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24 | </div> | |
25 | <div class="section"> | |
26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="boost_multiprecision.tut.mixed"></a><a class="link" href="mixed.html" title="Mixed Precision Arithmetic">Mixed Precision Arithmetic</a> | |
28 | </h3></div></div></div> | |
29 | <p> | |
30 | Mixed precision arithmetic is fully supported by the library. | |
31 | </p> | |
32 | <p> | |
33 | There are two different forms: | |
34 | </p> | |
35 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
36 | <li class="listitem"> | |
37 | Where the operands are of different precision. | |
38 | </li> | |
39 | <li class="listitem"> | |
40 | Where the operands are of the same precision, but yield a higher precision | |
41 | result. | |
42 | </li> | |
43 | </ul></div> | |
44 | <h5> | |
45 | <a name="boost_multiprecision.tut.mixed.h0"></a> | |
46 | <span class="phrase"><a name="boost_multiprecision.tut.mixed.mixing_operands_of_differing_pre"></a></span><a class="link" href="mixed.html#boost_multiprecision.tut.mixed.mixing_operands_of_differing_pre">Mixing | |
47 | Operands of Differing Precision</a> | |
48 | </h5> | |
49 | <p> | |
50 | If the arguments to a binary operator are of different precision, then the | |
51 | operation is allowed as long as there is an unambiguous implicit conversion | |
52 | from one argument type to the other. In all cases the arithmetic is performed | |
53 | "as if" the lower precision type is promoted to the higher precision | |
54 | type before applying the operator. However, particular backends may optimise | |
55 | this and avoid actually creating a temporary if they are able to do so. | |
56 | </p> | |
57 | <p> | |
58 | For example: | |
59 | </p> | |
60 | <pre class="programlisting"><span class="identifier">mpfr_float_50</span> <span class="identifier">a</span><span class="special">(</span><span class="number">2</span><span class="special">),</span> <span class="identifier">b</span><span class="special">;</span> | |
61 | <span class="identifier">mpfr_float_100</span> <span class="identifier">c</span><span class="special">(</span><span class="number">3</span><span class="special">),</span> <span class="identifier">d</span><span class="special">;</span> | |
62 | <span class="identifier">static_mpfr_float_50</span> <span class="identifier">e</span><span class="special">(</span><span class="number">5</span><span class="special">),</span> <span class="identifier">f</span><span class="special">;</span> | |
63 | <span class="identifier">mpz_int</span> <span class="identifier">i</span><span class="special">(</span><span class="number">20</span><span class="special">);</span> | |
64 | ||
65 | <span class="identifier">d</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">c</span><span class="special">;</span> <span class="comment">// OK, result of operand is an mpfr_float_100.</span> | |
66 | <span class="identifier">b</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">c</span><span class="special">;</span> <span class="comment">// Error, can't convert the result to an mpfr_float_50 as it will lose digits.</span> | |
67 | <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">e</span><span class="special">;</span> <span class="comment">// Error, operator is ambiguous, result could be of either type.</span> | |
68 | <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">e</span> <span class="special">*</span> <span class="identifier">i</span><span class="special">;</span> <span class="comment">// OK, unambiguous conversion from mpz_int to static_mpfr_float_50</span> | |
69 | </pre> | |
70 | <h5> | |
71 | <a name="boost_multiprecision.tut.mixed.h1"></a> | |
72 | <span class="phrase"><a name="boost_multiprecision.tut.mixed.operands_of_the_same_precision"></a></span><a class="link" href="mixed.html#boost_multiprecision.tut.mixed.operands_of_the_same_precision">Operands | |
73 | of the Same Precision</a> | |
74 | </h5> | |
75 | <p> | |
76 | Sometimes you want to apply an operator to two arguments of the same precision | |
77 | in such a way as to obtain a result of higher precision. The most common | |
78 | situation occurs with fixed precision integers, where you want to multiply | |
79 | two N-bit numbers to obtain a 2N-bit result. This is supported in this library | |
80 | by the following free functions: | |
81 | </p> | |
82 | <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">></span> | |
83 | <span class="identifier">ResultType</span><span class="special">&</span> <span class="identifier">add</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span> | |
84 | ||
85 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">></span> | |
86 | <span class="identifier">ResultType</span><span class="special">&</span> <span class="identifier">subtract</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span> | |
87 | ||
88 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">ResultType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Source1</span> <span class="keyword">class</span> <span class="identifier">Source2</span><span class="special">></span> | |
89 | <span class="identifier">ResultType</span><span class="special">&</span> <span class="identifier">multiply</span><span class="special">(</span><span class="identifier">ResultType</span><span class="special">&</span> <span class="identifier">result</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source1</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Source2</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span> | |
90 | </pre> | |
91 | <p> | |
92 | These functions apply the named operator to the arguments <span class="emphasis"><em>a</em></span> | |
93 | and <span class="emphasis"><em>b</em></span> and store the result in <span class="emphasis"><em>result</em></span>, | |
94 | returning <span class="emphasis"><em>result</em></span>. In all cases they behave "as | |
95 | if" arguments <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> were | |
96 | first promoted to type <code class="computeroutput"><span class="identifier">ResultType</span></code> | |
97 | before applying the operator, though particular backends may well avoid that | |
98 | step by way of an optimization. | |
99 | </p> | |
100 | <p> | |
101 | The type <code class="computeroutput"><span class="identifier">ResultType</span></code> must | |
102 | be an instance of class <code class="computeroutput"><span class="identifier">number</span></code>, | |
103 | and the types <code class="computeroutput"><span class="identifier">Source1</span></code> and | |
104 | <code class="computeroutput"><span class="identifier">Source2</span></code> may be either instances | |
105 | of class <code class="computeroutput"><span class="identifier">number</span></code> or native | |
106 | integer types. The latter is an optimization that allows arithmetic to be | |
107 | performed on native integer types producing an extended precision result. | |
108 | </p> | |
109 | <p> | |
110 | For example: | |
111 | </p> | |
112 | <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_int</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
113 | ||
114 | <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> | |
115 | <span class="special">{</span> | |
116 | <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span> | |
117 | ||
118 | <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint64_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint64_t</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)();</span> | |
119 | <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uint64_t</span> <span class="identifier">j</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> | |
120 | ||
121 | <span class="identifier">uint128_t</span> <span class="identifier">ui128</span><span class="special">;</span> | |
122 | <span class="identifier">uint256_t</span> <span class="identifier">ui256</span><span class="special">;</span> | |
123 | <span class="comment">//</span> | |
124 | <span class="comment">// Start by performing arithmetic on 64-bit integers to yield 128-bit results:</span> | |
125 | <span class="comment">//</span> | |
126 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">i</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
127 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">add</span><span class="special">(</span><span class="identifier">ui128</span><span class="special">,</span> <span class="identifier">i</span><span class="special">,</span> <span class="identifier">j</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
128 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">multiply</span><span class="special">(</span><span class="identifier">ui128</span><span class="special">,</span> <span class="identifier">i</span><span class="special">,</span> <span class="identifier">i</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
129 | <span class="comment">//</span> | |
130 | <span class="comment">// The try squaring a 128-bit integer to yield a 256-bit result:</span> | |
131 | <span class="comment">//</span> | |
132 | <span class="identifier">ui128</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">uint128_t</span><span class="special">>::</span><span class="identifier">max</span><span class="special">)();</span> | |
133 | <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">multiply</span><span class="special">(</span><span class="identifier">ui256</span><span class="special">,</span> <span class="identifier">ui128</span><span class="special">,</span> <span class="identifier">ui128</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> | |
134 | ||
135 | <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span> | |
136 | <span class="special">}</span> | |
137 | </pre> | |
138 | <p> | |
139 | Produces the output: | |
140 | </p> | |
141 | <pre class="programlisting"><span class="number">0</span><span class="identifier">xffffffffffffffff</span> | |
142 | <span class="number">0</span><span class="identifier">x10000000000000000</span> | |
143 | <span class="number">0</span><span class="identifier">xFFFFFFFFFFFFFFFE0000000000000001</span> | |
144 | <span class="number">0</span><span class="identifier">xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE00000000000000000000000000000001</span> | |
145 | </pre> | |
146 | <h5> | |
147 | <a name="boost_multiprecision.tut.mixed.h2"></a> | |
148 | <span class="phrase"><a name="boost_multiprecision.tut.mixed.backends_with_optimized_mixed_pr"></a></span><a class="link" href="mixed.html#boost_multiprecision.tut.mixed.backends_with_optimized_mixed_pr">Backends | |
149 | With Optimized Mixed Precision Arithmetic</a> | |
150 | </h5> | |
151 | <p> | |
152 | The following backends have at least some direct support for mixed precision | |
153 | arithmetic, and therefore avoid creating unnecessary temporaries when using | |
154 | the interfaces above. Therefore when using these types it's more efficient | |
155 | to use mixed precision arithmetic, than it is to explicitly cast the operands | |
156 | to the result type: | |
157 | </p> | |
158 | <p> | |
159 | <a class="link" href="floats/mpfr_float.html" title="mpfr_float">mpfr_float</a>, | |
160 | <a class="link" href="floats/gmp_float.html" title="gmp_float">gmp_float</a>, | |
161 | <a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>. | |
162 | </p> | |
163 | </div> | |
164 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
165 | <td align="left"></td> | |
166 | <td align="right"><div class="copyright-footer">Copyright © 2002-2013 John Maddock and Christopher Kormanyos<p> | |
167 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
168 | 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>) | |
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