is power of two

Checking whether a given number is an integer power of two is a classical demonstration showing how conclusions can be drawn from a number’s internal representation.

integer

uses
	// for math.log2
	math;

system.popCnt counts the number of set bits within an integer. The function’s name derives from “population count.” Population count, because sets are usually implemented by utilizing integers, where a set bit means a certain element is part of a set.

popCnt applied on an integer representing a set will return its cardinality (what the card function defined by Extended Pascal does).

(**
	\brief Determines whether a given number
	is an integer power of two.
	
	\param x the number to check
	\return true iff \f$2^n=x\f$ where \f$n \in \mathbb{Z}\f$
*)
function isPowerOfTwo(const x: qword): longbool;
begin
	isPowerOfTwo := popCnt(x) = 1;
end;

Now, you want to overload your function, since it is not necessarily the case you can limit your calculation to non-negative integers only. Ensure you are typecasting the given value, so the compiler chooses the right, our “base” function.

function isPowerOfTwo(const x: int64): longbool;
begin
	// a) 2^n is always positive.
	// b) There is no 2^n that equals zero.
	if x < 1 then
	begin
		isPowerOfTwo := false;
	end
	// c) Theoretically you could shortcut for x < 3
	else
	begin
		isPowerOfTwo := isPowerOfTwo(qword(x));
	end
end;

non-integer

Although integer operations are nice, sometimes you have to deal with floating point numbers. Since we wanna know the exponent in the expression [math]\displaystyle{ 2^n = x }[/math], and check it is an integer, we can use math.log2 in conjunction with system.frac.

function isPowerOfTwo(const x: float): longbool;
begin
	{$push}
	{$boolEval off}
	if (x <= 0) or isInfinite(x) or isNan(x) then
	{$pop}
	begin
		isPowerOfTwo := false;
	end
	else
	begin
		isPowerOfTwo := frac(log2(x)) = 0;
	end;
end;

Alternatively one could use the following implementation, utilizing math.frExp. It is even neater, since it does not really calculate the logarithm but just extracts the numbers (fxtract instruction on x86 platforms). However, (without writing inline assembly blocks) one needs to allocate two variables, where only one is actually needed.

function isPowerOfTwo(const x: float): longbool;
var
	mantissa: extended;
	exponent: longint;
begin
	{$push}
	{$boolEval off}
	if isInfinite(x) or isNan(x) then
	{$pop}
	begin
		isPowerOfTwo := false;
	end
	else
	begin
		frExp(x, mantissa, exponent);
		isPowerOfTwo := mantissa = 0.5;
	end;
end;

real popcnt

Note, despite its name the system.popCnt function does not simply use the popcnt x86 instruction, because it is not available on all CPUs and availability has to be checked first. Instead, the generic implementation of system.popCnt uses a table-lookup which is guaranteed to work on all platforms and is still really fast.

If you are checking millions or billions of numbers whether they are a power of two, the following would be of course [on most CPUs] faster:

function isPowerOfTwo(const x: qWord): longBool;
{$ifDef CPUx86_64}
{$asmMode Intel}
assembler;
asm
	// !!! you need to ensure popcnt is a legal instruction first
	popcnt rax, x   // rax := # of set bits in x
	dec al          //  al := al - 1;  ZF := al = 0
	setz al         //  al := ZF
end;
{$else}
begin
	isPowerOfTwo := popCnt(x) = 1;
end;
{$endIf}

An alternative algorithm not relying on popcnt’s availability would be (for example):

function isPowerOfTwo(const x: qWord): longBool; assembler;
asm
	mov rax, x                  // rax := rdi
	// Do not enter loop, because dividing zero will never emit a `1`.
	test rax, rax               //  ZF := rax = 0
	jz @is_power_of_zero_done   // if ZF then goto done
	
	// Repeated division by two: The first bit we've carried out (CF)
	// must have also been the _only_ set bit in the entire word (ZF).
@is_power_of_two_divide:
	shr rax, 1                  // rax := rax div 2
	jnc @is_power_of_two_divide // if not CF goto divide
	
	setz al                     //  al := ZF
@is_power_of_zero_done:
end;