## Integer square root

In [1] is a rather mysterious looking constant expression formula for an integer square root. This is a function that returns the smallest integer for which the square is less than the value to take the root of. Check out the black magic he used

// Stroustrup 10.4:  constexpr capable integer square root function
constexpr int isqrt_helper( int sq, int d, int n )
{
return sq <= n ? isqrt_helper( sq + d, d + 2, n ) : d ;
}

constexpr int isqrt( int n )
{
return isqrt_helper( 1, 3, n )/2 - 1 ;
}


The point of this construction was really to illustrate that it allows complex expressions to be used as compile time constants. I wonder at what point various compilers will give up trying to evaluate such expressions?

## Let’s take this apart a bit.

Consider the first few values of $$n > 0$$.

• $$n = 0$$. Here we have a call to $$\textrm{isqrt_helper}( 1, 3, 0 )$$ so the $$1 \le 0$$ predicate is false, and the return value is just $$3$$.

For that value we have (using integer arithmetic):

\label{eqn:isqrt:20}
\frac{3}{2} – 1 = 0,

as desired.

• $$n = 1$$. Here we have a call to $$\textrm{isqrt_helper}( 1, 3, 1 )$$ so the $$1 \le 1$$ predicate is true, resulting in a second call $$\textrm{isqrt_helper}( 4, 5, 1 )$$. For that call the $$4 \le 1$$ predicate is false, resulting in a return value of $$5$$.

This time we have a final result of

\label{eqn:isqrt:40}
\frac{5}{2} – 1 = 1,

as desired again. The result will be the same for any value $$n \in [1,3]$$.

• $$n = 4$$. We will end up with a call to $$\textrm{isqrt_helper}( 4, 5, 4 )$$ for which the $$4 \le 4$$ predicate is true, resulting in a followup call of $$\textrm{isqrt_helper}( 9, 7, 4 )$$. For that call the $$9 \le 4$$ predicate is false, resulting in a return value of $$7$$.

This time we have a final result of

\label{eqn:isqrt:45}
\frac{7}{2} – 1 = 2,

as expected. We get the same result for any value $$n \in [4,8]$$.

## Recurrence relations

The rough pattern of the magic involved can be seen. We have a sequence of calls

• $$\textrm{isqrt_helper}( 1, 3, n )$$,
• $$\textrm{isqrt_helper}( 4, 5, n )$$,
• $$\textrm{isqrt_helper}( 9, 7, n )$$,
• $$\textrm{isqrt_helper}( 16, 9, n )$$,

which terminates at the point where the first (square) parameter exceeds that value that we are taking the root of. Let the parameters of the sequence of calls be $$s_k$$, and $$d_k$$, so that with $$s_0 = 1, d_0 = 3$$ the $$k \in [0,…]$$ call to the helper function is $$q_k = \textrm{isqrt_helper}( s_k, d_k, n )$$.

The sequence for the second parameter, the eventual return value, can be summarized compactly as $$d_k = 3 + 2 k$$. It is not entirely obvious how we end up with a square for the values $$s_k = s_{k-1} + d_{k-1}$$, but this follows by summation. For $$k > 1$$ that is

\label{eqn:isqrt:60}
\begin{aligned}
s_k
&= s_{k-1} + d_{k-1} \\
&= s_0 + d_0 + d_1 + d_{k-1} \\
&= s_0 + \sum_{m=0}^{k-1} d_m \\
&= s_0 + \sum_{m=0}^{k-1} (3 + 2 m ) \\
&= s_0 + \sum_{m=1}^{k} (3 + 2 (m-1) ) \\
&= s_0 + \sum_{m=1}^{k} (1 + 2 m ) \\
&= 1 + k + 2 \sum_{m=1}^{k} m \\
&= 1 + k + 2 \frac{k(k+1)}{2} \\
&= k^2 + 2 k + 1 \\
&= (k+1)^2.
\end{aligned}

This clearly holds for the boundary cases $$k = 0,1$$ as well. This allows the helper function action to be summarized more compactly

\label{eqn:isqrt:80}
\textrm{isqrt_helper}(1, 3, n) = 3 + 2 k,

where $$k$$ is the smallest integer such that $$(k+1)^2 > n$$. After integer scaling the final result is

\label{eqn:isqrt:100}
(3 + 2 k)/2 -1 = k.

This little beastie makes sense after deconstruction, but it was very Jackson like to toss this into the book without comment or explanation.

As pointed out by Pramod Gupta, there’s a spooky appearance of collaboration between Stroustrup and Jackson’s publishers, not entirely limited to the book covers.

# References

[1] Bjarne Stroustrup. The C++ Programming Language, 4th Edition. Addison-Wesley, 2014.

## Notes on C++11 and C++14 from scientific computing for physicists

I recently wrapped up all the programming assignments for PHY1610, Scientific Computing for Physicists

In all the assignments, we were required to compile with either

-std=c++11

or

-std=c++14

It’s possible to use those options and still program using the older C++98 syntax, but I also used this as an opportunity to learn some new style C++.

With the cavaet that we were provided with boilerplate code for a number of assignments, there was a non-trivial amount of code written for this course:

$cloc cat f 2>&1 | tee o 186 text files. 177 unique files. 4 files ignored. http://cloc.sourceforge.net v 1.60 T=0.88 s (197.6 files/s, 16868.5 lines/s) ------------------------------------------------------------------------------- Language files blank comment code ------------------------------------------------------------------------------- C++ 111 1710 1159 7317 C/C++ Header 62 819 1525 2237 ------------------------------------------------------------------------------- SUM: 173 2529 2684 9554 -------------------------------------------------------------------------------  A lot of this code involved calling into external libraries (fftw3, cblas, lapack, gsl, netcdf, MPI, silo, boost exceptions, boost unittest, …) and was pretty fun to write. Looking through my submissions, here are some of the newer language features that ended up in my code. Keep in mind that new for me is relative to the C++ language features that I was able to use in DB2 code, which is restricted by the features made available by the very oldest compiler we were using accross all platform offerings. ## Using statements I had only seen using statements for namespace selection, as in using namespace std ;  This is, however, a more general construct, and also allows for what is effectively a scope limited typedef with a more natural syntax. Example: using carray = rarray<std::complex<double>, 1> ;  Compare this to typedef rarray<std::complex<double>, 1> carray ;  With the using syntax, the beginner programmer’s issue of remembering the order for the type,typename pair in a typedef statement is obliterated. I got quite used to using using by the end of the course. ## Testing language levels The following macros were helpful when experimenting with different language levels: #if defined __cplusplus && (__cplusplus >= 201103L) #define HAVE_CPLUSPLUS_11 #endif #if defined __cplusplus && (__cplusplus >= 201402L) #define HAVE_CPLUSPLUS_14 #endif  ## enum class C++11 introduces an ‘enum class’, different from an enum. For example, instead of writing: /** interval and derivative solver methods supplied by gsl */ enum solver { bisection, falsepos, brent, newton, secant, steffenson } ;  you would write: /** interval and derivative solver methods supplied by gsl */ enum class solver { bisection, falsepos, brent, newton, secant, steffenson } ;  The benefit of this compared to the non-class enum is that the enumeration names are not in the global scope. You would write void foo( const solver s ) { if ( s == solver::falsepos ) }  not void foo( const solver s ) { if ( s == falsepos ) }  This nicely avoids namespace clashes. That is not the only benefit to C++11 enums. C++11 enums can also be forward referenced, provided the storage class of the enum is also specified. If you have ever worked on code that is massively coupled and interdependent (such as DB2), you have seen places where piles of headers have to get dragged in for enum bodies, because it is not possible to forward reference an enum portably. This is a very nice feature! A simple example of a forward declared C++11 enum is: enum solver : int ; void foo( const solver s ) ; enum solver : int { x = 0, y = 1 } ;  Or, using the non-global enum class syntax: enum class what : int ; void foo( const what s ) ; enum class what : int { x = 0, y = 1 } ;  I didn’t actually use enum classes for enum forward referencing in my phy1610 assignments, because they were too simple to require that. There is huge potential for using enums with storage classes in DB2 code. I expect that is also true for many other huge scale C++ codebases. The fact that this feature does not have appear to be tied to a requirement to also use ‘enum class’ is very nice for transforming legacy code. I left IBM before the day of seeing the use of compilers that allowed that on all platforms, but can imagine there will be some huge potential build time savings once C++11 compilers are uniformly available for DB2 code (and the code is ported to compile with C++11 enabled on all platforms). As a side note, the storage class qualification, even if not being used for forward referencing is quite nice. I used it for return codes from main, which have to fit within one byte (i.e. within the waitpid waitstatus byte). For example: enum class RETURNCODES : unsigned char { SUCCESS ///< exit code for successful exectution ,HELP ///< exit code when -help (or bad option is supplied) ,PARSE_ERROR ///< exit code if there's a parse error */ ,EXCEPTION ///< exit code if there's an unexpected exception thrown */ } ;  ## Uniform initialization A new initialization paradigm is available in C++11. Instead of using constructor syntax for initialization, as in /** Input parameters for gsl solver iteration. */ struct iterationParameters { const Uint m_max_iter ; ///< Maximum number of iterations before giving up. const double m_abserr ; ///< the absolute error criteria for convergence. const double m_relerr ; ///< the relative error criteria for convergence. const bool m_verbose ; ///< verbose output iterationParameters( const Uint max_iter, const double abserr, const double relerr, const bool verbose ) : m_max_iter(max_iter), m_abserr(abserr), m_relerr(relerr), m_verbose(verbose) { } } ;  one could write /** Input parameters for gsl solver iteration. */ struct iterationParameters { const Uint m_max_iter ; ///< Maximum number of iterations before giving up. const double m_abserr ; ///< the absolute error criteria for convergence. const double m_relerr ; ///< the relative error criteria for convergence. const bool m_verbose ; ///< verbose output iterationParameters( const Uint max_iter, const double abserr, const double relerr, const bool verbose ) : m_max_iter{max_iter}, m_abserr{abserr}, m_relerr{relerr}, m_verbose{verbose} { } } ;  This is a little foreign looking and it is easy to wonder what the advantage is. One of the advantages is that this syntax can be used for container initialization. For example, instead of std::vector<int> v ; v.push_back( 1 ) ; v.push_back( 2 ) ; v.push_back( 3 ) ;  you can just do std::vector<int> v{ 1, 2, 3 } ;  This is called uniform initialization, since this mechanism was extended to basic types as well. For example, instead of initializing an array with an assignment operator, as in  constexpr struct option long_options[] = { { "help", 0, NULL, 'h' }, { "number", 1, NULL, 'n' }, { "lower", 1, NULL, 'l' }, { "upper", 1, NULL, 'u' }, { NULL, 0, NULL, 0 } } ;  you can write  constexpr struct option long_options[]{ { "help", 0, NULL, 'h' }, { "number", 1, NULL, 'n' }, { "lower", 1, NULL, 'l' }, { "upper", 1, NULL, 'u' }, { NULL, 0, NULL, 0 } } ;  Instead of just providing a special mechanism to initialize container class objects, the language was extended to provide a new initialization syntax that could be used to initialize contain those objects and all others. However, this is not just a different syntax for initialization, because there the types have to match strictly. For example this init of a couple stack variables will not compile  int more{3} ; float x1{-2.0} ; size_t size{meta.numThreads*20} ;  What is required is one of  float x1{-2.0f} ; // or double x1{-2.0} ;  Additionally, suppose that meta.numThreads has int type. Such a uniform initialization attempt will not compile, since the product is not of type size_t. That line can be written as:  size_t size{(size_t)meta.numThreads*20} ; // or: size_t size = meta.numThreads*20 ;  I found uniform initialization hard on the eyes because it looked so foreign, but did eventually get used to it, with one exception. It seems to me that a longer initialization expression like the following is harder to read double x{ midpoint( x1, x1 + intervalWidth ) } ;  than double x = midpoint( x1, x1 + intervalWidth ) ;  There were also cases with -std=c++11 where uniform init and auto variables (see below) did not interact well, producing errors later when my auto-uniform-init’ed variables got interpreted as initializer lists instead of the types I desired. All such errors seemed to go away with -std=c++14, which seemed to generally provide a more stable language environment. ## New string to integer functions The c++11 standard library has new string to integer functions http://en.cppreference.com/w/cpp/string/basic_string/stoul which are more convenient than the strtoul functions. These throw exceptions on error, but still allow the collection of errno and error position if you want them. using Uint = std::uintptr_t ; /** Register sized signed integer type for loop counters and so forth. */ using Sint = std::intptr_t ; /** wrapper for stoul to match the type of Uint above. */ #if defined _WIN64 #define strToUint std::stoull #else #define strToUint std::stoul #endif  There are other similar functions like std::stod, for string to double conversion. There were also opposite convertors, such as to_string, for converting integer types to strings. For example: const std::string filename{ fileBaseName + "_" + std::to_string( rank ) + ".out" } ;  ## Static assertions. DB2 had a static assertion implementation (OSS_CTASSERT, or sqlzStaticAssert?) but there is now one in the standard. Here’s an example using the Uint “typedef” above: /** Force a compilation error if size assumptions are invalid. */ inline void strToUintAssumptions() { #if defined _WIN64 static_assert( sizeof(Uint) == sizeof(unsigned long long), "bad assumptions about sizeof uintptr_t, long long" ) ; #else static_assert( sizeof(Uint) == sizeof(unsigned long), "bad assumptions about sizeof uintptr_t, long" ) ; #endif }  The advantage of static_assert over a typedef (variable sized array) implementation like DB2 HAD is that compilers likely produce a better error message when it fails (instead of something unintuitive like “reference of array location at offset -1 is invalid”). ## Boost exceptions. While not part of c++11, the boost exception classes were available for my assignments. These are pretty easy to use. As setup you define some helper classes, which really just provide a name for the exception, and a name to identify any of the data that you’d like to throw along with the underlying exception. This could look like the following for example: #include <boost/exception/exception.hpp> #include <boost/exception/info.hpp> struct error : virtual std::exception, virtual boost::exception { } ; struct regex_match_error : virtual error { } ; struct tag_match_input ; typedef boost::error_info<tag_match_input,std::string> match_info ; struct tag_match_re ; typedef boost::error_info<tag_match_re,std::string> re_info ; struct tag_intdata ; typedef boost::error_info<tag_intdata,long> intdata_info ;  Such classes would be best in a namespace since they are generic, but I didn’t bother for all these assignments. I used the boost exceptions for a couple things. One of which, of course, was throwing exceptions, but the other was as an assert-with-data backend: #define ASSERT_NO_ERROR (static_cast<void>(0)) #ifdef NDEBUG #define ASSERT_DATA_INT( expr, v1 ) ASSERT_NO_ERROR #define ASSERT_DATA_INT_INT( expr, v1, v2 ) ASSERT_NO_ERROR #else #define ASSERT_DATA_INT( expr, v1 ) \ ( (expr) \ ? ASSERT_NO_ERROR \ : BOOST_THROW_EXCEPTION( \ assert_error() \ << intdata_info( v1 ) ) ) //... #endif  This allowed me to assert with data as in ASSERT_DATA_INT( sz > 0, sz ) ; ASSERT_DATA_INT_INT( taskNumber < numTasks, taskNumber, numTasks ) ;  This way I get not just the abort from the assert, but also the underlying reason, and can dump those to the console with no additional effort than catching any other boost exception: //... #include <boost/exception/diagnostic_information.hpp> int main( int argc, char ** argv ) { try { auto expected{7} ; ASSERT_DATA_INT_INT( argc == expected, argc, expected ) ; } catch ( boost::exception & e ) { auto s { boost::diagnostic_information( e ) } ; std::cout << s << std::endl ; // ...  This generates something like: $ ./bassert
bassert.cc(11): Throw in function int main(int, char**)
Dynamic exception type: boost::exception_detail::clone_impl<assert_error>
std::exception::what: std::exception
[tag_intdata*] = 1
[tag_intdata2*] = 7



I wonder how efficient constructing such an exception object is? When pre-processed the assertion above expands to

      ( (argc == expected) ? (static_cast<void>(0)) :
::boost::exception_detail::throw_exception_(
assert_error() << intdata_info( argc ) << intdata2_info( expected )
,__PRETTY_FUNCTION__,"bassert.cc",11)
) ;


Stepping through this in the debugger I see some interesting stuff, but it included heap (i.e. new) allocations. This means that this sort of Boost exception may malfunction very badly in out of memory conditions where it is conceivable that one would want to throw an exception.

The runtime cost can’t be that inexpensive either (when the assert is triggered). I see four function calls even before the throw is processed:

assert_error const& boost::exception_detail::set_info(assert_error const&, boost::error_info const&)-0x4
assert_error const& boost::exception_detail::set_info(assert_error const&, boost::error_info const&)-0x4
assert_error::assert_error(assert_error const&)-0x4
void boost::throw_exception(assert_error const&)-0x4


and the total instruction count goes up to ~140 from 4 for the NDEBUG case (with optimization). Only 5 instructions get executed in the happy codepath. This is what we want in exception handling code: very cheap when it’s not triggered, with all the expense moved to the unhappy codepath.

The negative side effect of this sort of error handling looks like a lot of instruction cache bloat.

## Boost test

The boost test library is also not a C++11 feature, but new for me, and learned in this course. Here’s a fragment of how it is used

#define BOOST_TEST_MAIN
#define BOOST_TEST_MODULE test

#include <boost/test/unit_test.hpp>
#include <vector>

BOOST_AUTO_TEST_CASE( testExample )
{
std::vector<int> v(3) ;

BOOST_REQUIRE( 3 == v.size() ) ;
BOOST_REQUIRE_MESSAGE( 3 == v.size(), "size: " + std::to_string( v.size() ) ) ;
}


A boost test after being run looks like:

$./test --report_level=detailed --log_level=all Running 1 test case... Entering test module "test" test.cc:9: Entering test case "testExample" test.cc:13: info: check 3 == v.size() has passed test.cc:14: info: check 'size: 3' has passed test.cc:9: Leaving test case "testExample"; testing time: 87us Leaving test module "test"; testing time: 103us Test module "test" has passed with: 1 test case out of 1 passed 2 assertions out of 2 passed Test case "testExample" has passed with: 2 assertions out of 2 passed  ## Range for and auto type The range for is much like perl’s foreach. For example, in perl you could write my @a = ( 1, 2, 3 ) ; foreach my$v ( @a )
{
foo( \$v ) ;
}


An equivalent C++ loop like this can be as simple as

std::vector<int> a{1, 2, 3 } ;
for ( auto v : a )
{
foo( v ) ;
}


You can also declare the list of items to iterate over inline, as in

using iocfg = iohandler::cfg ;
for ( auto c : { iocfg::graphics, iocfg::ascii, iocfg::netcdf, iocfg::noop } )
{
// ...
}


Observe that, just like perl, C++ no longer requires any explicit type for the loop variable, as it is deduced when auto is specified. It is still strongly typed, but you can write code that doesn’t explicitly depend on that type. I see lots of benefits to this, as you can have additional freedom to change type definitions and not have to adjust everything that uses it.

I can imagine that it could potentially get confusing if all variables in a function get declared auto, but did not find that to be the case for any of the code I produced in these assignments.

One gotcha with auto that I did hit was that care is required in computed expressions. I’d used auto in one case and the result got stored as a large unsigned value, instead of signed as desired (i.e. negative values got stored in unsigned auto variables). In that case I used an explicit type. Extensive use of auto may end up requiring more unit and other test if the types picked are not those that are desired.

## std::chrono (ticks.h)

This is a nice portability layer for fine grain time measurements, allowing you to avoid platform specific functions like gettimeofday, and also avoid any composition of the seconds/subseconds data that many such interfaces provide.

Here’s a fragment of a class that allows interval time measurements and subsequent conversion:

class ticks
{
using clock      = std::chrono::high_resolution_clock ;

clock::time_point m_sample ;
public:

static inline ticks sample()
{
ticks t ;
t.m_sample = clock::now() ;

return t ;
}

using duration   = decltype( m_sample - m_sample ) ;

friend duration operator -( const ticks & a, const ticks & b ) ;
} ;

inline ticks::duration operator -( const ticks & a, const ticks & b )
{
return a.m_sample - b.m_sample ;
}

inline auto durationToMicroseconds( const ticks::duration & diff )
{
return std::chrono::duration_cast<std::chrono::microseconds>( diff ).count() ;
}


Note that the last function is using c++14 return type deduction. That does not work without coersion
in c++11, requiring:

inline auto durationToMicroseconds( const ticks::duration & diff )
-> decltype(std::chrono::duration_cast<std::chrono::microseconds>( diff ).count())
{
return std::chrono::duration_cast<std::chrono::microseconds>( diff ).count() ;
}


which is very ugly.

## Random numbers

/**
A random number generator that produces integer uniformly
distributed in the interval:

[a, a + delta N]

with separation delta between values returned.
*/
template <int a, int delta, int N>
class RandomIntegers
{
std::random_device                        m_rd ;A
//std::default_random_engine                m_engine ;
std::mt19937                              m_engine ;
std::uniform_int_distribution<unsigned>   m_uniform ;

public:
/** constuct a uniform random number generator for the specified range */
RandomIntegers( )
: m_rd()
, m_engine( m_rd() )
, m_uniform( 0, N )
{
static_assert( N > 0, "Integer N > 0 expected" ) ;
static_assert( delta > 0, "Integer delta > 0 expected" ) ;
}

/**
return a uniform random number sample from {a, a + delta, ..., a + delta N}
*/
int sample()
{
auto p = m_uniform( m_engine ) ;

return a + p * delta ;
}
} ;


## constexpr

Instead of using #defines, one can use completely typed declarations, but still constant using the constexpr keyword. An example

constexpr size_t N{3} ;
std::tuple<int, N> t ;


## nullptr

The days of not knowing what header defines NULL and dealing with conflicting definitions are over. Instead of using NULL, we now have a builtin language construct nullptr available.

## Lambdas and sort

Custom sorting is really simple in c++ now. Here’s an example of a partial sort (sorting the top N elements, and leaving the rest unspecified). The sort function no longer has to be a function call, and can be specified inline

auto second_greater = [](auto & left, auto & right) { return left.second > right.second ; } ;
std::partial_sort( cvec.begin(),
cvec.begin() + N,
cvec.end(),
second_greater ) ;


The “inline” sort function here is using c++14 lambda syntax. For c++11, the parameter types can’t be auto, so something such as the following might be required

auto second_greater = [](const results_pair & left, const results_pair & right) { return left.second > right.second ; } ;


## Useful standard helper methods

The standard library has lots of useful utility functions. I’m sure I only scratched the surface discovering some of those. Some I used were:

std::swap( m_sz, other.m_sz ) ;
std::fill( m_storage.begin(), m_storage.end(), v ) ;
std::copy( b.m_storage.begin(), b.m_storage.end(), m_storage.begin() ) ;
r.first  = std::max( l, m_myFirstGlobalElementIndex ) ;
r.second = std::min( u, m_myLastGlobalElementIndex ) ;


I also liked the copysign function, allowing easy access to the sign bit of a float or double without messing around with extracting the bit, or explicit predicates:

inline double signof( const double v )
{
return std::copysign( 1.0, v ) ;
}


Mean and standard deviation were also really easy to calculate. Here’s an example that used a lambda function to calculate the difference from the mean to get at the squared difference from the mean:


m_sum = std::accumulate( v.begin(), v.end(), 0.0 ) ;
m_mean = m_sum / v.size() ;
double mean = m_mean ; // for lambda capture

std::vector<double> diff( v.size() ) ;

std::transform( v.begin(), v.end(), diff.begin(), [mean](double x) { return x - mean; } ) ;

m_sq_sum = std::inner_product( diff.begin(), diff.end(), diff.begin(), 0.0 ) ;


## decltype

Attempting to mix auto with g++’s ‘-Wall -Werror’ causes some trouble. For example, this doesn’t work

void foo ( const size_t size )
{
for ( auto i{0} ; i < size ; i++ )
{
// ...
}
}


This doesn’t compile since the i < size portion generates sign vs unsigned comparison warnings. There are a few ways to fix this.

   // specify the type explicitly:
for ( size_t i{0} ; i < size ; i++ )

// let the compiler use the type of the size variable:
for ( decltype(size) i{0} ; i < size ; i++ )


The decltype method is probably of more use in template code. For non-template code, I found that explicitly specifying the type was more readable.

## std::valarray (myrarray.h)

The standard library has a vectored array construct, but I was disappointed with the quality of the generated code that I observed. It also turned out to be faster not to use it. For example:

void SineCosineVecOps( std::valarray<float> & s, std::valarray<float> & c, const std::valarray<float> & v )
{
s = std::sin( v ) ;
c = std::cos( v ) ;
}

void SineCosineManOps( std::valarray<float> & s, std::valarray<float> & c, const std::valarray<float> & v )
{
for ( Uint i{0} ; i < ASIZE ; i++ )
{
float theta = v[i] ;

s[i] = std::sin( theta ) ;
c[i] = std::cos( theta ) ;
}
}


when run on a 300 element array executed close to 1.5x slower using the valarray vector assignment operation, and had close to 3x times the instructions (with optimization)!

Perhaps other compilers do better with valarray. g++ 5.3 is certainly not worth using with that container type.