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HTML
749 lines
34 KiB
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<div id="org-div-home-and-up">
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<a accesskey="h" href=""> UP </a>
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|
|
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<a accesskey="H" href="index.html"> HOME </a>
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</div><div id="content">
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<div id="table-of-contents">
|
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<h2>Table of Contents</h2>
|
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<div id="text-table-of-contents">
|
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<ul>
|
||
<li><a href="#org6dc4c21">1. Control of the numerical precision</a></li>
|
||
<li><a href="#org9ccfce5">2. Precision</a></li>
|
||
<li><a href="#orge3baf56">3. Range</a></li>
|
||
<li><a href="#orgaecc549">4. Helper functions</a>
|
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<ul>
|
||
<li><a href="#org8481318">4.1. Epsilon</a></li>
|
||
<li><a href="#org1b092d5">4.2. Testing the number of unchanged bits</a></li>
|
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</li>
|
||
<li><a href="#orgcf44399">5. Approximate functions</a>
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|
||
<li><a href="#orgbd1af10">5.1. Exponential</a></li>
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<p>
|
||
3+TITLE: Numerical precision
|
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</p>
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<div id="outline-container-org6dc4c21" class="outline-2">
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<h2 id="org6dc4c21"><span class="section-number-2">1</span> Control of the numerical precision</h2>
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||
<div class="outline-text-2" id="text-1">
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||
<p>
|
||
Controlling numerical precision enables optimizations. Here, the
|
||
default parameters determining the target numerical precision and
|
||
range are defined. Following the IEEE Standard for Floating-Point
|
||
Arithmetic (IEEE 754),
|
||
<i>precision</i> refers to the number of significand bits (including the
|
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sign bit) and <i>range</i> refers to the number of exponent bits.
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</p>
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|
||
<td class="org-left"><code>QMCKL_DEFAULT_PRECISION</code></td>
|
||
<td class="org-right">53</td>
|
||
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|
||
|
||
<tr>
|
||
<td class="org-left"><code>QMCKL_DEFAULT_RANGE</code></td>
|
||
<td class="org-right">11</td>
|
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||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #a020f0;">typedef</span> <span style="color: #a020f0;">struct</span> <span style="color: #228b22;">qmckl_numprec_struct</span> {
|
||
<span style="color: #228b22;">uint32_t</span> <span style="color: #a0522d;">precision</span>;
|
||
<span style="color: #228b22;">uint32_t</span> <span style="color: #a0522d;">range</span>;
|
||
} <span style="color: #228b22;">qmckl_numprec_struct</span>;
|
||
</pre>
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</div>
|
||
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||
<p>
|
||
The following functions set and get the required precision and
|
||
range. <code>precision</code> is an integer between 2 and 53, and <code>range</code> is an
|
||
integer between 2 and 11.
|
||
</p>
|
||
|
||
<p>
|
||
The setter functions functions return a new context as a 64-bit
|
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integer. The getter functions return the value, as a 32-bit
|
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integer. The update functions return <code>QMCKL_SUCCESS</code> or
|
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<code>QMCKL_FAILURE</code>.
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</p>
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<div id="outline-container-org9ccfce5" class="outline-2">
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<h2 id="org9ccfce5"><span class="section-number-2">2</span> Precision</h2>
|
||
<div class="outline-text-2" id="text-2">
|
||
<p>
|
||
<code>qmckl_context_set_numprec_precision</code> modifies the parameter for the
|
||
numerical precision in the context.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_numprec_precision</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #a020f0;">const</span> <span style="color: #228b22;">int</span> <span style="color: #a0522d;">precision</span>) {
|
||
|
||
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
|
||
<span style="color: #a020f0;">return</span> QMCKL_INVALID_CONTEXT;
|
||
|
||
<span style="color: #a020f0;">if</span> (precision < 2) {
|
||
<span style="color: #a020f0;">return</span> qmckl_failwith(context,
|
||
QMCKL_INVALID_ARG_2,
|
||
<span style="color: #8b2252;">"qmckl_update_numprec_precision"</span>,
|
||
<span style="color: #8b2252;">"precision < 2"</span>);
|
||
}
|
||
|
||
<span style="color: #a020f0;">if</span> (precision > 53) {
|
||
<span style="color: #a020f0;">return</span> qmckl_failwith(context,
|
||
QMCKL_INVALID_ARG_2,
|
||
<span style="color: #8b2252;">"qmckl_update_numprec_precision"</span>,
|
||
<span style="color: #8b2252;">"precision > 53"</span>);
|
||
}
|
||
|
||
<span style="color: #228b22;">qmckl_context_struct</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ctx</span> = (<span style="color: #228b22;">qmckl_context_struct</span>*) context;
|
||
|
||
/* <span style="color: #b22222;">This should be always true because the context is valid</span> */
|
||
assert (ctx != <span style="color: #008b8b;">NULL</span>);
|
||
|
||
qmckl_lock(context);
|
||
{
|
||
ctx->numprec.precision = (<span style="color: #228b22;">uint32_t</span>) precision;
|
||
}
|
||
qmckl_unlock(context);
|
||
|
||
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
||
}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">interface</span>
|
||
<span style="color: #228b22;">integer</span> (qmckl_exit_code) <span style="color: #a020f0;">function</span> <span style="color: #0000ff;">qmckl_set_numprec_precision</span>(context, precision) <span style="color: #a020f0;">bind</span>(C)
|
||
<span style="color: #a020f0;">use</span>, <span style="color: #a020f0;">intrinsic</span> :: <span style="color: #0000ff;">iso_c_binding</span>
|
||
<span style="color: #a020f0;">import</span>
|
||
<span style="color: #228b22;">integer</span> (qmckl_context), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> context</span>
|
||
<span style="color: #228b22;">integer</span> (<span style="color: #008b8b;">c_int32_t</span>), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> precision</span>
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">qmckl_set_numprec_precision</span>
|
||
<span style="color: #a020f0;">end interface</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<code>qmckl_get_numprec_precision</code> returns the value of the numerical precision in the context.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">int</span> <span style="color: #0000ff;">qmckl_get_numprec_precision</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>) {
|
||
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
|
||
<span style="color: #a020f0;">return</span> qmckl_failwith(context,
|
||
QMCKL_INVALID_CONTEXT,
|
||
<span style="color: #8b2252;">"qmckl_get_numprec_precision"</span>,
|
||
<span style="color: #8b2252;">""</span>);
|
||
}
|
||
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context_struct</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ctx</span> = (<span style="color: #228b22;">qmckl_context_struct</span>*) context;
|
||
<span style="color: #a020f0;">return</span> ctx->numprec.precision;
|
||
}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">interface</span>
|
||
<span style="color: #228b22;">integer</span> (qmckl_exit_code) <span style="color: #a020f0;">function</span> <span style="color: #0000ff;">qmckl_get_numprec_precision</span>(context) <span style="color: #a020f0;">bind</span>(C)
|
||
<span style="color: #a020f0;">use</span>, <span style="color: #a020f0;">intrinsic</span> :: <span style="color: #0000ff;">iso_c_binding</span>
|
||
<span style="color: #a020f0;">import</span>
|
||
<span style="color: #228b22;">integer</span> (qmckl_context), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> context</span>
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">qmckl_get_numprec_precision</span>
|
||
<span style="color: #a020f0;">end interface</span>
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orge3baf56" class="outline-2">
|
||
<h2 id="orge3baf56"><span class="section-number-2">3</span> Range</h2>
|
||
<div class="outline-text-2" id="text-3">
|
||
<p>
|
||
<code>qmckl_set_numprec_range</code> modifies the parameter for the numerical
|
||
range in a given context.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">qmckl_exit_code</span> <span style="color: #0000ff;">qmckl_set_numprec_range</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>, <span style="color: #a020f0;">const</span> <span style="color: #228b22;">int</span> <span style="color: #a0522d;">range</span>) {
|
||
|
||
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
|
||
<span style="color: #a020f0;">return</span> QMCKL_INVALID_CONTEXT;
|
||
|
||
<span style="color: #a020f0;">if</span> (range < 2) {
|
||
<span style="color: #a020f0;">return</span> qmckl_failwith(context,
|
||
QMCKL_INVALID_ARG_2,
|
||
<span style="color: #8b2252;">"qmckl_set_numprec_range"</span>,
|
||
<span style="color: #8b2252;">"range < 2"</span>);
|
||
}
|
||
|
||
<span style="color: #a020f0;">if</span> (range > 11) {
|
||
<span style="color: #a020f0;">return</span> qmckl_failwith(context,
|
||
QMCKL_INVALID_ARG_2,
|
||
<span style="color: #8b2252;">"qmckl_set_numprec_range"</span>,
|
||
<span style="color: #8b2252;">"range > 11"</span>);
|
||
}
|
||
|
||
<span style="color: #228b22;">qmckl_context_struct</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ctx</span> = (<span style="color: #228b22;">qmckl_context_struct</span>*) context;
|
||
|
||
/* <span style="color: #b22222;">This should be always true because the context is valid</span> */
|
||
assert (ctx != <span style="color: #008b8b;">NULL</span>);
|
||
|
||
qmckl_lock(context);
|
||
{
|
||
ctx->numprec.range = (<span style="color: #228b22;">uint32_t</span>) range;
|
||
}
|
||
qmckl_unlock(context);
|
||
|
||
<span style="color: #a020f0;">return</span> QMCKL_SUCCESS;
|
||
}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-f90"><span style="color: #a020f0;">interface</span>
|
||
<span style="color: #228b22;">integer</span> (qmckl_exit_code) <span style="color: #a020f0;">function</span> <span style="color: #0000ff;">qmckl_set_numprec_range</span>(context, range) <span style="color: #a020f0;">bind</span>(C)
|
||
<span style="color: #a020f0;">use</span>, <span style="color: #a020f0;">intrinsic</span> :: <span style="color: #0000ff;">iso_c_binding</span>
|
||
<span style="color: #a020f0;">import</span>
|
||
<span style="color: #228b22;">integer</span> (qmckl_context), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> context</span>
|
||
<span style="color: #228b22;">integer</span> (<span style="color: #008b8b;">c_int32_t</span>), <span style="color: #a020f0;">intent</span>(in), <span style="color: #a020f0;">value</span> ::<span style="color: #a0522d;"> range</span>
|
||
<span style="color: #a020f0;">end function</span> <span style="color: #0000ff;">qmckl_set_numprec_range</span>
|
||
<span style="color: #a020f0;">end interface</span>
|
||
</pre>
|
||
</div>
|
||
|
||
<p>
|
||
<code>qmckl_get_numprec_range</code> returns the value of the numerical range in the context.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">int</span> <span style="color: #0000ff;">qmckl_get_numprec_range</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>) {
|
||
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
|
||
<span style="color: #a020f0;">return</span> qmckl_failwith(context,
|
||
QMCKL_INVALID_CONTEXT,
|
||
<span style="color: #8b2252;">"qmckl_get_numprec_range"</span>,
|
||
<span style="color: #8b2252;">""</span>);
|
||
}
|
||
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context_struct</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ctx</span> = (<span style="color: #228b22;">qmckl_context_struct</span>*) context;
|
||
<span style="color: #a020f0;">return</span> ctx->numprec.range;
|
||
}
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-orgaecc549" class="outline-2">
|
||
<h2 id="orgaecc549"><span class="section-number-2">4</span> Helper functions</h2>
|
||
<div class="outline-text-2" id="text-4">
|
||
</div>
|
||
<div id="outline-container-org8481318" class="outline-3">
|
||
<h3 id="org8481318"><span class="section-number-3">4.1</span> Epsilon</h3>
|
||
<div class="outline-text-3" id="text-4-1">
|
||
<p>
|
||
<code>qmckl_get_numprec_epsilon</code> returns \(\epsilon = 2^{1-n}\) where <code>n</code> is the precision.
|
||
We need to remove the sign bit from the precision.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">double</span> <span style="color: #0000ff;">qmckl_get_numprec_epsilon</span>(<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context</span> <span style="color: #a0522d;">context</span>) {
|
||
<span style="color: #a020f0;">if</span> (qmckl_context_check(context) == QMCKL_NULL_CONTEXT)
|
||
<span style="color: #a020f0;">return</span> QMCKL_INVALID_CONTEXT;
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">qmckl_context_struct</span>* <span style="color: #a020f0;">const</span> <span style="color: #a0522d;">ctx</span> = (<span style="color: #228b22;">qmckl_context_struct</span>*) context;
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">int</span> <span style="color: #a0522d;">precision</span> = ctx->numprec.precision;
|
||
<span style="color: #a020f0;">return</span> 1. / (<span style="color: #228b22;">double</span>) ( ((<span style="color: #228b22;">uint64_t</span>) 1) << (precision-2));
|
||
}
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="outline-container-org1b092d5" class="outline-3">
|
||
<h3 id="org1b092d5"><span class="section-number-3">4.2</span> Testing the number of unchanged bits</h3>
|
||
<div class="outline-text-3" id="text-4-2">
|
||
<p>
|
||
To test that a given approximation keeps a given number of bits
|
||
unchanged, we need a function that returns the number of unchanged
|
||
bits in the range, and in the precision.
|
||
</p>
|
||
|
||
<p>
|
||
For this, we first count by how many units in the last place (ulps) two
|
||
numbers differ.
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">int64_t</span> <span style="color: #0000ff;">countUlpDifference_64</span>(<span style="color: #228b22;">double</span> <span style="color: #a0522d;">a</span>, <span style="color: #228b22;">double</span> <span style="color: #a0522d;">b</span>) {
|
||
|
||
<span style="color: #a020f0;">union</span> <span style="color: #228b22;">int_or_float</span> {
|
||
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">i</span>;
|
||
<span style="color: #228b22;">double</span> <span style="color: #a0522d;">f</span>;
|
||
} <span style="color: #a0522d;">x</span>, <span style="color: #a0522d;">y</span>;
|
||
|
||
x.f = a;
|
||
y.f = b;
|
||
|
||
// <span style="color: #b22222;">Handle sign bit discontinuity: if the signs are different and either value is not zero</span>
|
||
<span style="color: #a020f0;">if</span> ((x.i < 0) != (y.i < 0) && (x.f != 0.0) && (y.f != 0.0)) {
|
||
// <span style="color: #b22222;">Use the absolute values and add the distance to zero for both numbers</span>
|
||
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">distanceToZeroForX</span> = x.i < 0 ? INT64_MAX + x.i : INT64_MAX - x.i;
|
||
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">distanceToZeroForY</span> = y.i < 0 ? INT64_MAX + y.i : INT64_MAX - y.i;
|
||
<span style="color: #a020f0;">return</span> distanceToZeroForX + distanceToZeroForY;
|
||
}
|
||
|
||
// <span style="color: #b22222;">Calculate the difference in their binary representations</span>
|
||
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">result</span> = x.i - y.i;
|
||
result = result > 0 ? result : -result;
|
||
<span style="color: #a020f0;">return</span> result;
|
||
}
|
||
</pre>
|
||
</div>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">int32_t</span> <span style="color: #0000ff;">qmckl_test_precision_64</span>(<span style="color: #228b22;">double</span> <span style="color: #a0522d;">a</span>, <span style="color: #228b22;">double</span> <span style="color: #a0522d;">b</span>) {
|
||
|
||
<span style="color: #228b22;">int64_t</span> <span style="color: #a0522d;">diff</span> = countUlpDifference_64(a,b);
|
||
|
||
<span style="color: #a020f0;">if</span> (diff == 0) <span style="color: #a020f0;">return</span> 53;
|
||
|
||
<span style="color: #228b22;">int32_t</span> <span style="color: #a0522d;">result</span> = 53;
|
||
|
||
<span style="color: #a020f0;">for</span> (<span style="color: #228b22;">int</span> <span style="color: #a0522d;">i</span>=0 ; i<53 && diff != 0 ; ++i) {
|
||
diff >>= 1;
|
||
result--;
|
||
}
|
||
|
||
<span style="color: #a020f0;">return</span> result;
|
||
}
|
||
</pre>
|
||
</div>
|
||
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">int32_t</span> <span style="color: #0000ff;">qmckl_test_precision_32</span>(<span style="color: #228b22;">float</span> <span style="color: #a0522d;">a</span>, <span style="color: #228b22;">float</span> <span style="color: #a0522d;">b</span>) {
|
||
<span style="color: #a020f0;">return</span> qmckl_test_precision_64( (<span style="color: #228b22;">double</span>) a, (<span style="color: #228b22;">double</span>) b );
|
||
}
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div id="outline-container-orgcf44399" class="outline-2">
|
||
<h2 id="orgcf44399"><span class="section-number-2">5</span> Approximate functions</h2>
|
||
<div class="outline-text-2" id="text-5">
|
||
</div>
|
||
<div id="outline-container-orgbd1af10" class="outline-3">
|
||
<h3 id="orgbd1af10"><span class="section-number-3">5.1</span> Exponential</h3>
|
||
<div class="outline-text-3" id="text-5-1">
|
||
<p>
|
||
Fast exponential function, adapted from Johan Rade's implementation
|
||
(<a href="https://gist.github.com/jrade/293a73f89dfef51da6522428c857802d">https://gist.github.com/jrade/293a73f89dfef51da6522428c857802d</a>). It
|
||
is based on Schraudolph's paper:
|
||
</p>
|
||
|
||
<p>
|
||
N. Schraudolph, "A Fast, Compact Approximation of the Exponential Function",
|
||
<i>Neural Computation</i> <b>11</b>, 853–862 (1999).
|
||
(available at <a href="https://nic.schraudolph.org/pubs/Schraudolph99.pdf">https://nic.schraudolph.org/pubs/Schraudolph99.pdf</a>)
|
||
</p>
|
||
|
||
<div class="org-src-container">
|
||
<pre class="src src-c"><span style="color: #228b22;">float</span> <span style="color: #0000ff;">fastExpf</span>(<span style="color: #228b22;">float</span> <span style="color: #a0522d;">x</span>)
|
||
{
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">float</span> <span style="color: #a0522d;">a</span> = 12102203.0;
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">float</span> <span style="color: #a0522d;">b</span> = 1064986816.0;
|
||
x = a * x + b;
|
||
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">float</span> <span style="color: #a0522d;">c</span> = 8388608.0;
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">float</span> <span style="color: #a0522d;">d</span> = 2139095040.0;
|
||
<span style="color: #a020f0;">if</span> (x < c || x > d)
|
||
x = (x < c) ? 0.0f : d;
|
||
|
||
<span style="color: #228b22;">uint32_t</span> <span style="color: #a0522d;">n</span> = (<span style="color: #228b22;">uint32_t</span>) x;
|
||
memcpy(&x, &n, 4);
|
||
<span style="color: #a020f0;">return</span> x;
|
||
}
|
||
|
||
|
||
<span style="color: #228b22;">double</span> <span style="color: #0000ff;">fastExp</span>(<span style="color: #228b22;">double</span> <span style="color: #a0522d;">x</span>)
|
||
{
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">a</span> = 6497320848556798.0;
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">b</span> = 4606985713057410560.0;
|
||
x = a * x + b;
|
||
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">c</span> = 4503599627370496.0;
|
||
<span style="color: #a020f0;">const</span> <span style="color: #228b22;">double</span> <span style="color: #a0522d;">d</span> = 9218868437227405312.0;
|
||
<span style="color: #a020f0;">if</span> (x < c || x > d)
|
||
x = (x < c) ? 0.0 : d;
|
||
|
||
<span style="color: #228b22;">uint64_t</span> <span style="color: #a0522d;">n</span> = (<span style="color: #228b22;">uint64_t</span>) x;
|
||
memcpy(&x, &n, 8);
|
||
<span style="color: #a020f0;">return</span> x;
|
||
}
|
||
</pre>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="postamble" class="status">
|
||
<p class="author">Author: TREX CoE</p>
|
||
<p class="date">Created: 2024-07-11 Thu 08:55</p>
|
||
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
|
||
</div>
|
||
</body>
|
||
</html>
|