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< body >
< div id = "org-div-home-and-up" >
< a accesskey = "h" href = "" > UP < / a >
|
< a accesskey = "H" href = "index.html" > HOME < / a >
< / div > < div id = "content" >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orgdd3dd3f" > 1. Control of the numerical precision< / a > < / li >
< li > < a href = "#org4c64608" > 2. Precision< / a > < / li >
< li > < a href = "#org0d3b29e" > 3. Range< / a > < / li >
< li > < a href = "#org7188ecd" > 4. Helper functions< / a >
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< ul >
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< li > < a href = "#org5628f33" > 4.1. Epsilon< / a > < / li >
< li > < a href = "#org01529c4" > 4.2. Testing the number of unchanged bits< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org6442b59" > 5. Approximate functions< / a >
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< ul >
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< li > < a href = "#orgcf7f3a2" > 5.1. Exponential< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / div >
< / div >
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< p >
3+TITLE: Numerical precision
< / p >
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< div id = "outline-container-orgdd3dd3f" class = "outline-2" >
< h2 id = "orgdd3dd3f" > < 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" >
< 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),
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< i > precision< / i > refers to the number of significand bits (including the
sign bit) and < i > range< / i > refers to the number of exponent bits.
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< / p >
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< table id = "org1b4b9aa" border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
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< colgroup >
< col class = "org-left" / >
< col class = "org-right" / >
< / colgroup >
< tbody >
< tr >
< td class = "org-left" > < code > QMCKL_DEFAULT_PRECISION< / code > < / td >
< td class = "org-right" > 53< / td >
< / tr >
< tr >
< td class = "org-left" > < code > QMCKL_DEFAULT_RANGE< / code > < / td >
< td class = "org-right" > 11< / td >
< / tr >
< / tbody >
< / table >
< 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 >
< / div >
< 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
integer. The getter functions return the value, as a 32-bit
integer. The update functions return < code > QMCKL_SUCCESS< / code > or
< code > QMCKL_FAILURE< / code > .
< / p >
< / div >
< / div >
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< div id = "outline-container-org4c64608" class = "outline-2" >
< h2 id = "org4c64608" > < span class = "section-number-2" > 2< / span > Precision< / h2 >
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< 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;
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/* < span style = "color: #b22222;" > This should be always true because the context is valid< / span > */
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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 >
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< div id = "outline-container-org0d3b29e" class = "outline-2" >
< h2 id = "org0d3b29e" > < span class = "section-number-2" > 3< / span > Range< / h2 >
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< 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;
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/* < span style = "color: #b22222;" > This should be always true because the context is valid< / span > */
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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 >
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< div id = "outline-container-org7188ecd" class = "outline-2" >
< h2 id = "org7188ecd" > < span class = "section-number-2" > 4< / span > Helper functions< / h2 >
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< div class = "outline-text-2" id = "text-4" >
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< / div >
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< div id = "outline-container-org5628f33" class = "outline-3" >
< h3 id = "org5628f33" > < span class = "section-number-3" > 4.1< / span > Epsilon< / h3 >
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< div class = "outline-text-3" id = "text-4-1" >
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< 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 > ) {
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< 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;
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< span style = "color: #a020f0;" > return< / span > 1. / (< span style = "color: #228b22;" > double< / span > ) ( ((< span style = "color: #228b22;" > uint64_t< / span > ) 1) < < (precision-2));
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}
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org01529c4" class = "outline-3" >
< h3 id = "org01529c4" > < span class = "section-number-3" > 4.2< / span > Testing the number of unchanged bits< / h3 >
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< 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 >
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< div id = "outline-container-org6442b59" class = "outline-2" >
< h2 id = "org6442b59" > < span class = "section-number-2" > 5< / span > Approximate functions< / h2 >
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< div class = "outline-text-2" id = "text-5" >
< / div >
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< div id = "outline-container-orgcf7f3a2" class = "outline-3" >
< h3 id = "orgcf7f3a2" > < span class = "section-number-3" > 5.1< / span > Exponential< / h3 >
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< 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–
(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 >
2022-09-28 18:04:21 +02:00
< / div >
< div id = "postamble" class = "status" >
< p class = "author" > Author: TREX CoE< / p >
2024-07-11 13:54:19 +02:00
< p class = "date" > Created: 2024-07-11 Thu 11:53< / p >
2022-09-28 18:04:21 +02:00
< p class = "validation" > < a href = "http://validator.w3.org/check?uri=referer" > Validate< / a > < / p >
< / div >
< / body >
< / html >
