Is strict template evaluation in principle impossible in C++? - c++

I think I understand how templates are evaluated lazily in C++ e.g. a la recursive replacements and a final simplification of the expansion. This typically limits the recursion depth available. I wonder if with the features new in C++11 (e.g. variadic templates or template packs) or with some Boost it is possible to force strict template evaluation. Or is this in principle impossible in C++?
Consider for example a template which sums all integer values 0..n:
template <int n>
struct sumAll { enum { value = n + sumAll<n-1>::value }; };
template <>
struct sumAll<0> { enum { value = 0 }; };
#include <iostream>
int main() { std::cout << sumAll<10000>::value << std::endl; }
Here sumAll<10>::value would be expanded to
sumAll<10>::value = 10 + sumAll<9>::value
= 10 + 9 + sumAll<8>::value
= 10 + 9 + 8 + sumAll<7>::value
= ...
= 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0
and the final summation would only be performed once the template has been completely expanded. If that final expansion gets too long (e.g. in complex series expansions with many terms) the compiler will ultimately run out of space to store additional terms.
My question was in essence if there was a way to perform simplifications (like above summation) earlier.

You decide the recursion depth yourself. And just like normal recursion can cause stack overflows, template recursion can. But that's often fixable by a better recursive algorithm. Trivially:
template <int n>
struct sumAll { enum { value = n + n-1 + sumAll<n-2>::value }; };
template <>
struct sumAll<1> { enum { value = 0 }; };
template <>
struct sumAll<0> { enum { value = 0 }; };
Smarter:
template <int n>
struct sumAll { enum { value = (n*n+2)/2; };
Of course, you may complain that the latter is just being silly and real examples are more complex. But isn't that the whole problem? The compiler can't magically make that complexity go away for you.

C++ templates are turing-complete, which means that you use them to evaluate every computable function at compile time. It then follows from the halting theorem that
You cannot, in general, compute the amount of memory require to compile of a C++ program in advance. (I.e., there is no computable function which maps every C++ program to a memory bound for its compilation)
You cannot, in general, decide whether the compiler will ever finish instantiating template, or will go on forever.
So while you might be able to tweak a compiler to use less memory in some cases, you cannot solve the general problem of it running out of memory sometimes.

Related

How to enumerate a constant array at compile time in C++?

I am trying to generate a hash at COMPILE TIME from a literal string (array of characters). For example:
unsigned long long compiledHash = ComputeHash("literal string");
I am currently stuck on finding a way to enumerate all characters in the string and creating a unique hash. If I use a for loop like I would normally, the compiler won't generate the hash at compile time, which is not what I want.
I might of found a way to do so, but the compiler is stuck in an infinite loop when calculating the hash.
template <size_t _length, typename T, int n> struct CostructHash {
unsigned long long Value;
constexpr __forceinline CostructHash(const T(&str)[_length]) :
Value(str[n] ^ n + (n > 0 ? CostructHash<_length, T, n - 1>(str).Value : 0)) {}
};
template<size_t _length>
constexpr __forceinline unsigned long long ComputeHash(const char(&str)[_length]) {
return CostructHash<_length, char, _length - 1>(str).Value;
}
As you can see I use recursion to go through all characters in the string, but I must of messed up somewhere, because as I said the compiler freezes forever when it calls ComputeHash.
I understand that I must be missing the base case that stops the recursion, but as far as I understand (n > 0 ? CostructHash<_length, T, n - 1>(str).Value : 0) should do the job since I am always decreasing n by 1 and checking if n is bigger than 0. So why is the recursion not stopping?
Also, there may be an easier way to do what I am trying?
Do you see the problem in the code
The recursion is infinite because there is no base case for the template instantiations.
but as far as I understand (n > 0 ? CostructHash<_length, T, n - 1>(str).Value : 0) should do the job since I am always decreasing n by 1 and checking if n is bigger than 0. So why is the recursion not stopping?
The template is instantiated before the compiler decides whether that branch will be taken. You have to use if constexpr instead of the ternary conditional, or you have to specialise the template for the base case.
Also, there may be an easier way to do what I am trying?
This seems to work fine:
constexpr std::size_t
ComputeHash(std::string_view str) {
std::size_t result = 0;
std::size_t i = 0;
for(auto c : str) {
result += c ^ i++;
}
return result;
}

Compile-Time Creation of Array of Templated Objects in High Level Synthesis

I'm trying to accomplish this with HLS, not with "normal" C++, so most libraries (STL, boost, etc.) won't work as they can't be synthesized (manual memory management is not allowed). I think this should be possible with template metaprogramming, but I'm a little stuck.
I want to create an array of shift registers, each with a variable depth. I have N inputs, and I want to create N shift registers, with depths 1 to N, where N is known at compile time. My shift register class basically looks like
template<int DEPTH>
class shift_register{
int registers[DEPTH];
...
};
I tried following this and adapting it: Programmatically create static arrays at compile time in C++ , however, the issue is with the last line. Each templated shift register is going to be a different type, and so can't be put together in an array. But I do need an array, as there wouldn't be a way to access each shift register.
Any help would be appreciated!
Just to clarify, my problem was the following: generate N shift_registers, templated from 1 to N, where N is a compile time constant.
For example, if I had N=4, I could easily write this as:
shift_register<1> sr1;
shift_register<2> sr2;
shift_register<3> sr3;
shift_register<4> sr4;
But this wouldn't be easy to change, if I wanted a different value for N in the future.
I ended up using the preprocessor and took the solution from here: How do I write a recursive for-loop "repeat" macro to generate C code with the CPP preprocessor?
I used the macros from that solution like this:
#define CAT(a, ...) PRIMITIVE_CAT(a, __VA_ARGS__)
#define PRIMITIVE_CAT(a, ...) a ## __VA_ARGS__
#define BODY(i) shift_register<i> CAT(sr,i)
REPEAT_ADD_ONE(BODY, N, 1);
And then something similar to that in order to access the shift registers, in a sort of array fashion.
This let me achieve the compile time generation that I was looking for, and get the array type access I needed.
Your question was somewhat difficult to understand but I'll do my best...
template <typename ... Args>
constexpr auto make_array(Args && ... pArgs)
{
using type = std::common_type_t<std::decay_t<Args>...>;
return std::array<type, sizeof...(Args)>{ (type)pArgs ... };
}
Then use it like this:
auto constexpr var_array_of_arrays = std::make_tuple
(
make_array(1, 2, 3, 3),
make_array(2, 3, 4),
make_array(1, 2, 3 ,4 ,3, 5)
);
To get the M'th element you access it like this, n has to actually be a compile-time constant:
std::get<M>(var_array_of_arrays);
To access the Nth element in the Mth array:
auto constexpr value = std::get<M>(var_array_of_arrays)[N]
An to improve the interface:
template <size_t M, size_t N, typename T >
constexpr decltype(auto) get_element(T && pInput)
{
return std::get<M>(std::forward<T>(pInput))[N];
}
Used like this:
auto constexpr element0_1 = get_element<0, 1>(var_array_of_arrays);
This will allow you to use an array of variable length arrays, or atleast something that behaves like that and is identical to that in memory.
A full example is here:
Online compiler
Whenever I hear "compile time number sequence" I think std::index_sequence
namespace detail {
template <typename>
struct shift_registers;
template <std::size_t ... Is> // 0, 1, ... N-1
struct shift_registers<std::index_sequence<Is...> > {
using type = std::tuple<shift_register<Is + 1>...>;
};
template <typename T>
using shift_registers_t = typename shift_registers<T>::type
}
template <std::size_t N>
using shift_registers = detail::shift_registers_t<std::make_index_sequence<N>>;

template metaprogramming Ternary value not hitting base case

I am writing a simple test program using TMP to calculate the Nth fibonacci number. I have already found many ways to do this, but I'm just trying out a bunch of ways to get my understanding better. The way I am having a problem with is this:
template<int A>
struct fib
{
static const bool value = (A<2);
static const int num = (value?A:(fib<A-1>::num + fib<A-2>::num));
};
The error message I am getting is:
error: template instantiation depth exceeds maximum of 900 (use -ftemplate-depth= to increase the maximum) instantiating 'fib<-1796>::value'|
I have tried substituting many values into the "false" field of the ternary, just to play with it and see what it does. I still do not understand why this does not work. Can anyone help me and tell me why? Thanks.
EDIT: My guess is that the compiler might be evaluating the T/F fields of the ternary before checking to see if the value is true or false, but I'm not sure since that's not how an if statement is supposed to work at all, and these are supposed to roughly emulate if statements
I must admit that I'm not that experienced concerning template programming. But in OP's case, a simple solution would be template specialization.
Sample code:
#include <iostream>
template<int A>
struct fib
{
static const int num = fib<A-1>::num + fib<A-2>::num;
};
template<>
struct fib<1>
{
static const int num = 1;
};
template<>
struct fib<2>
{
static const int num = 1;
};
int main()
{
fib<10> fib10;
std::cout << "fib<10>: " << fib10.num << '\n';
return 0;
}
Output:
fib<10>: 55
Live Demo on coliru
One way to write this in a more straightforward manner is to use if constexpr. Unlike with regular if (and with the ternary operator), templates in the not taken branch are not instantiated.
template <int n>
struct fib {
constexpr static int eval() {
if constexpr (n < 2)
return n;
else
return fib<n-1>::eval() + fib<n-2>::eval();
}
};
Of course once you have if constexpr you don't really need templates to make a compile-time function of this type. A constexpr non-template function will do just fine. This is just an illustration of the technique.
The first comment on my original post is the correct answer. The author is (n.m.). Template type instantiations are evaluated in both fields of the ternary, while the object instantiation itself is not. To solve this, I will look into std::condiditional or std::enable_if

How does int*** in this template metaprogram work?

How is template metaprogramming working here (static const int value = 1 + StarCounter<\U>::value;) to print out 3 ?
#include <iostream>
template <typename T>
struct StarCounter
{
static const int value = 0;
};
template <typename U>
struct StarCounter<U*>
{
static const int value = 1 + StarCounter<U>::value;
};
int main()
{
std::cout << StarCounter<int***>::value << std::endl;//How is it printing 3?
return 0;
}
The first template creates a struct that will always return 0 when you call StarCounter<U>::value.
The second template specialises the first one for cases where a pointer is used. So when you call it with StarCounter<U*>::value, the second template is used, not the first and it will return StarCounter<U>::value + 1. Note that it removes the pointer at each recursion step.
So the call to StarCounter<int***>::value will expend to:
StarCounter<int***>::value // second template is used due to pointer
1 + StarCounter<int**>::value // second template is used due to pointer
1 + 1 + StarCounter<int*>::value // second template is used due to pointer
1 + 1 + 1 + StarCounter<int>::value // no pointer here, so first template is used
1 + 1 + 1 + 0
3
StarCounter<int>::value
equals 0, because it's matched with first instantiation of the template, where value is explicitly defined.
StarCounter<int*>::value = 1 + StarCounter<int>::value
equals 1, because StarCounter<int*> is matched with StarCounter<U*>. Yes, StarCounter<T> can also be considered as a match, but StarCounter<U*> is more specific and that's why this one is preferred.
Similarly,
StarCounter<int**>::value = 1 + StarCounter<int*>::value
equals 2 and
StarCounter<int***>::value = 1 + StarCounter<int**>::value
equals 3.
I find it helps to think of runtime equivalents when it comes to metaprogramming. In template metaprogramming, we use partial specialization, as in runtime programming, we use recursion. The primary template functions as the base case and the specializations function as the recursive cases.
Consider the following recursive version of determining the size of a container:
def size(x):
if empty(x):
return 0
else:
return 1 + size(tail(x))
This is the equivalent of the template code you present. The primary template, StarCounter<T>, is the base case. The empty case. It has size (value) zero. The specialization, StarCounter<U*>, is the recursive case. It has size (value) 1 plus the size of recursing with the tail (StarCounter<U>).
In C++17, we can even more explicitly make the metaprogramming version equivalent to the runtime recursive version (this is presented solely as an example and not as a way that this code should be written):
template <class T>
struct StarCounter {
static constexpr int calc_value() {
if constexpr (!std::is_pointer<T>::value) {
return 0;
}
else {
return 1 + StarCounter<std::remove_pointer_t<T>>::value;
}
}
static constexpr int value = calc_value();
};
There is a template StarCounter which in it's more general form has constant value equal to 0. So when you use this template for most of the types and ask for the value you will get 0.
This template has also a specialized version which accepts pointers.
It's implementation also has constant value which is equal to 1 (as we have a pointer which means we have at least one star) plus value of value of type which this pointer points to.
In case of three stars we have:
StarCounter<int***>::value = 1 + StarCounter<int**>::value (1) + StarCounter<int*>::value (1) + StarCounter<int>::value (0)

What is metaprogramming?

With reference to this question, could anybody please explain and post example code of metaprogramming? I googled the term up, but I found no examples to convince me that it can be of any practical use.
On the same note, is Qt's Meta Object System a form of metaprogramming?
jrh
Most of the examples so far have operated on values (computing digits of pi, the factorial of N or similar), and those are pretty much textbook examples, but they're not generally very useful. It's just hard to imagine a situation where you really need the compiler to comput the 17th digit of pi. Either you hardcode it yourself, or you compute it at runtime.
An example that might be more relevant to the real world could be this:
Let's say we have an array class where the size is a template parameter(so this would declare an array of 10 integers: array<int, 10>)
Now we might want to concatenate two arrays, and we can use a bit of metaprogramming to compute the resulting array size.
template <typename T, int lhs_size, int rhs_size>
array<T, lhs_size + rhs_size> concat(const array<T, lhs_size>& lhs, const array<T, rhs_size>& rhs){
array<T, lhs_size + rhs_size> result;
// copy values from lhs and rhs to result
return result;
}
A very simple example, but at least the types have some kind of real-world relevance. This function generates an array of the correct size, it does so at compile-time, and with full type safety. And it is computing something that we couldn't easily have done either by hardcoding the values (we might want to concatenate a lot of arrays with different sizes), or at runtime (because then we'd lose the type information)
More commonly, though, you tend to use metaprogramming for types, rather than values.
A good example might be found in the standard library. Each container type defines its own iterator type, but plain old pointers can also be used as iterators.
Technically an iterator is required to expose a number of typedef members, such as value_type, and pointers obviously don't do that. So we use a bit of metaprogramming to say "oh, but if the iterator type turns out to be a pointer, its value_type should use this definition instead."
There are two things to note about this. The first is that we're manipulating types, not values We're not saying "the factorial of N is so and so", but rather, "the value_type of a type T is defined as..."
The second thing is that it is used to facilitate generic programming. (Iterators wouldn't be a very generic concept if it didn't work for the simplest of all examples, a pointer into an array. So we use a bit of metaprogramming to fill in the details required for a pointer to be considered a valid iterator).
This is a fairly common use case for metaprogramming. Sure, you can use it for a wide range of other purposes (Expression templates are another commonly used example, intended to optimize expensive calculations, and Boost.Spirit is an example of going completely overboard and allowing you to define your own parser at compile-time), but probably the most common use is to smooth over these little bumps and corner cases that would otherwise require special handling and make generic programming impossible.
The concept comes entirely from the name Meta- means to abstract from the thing it is prefixed on.
In more 'conversational style' to do something with the thing rather than the thing itself.
In this regard metaprogramming is essentially writing code, which writes (or causes to be written) more code.
The C++ template system is meta programming since it doesn't simply do textual substitution (as the c preprocessor does) but has a (complex and inefficient) means of interacting with the code structure it parses to output code that is far more complex. In this regard the template preprocessing in C++ is Turing complete. This is not a requirement to say that something is metaprogramming but is almost certainly sufficient to be counted as such.
Code generation tools which are parametrizable may be considered metaprogramming if their template logic is sufficiently complex.
The closer a system gets to working with the abstract syntax tree that represents the language (as opposed to the textual form we represent it in) the more likely it is to be considered metaprogramming.
From looking at the QT MetaObjects code I would not (from a cursory inspection) call it meta programming in the sense usually reserved for things like the C++ template system or Lisp macros. It appears to simply be a form of code generation which injects some functionality into existing classes at the compile stage (it can be viewed as a precursor to the sort of Aspect Oriented Programming style currently in vogue or the prototype based object systems in languages like JavaScripts
As example of the sort of extreme lengths you can take this in C++ there is Boost MPL whose tutorial shows you how to get:
Dimensioned types (Units of Measure)
quantity<float,length> l( 1.0f );
quantity<float,mass> m( 2.0f );
m = l; // compile-time type error
Higher Order Metafunctions
twice(f, x) := f(f(x))
template <class F, class X>
struct twice
: apply1<F, typename apply1<F,X>::type>
{};
struct add_pointer_f
{
template <class T>
struct apply : boost::add_pointer<T> {};
};
Now we can use twice with add_pointer_f to build pointers-to-pointers:
BOOST_STATIC_ASSERT((
boost::is_same<
twice<add_pointer_f, int>::type
, int**
>::value
));
Although it's large (2000loc) I made a reflexive class system within c++ that is compiler independant and includes object marshalling and metadata but has no storage overhead or access time penalties. It's hardcore metaprogramming, and being used in a very big online game for mapping game objects for network transmission and database-mapping (ORM).
Anyways it takes a while to compile, about 5 minutes, but has the benefit of being as fast as hand tuned code for each object. So it saves lots of money by reducing significant CPU time on our servers (CPU usage is 5% of what it used to be).
Here's a common example:
template <int N>
struct fact {
enum { value = N * fact<N-1>::value };
};
template <>
struct fact<1> {
enum { value = 1 };
};
std::cout << "5! = " << fact<5>::value << std::endl;
You're basically using templates to calculate a factorial.
A more practical example I saw recently was an object model based on DB tables that used template classes to model foreign key relationships in the underlying tables.
Another example: in this case the metaprogramming tecnique is used to get an arbitrary-precision value of PI at compile-time using the Gauss-Legendre algorithm.
Why should I use something like that in real world? For example to avoid repeating computations, to obtain smaller executables, to tune up code for maximizing performance on a specific architecture, ...
Personally I love metaprogramming because I hate repeating stuff and because I can tune up constants exploiting architecture limits.
I hope you like that.
Just my 2 cents.
/**
* FILE : MetaPI.cpp
* COMPILE : g++ -Wall -Winline -pedantic -O1 MetaPI.cpp -o MetaPI
* CHECK : g++ -Wall -Winline -pedantic -O1 -S -c MetaPI.cpp [read file MetaPI.s]
* PURPOSE : simple example template metaprogramming to compute the
* value of PI using [1,2].
*
* TESTED ON:
* - Windows XP, x86 32-bit, G++ 4.3.3
*
* REFERENCES:
* [1]: http://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm
* [2]: http://www.geocities.com/hjsmithh/Pi/Gauss_L.html
* [3]: http://ubiety.uwaterloo.ca/~tveldhui/papers/Template-Metaprograms/meta-art.html
*
* NOTE: to make assembly code more human-readable, we'll avoid using
* C++ standard includes/libraries. Instead we'll use C's ones.
*/
#include <cmath>
#include <cstdio>
template <int maxIterations>
inline static double compute(double &a, double &b, double &t, double &p)
{
double y = a;
a = (a + b) / 2;
b = sqrt(b * y);
t = t - p * ((y - a) * (y - a));
p = 2 * p;
return compute<maxIterations - 1>(a, b, t, p);
}
// template specialization: used to stop the template instantiation
// recursion and to return the final value (pi) computed by Gauss-Legendre algorithm
template <>
inline double compute<0>(double &a, double &b, double &t, double &p)
{
return ((a + b) * (a + b)) / (4 * t);
}
template <int maxIterations>
inline static double compute()
{
double a = 1;
double b = (double)1 / sqrt(2.0);
double t = (double)1 / 4;
double p = 1;
return compute<maxIterations>(a, b, t, p); // call the overloaded function
}
int main(int argc, char **argv)
{
printf("\nTEMPLATE METAPROGRAMMING EXAMPLE:\n");
printf("Compile-time PI computation based on\n");
printf("Gauss-Legendre algorithm (C++)\n\n");
printf("Pi=%.16f\n\n", compute<5>());
return 0;
}
The following example is lifted from the excellent book C++ Templates - The complete guide.
#include <iostream>
using namespace std;
template <int N> struct Pow3 {
enum { pow = 3 * Pow3<N-1>::pow };
}
template <> struct Pow3<0> {
enum { pow = 1 };
}
int main() {
cout << "3 to the 7 is " << Pow<7>::pow << "\n";
}
The point of this code is that the recursive calculation of the 7th power of 3 takes place at compile time rather than run time. It is thus extremely efficient in terms of runtime performance, at the expense of slower compilation.
Is this useful? In this example, probably not. But there are problems where performing calculations at compile time can be an advantage.
It's hard to say what C++ meta-programming is. More and more I feel it is much like introducing 'types' as variables, in the way functional programming has it. It renders declarative programming possible in C++.
It's way easier to show examples.
One of my favorites is a 'trick' (or pattern:) ) to flatte multiply nested switch/case blocks:
#include <iostream>
using namespace std;
enum CCountry { Belgium, Japan };
enum CEra { ancient, medieval, future };
// nested switch
void historic( CCountry country, CEra era ) {
switch( country ) {
case( Belgium ):
switch( era ) {
case( ancient ): cout << "Ambiorix"; break;
case( medieval ): cout << "Keizer Karel"; break;
}
break;
case( Japan ):
switch( era ) {
case( future ): cout << "another Ruby?"; break;
case( medieval ): cout << "Musashi Mijamoto"; break;
}
break;
}
}
// the flattened, metaprogramming way
// define the conversion from 'runtime arguments' to compile-time arguments (if needed...)
// or use just as is.
template< CCountry country, CEra era > void thistoric();
template<> void thistoric<Belgium, ancient> () { cout << "Ambiorix"; }
template<> void thistoric<Belgium, medieval>() { cout << "Keizer Karel"; }
template<> void thistoric<Belgium, future >() { cout << "Beer, lots of it"; }
template<> void thistoric<Japan, ancient> () { cout << "wikipedia"; }
template<> void thistoric<Japan, medieval>() { cout << "Musashi"; }
template<> void thistoric<Japan, future >() { cout << "another Ruby?"; }
// optional: conversion from runtime to compile-time
//
template< CCountry country > struct SelectCountry {
static void select( CEra era ) {
switch (era) {
case( medieval ): thistoric<country, medieval>(); break;
case( ancient ): thistoric<country, ancient >(); break;
case( future ): thistoric<country, future >(); break;
}
}
};
void Thistoric ( CCountry country, CEra era ) {
switch( country ) {
case( Belgium ): SelectCountry<Belgium>::select( era ); break;
case( Japan ): SelectCountry<Japan >::select( era ); break;
}
}
int main() {
historic( Belgium, medieval ); // plain, nested switch
thistoric<Belgium,medieval>(); // direct compile time switch
Thistoric( Belgium, medieval );// flattened nested switch
return 0;
}
The only time I needed to use Boost.MPL in my day job was when I needed to convert boost::variant to and from QVariant.
Since boost::variant has an O(1) visitation mechanism, the boost::variant to QVariant direction is near-trivial.
However, QVariant doesn't have a visitation mechanism, so in order to convert it into a boost::variant, you need to iterate over the mpl::list of types that the specific boost::variant instantiation can hold, and for each type ask the QVariant whether it contains that type, and if so, extract the value and return it in a boost::variant. It's quite fun, you should try it :)
QtMetaObject basically implements reflection (Reflection) and IS one of the major forms of metaprogramming, quite powerful actually. It is similar to Java's reflection and it's also commonly used in dynamic languages (Python, Ruby, PHP...). It's more readable than templates, but both have their pros and cons.
This is a simple "value computation" along the lines of Factorial. However, it's one you are much more likely to actually use in your code.
The macro CT_NEXTPOWEROFTWO2(VAL) uses template metaprogramming to compute the next power of two greater than or equal to a value for values known at compile time.
template<long long int POW2VAL> class NextPow2Helper
{
enum { c_ValueMinusOneBit = (POW2VAL&(POW2VAL-1)) };
public:
enum {
c_TopBit = (c_ValueMinusOneBit) ?
NextPow2Helper<c_ValueMinusOneBit>::c_TopBit : POW2VAL,
c_Pow2ThatIsGreaterOrEqual = (c_ValueMinusOneBit) ?
(c_TopBit<<1) : c_TopBit
};
};
template<> class NextPow2Helper<1>
{ public: enum { c_TopBit = 1, c_Pow2ThatIsGreaterOrEqual = 1 }; };
template<> class NextPow2Helper<0>
{ public: enum { c_TopBit = 0, c_Pow2ThatIsGreaterOrEqual = 0 }; };
// This only works for values known at Compile Time (CT)
#define CT_NEXTPOWEROFTWO2(VAL) NextPow2Helper<VAL>::c_Pow2ThatIsGreaterOrEqual