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<title>Mixed Precision Arithmetic</title>
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_multiprecision.tut.mixed"></a><a class="link" href="mixed.html" title="Mixed Precision Arithmetic">Mixed Precision Arithmetic</a>
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</h3></div></div></div>
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<p>
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Mixed precision arithmetic is fully supported by the library.
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</p>
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<p>
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There are two different forms:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Where the operands are of different precision.
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</li>
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<li class="listitem">
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Where the operands are of the same precision, but yield a higher precision
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result.
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</li>
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</ul></div>
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<h5>
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<a name="boost_multiprecision.tut.mixed.h0"></a>
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<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
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Operands of Differing Precision</a>
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</h5>
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<p>
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If the arguments to a binary operator are of different precision, then the
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operation is allowed as long as there is an unambiguous implicit conversion
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from one argument type to the other. In all cases the arithmetic is performed
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"as if" the lower precision type is promoted to the higher precision
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type before applying the operator. However, particular backends may optimise
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this and avoid actually creating a temporary if they are able to do so.
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</p>
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<p>
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For example:
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</p>
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<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>
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<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>
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<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>
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<span class="identifier">mpz_int</span> <span class="identifier">i</span><span class="special">(</span><span class="number">20</span><span class="special">);</span>
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<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>
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<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>
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<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>
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<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>
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</pre>
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<h5>
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<a name="boost_multiprecision.tut.mixed.h1"></a>
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<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
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of the Same Precision</a>
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</h5>
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<p>
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Sometimes you want to apply an operator to two arguments of the same precision
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in such a way as to obtain a result of higher precision. The most common
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situation occurs with fixed precision integers, where you want to multiply
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two N-bit numbers to obtain a 2N-bit result. This is supported in this library
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by the following free functions:
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</p>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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</pre>
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<p>
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These functions apply the named operator to the arguments <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>b</em></span> and store the result in <span class="emphasis"><em>result</em></span>,
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returning <span class="emphasis"><em>result</em></span>. In all cases they behave "as
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if" arguments <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> were
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first promoted to type <code class="computeroutput"><span class="identifier">ResultType</span></code>
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before applying the operator, though particular backends may well avoid that
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step by way of an optimization.
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</p>
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<p>
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The type <code class="computeroutput"><span class="identifier">ResultType</span></code> must
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be an instance of class <code class="computeroutput"><span class="identifier">number</span></code>,
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and the types <code class="computeroutput"><span class="identifier">Source1</span></code> and
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<code class="computeroutput"><span class="identifier">Source2</span></code> may be either instances
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of class <code class="computeroutput"><span class="identifier">number</span></code> or native
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integer types. The latter is an optimization that allows arithmetic to be
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performed on native integer types producing an extended precision result.
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</p>
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<p>
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For example:
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</p>
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<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>
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<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
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<span class="special">{</span>
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<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>
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<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>
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<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>
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<span class="identifier">uint128_t</span> <span class="identifier">ui128</span><span class="special">;</span>
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<span class="identifier">uint256_t</span> <span class="identifier">ui256</span><span class="special">;</span>
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<span class="comment">//</span>
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<span class="comment">// Start by performing arithmetic on 64-bit integers to yield 128-bit results:</span>
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<span class="comment">//</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">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>
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<span class="comment">//</span>
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<span class="comment">// The try squaring a 128-bit integer to yield a 256-bit result:</span>
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<span class="comment">//</span>
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<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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">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>
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<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
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<span class="special">}</span>
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</pre>
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<p>
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Produces the output:
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</p>
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<pre class="programlisting"><span class="number">0</span><span class="identifier">xffffffffffffffff</span>
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<span class="number">0</span><span class="identifier">x10000000000000000</span>
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<span class="number">0</span><span class="identifier">xFFFFFFFFFFFFFFFE0000000000000001</span>
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<span class="number">0</span><span class="identifier">xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE00000000000000000000000000000001</span>
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</pre>
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<h5>
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<a name="boost_multiprecision.tut.mixed.h2"></a>
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<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
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With Optimized Mixed Precision Arithmetic</a>
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</h5>
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<p>
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The following backends have at least some direct support for mixed precision
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arithmetic, and therefore avoid creating unnecessary temporaries when using
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the interfaces above. Therefore when using these types it's more efficient
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to use mixed precision arithmetic, than it is to explicitly cast the operands
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to the result type:
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</p>
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<p>
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<a class="link" href="floats/mpfr_float.html" title="mpfr_float">mpfr_float</a>,
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<a class="link" href="floats/gmp_float.html" title="gmp_float">gmp_float</a>,
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<a class="link" href="ints/cpp_int.html" title="cpp_int">cpp_int</a>.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2002-2019 John Maddock
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and Christopher Kormanyos<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div></td>
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