <span class="number">2</span><span class="special">^</span><span class="identifier">shift</span> <span class="special">*</span> <span class="number">10</span><span class="special">^</span><span class="identifier">E</span></code>
- where <code class="computeroutput"><span class="identifier">E</span></code> is the decimal exponent
- and <code class="computeroutput"><span class="identifier">shift</span></code> is calculated so
- that the result is an N bit integer assuming we want N digits printed in
- the result. As before we use limited precision arithmetic to calculate the
- result and up the precision as necessary until the result is unambiguously
- correctly rounded. In addition our initial calculation of the decimal exponent
- may be out by 1, so we have to correct that and loop as well in the that
- case.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2002-2019 John Maddock
- and Christopher Kormanyos<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
- </p>
- </div></td>
- </tr></table>
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