# is power of two

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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

3 uses
4 	// for math.log2
5 	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).

 7 (**
8 	\brief Determines whether a given number
9 	is an integer power of two.
10
11 	\param x the number to check
12 	\return true iff \f$2^n=x\f$ where \f$n \in \mathbb{Z}\f$
13 *)
14 function isPowerOfTwo(const x: qword): longbool;
15 begin
16 	isPowerOfTwo := popCnt(x) = 1;
17 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.

19 function isPowerOfTwo(const x: int64): longbool;
20 begin
21 	// a) 2^n is always positive.
22 	// b) There is no 2^n that equals zero.
23 	if x < 1 then
24 	begin
25 		isPowerOfTwo := false;
26 	end
27 	// c) Theoretically you could shortcut for x < 3
28 	else
29 	begin
30 		isPowerOfTwo := isPowerOfTwo(qword(x));
31 	end
32 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 $2^n = x$, and check it is an integer, we can use math.log2 in conjunction with system.frac.

34 function isPowerOfTwo(const x: float): longbool;
35 begin
36 	{$push} 37 {$boolEval off}
38 	if (x <= 0) or isInfinite(x) or isNan(x) then
39 	{$pop} 40 begin 41 isPowerOfTwo := false; 42 end 43 else 44 begin 45 isPowerOfTwo := frac(log2(x)) = 0; 46 end; 47 end;  Alternatively one could use the following implementation, utilizing math.frExp. It is even neater, since it does not really calculate the logarithm but just extracts the numbers (fxtract instruction on x86 platforms). However, (without writing inline assembly blocks) one needs to allocate two variables, where only one is actually needed. 34 function isPowerOfTwo(const x: float): longbool; 35 var 36 mantissa: extended; 37 exponent: longint; 38 begin 39 {$push}
40 	{$boolEval off} 41 if isInfinite(x) or isNan(x) then 42 {$pop}
43 	begin
44 		isPowerOfTwo := false;
45 	end
46 	else
47 	begin
48 		frExp(x, mantissa, exponent);
49 		isPowerOfTwo := mantissa = 0.5;
50 	end;
51 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:

 7 function isPowerOfTwo(const x: qWord): longBool;
8 {$ifDef CPUx86_64} 9 {$asmMode Intel}
10 assembler;
11 asm
12 	// !!! you need to ensure popcnt is a legal instruction first
13 	popcnt rax, x   // rax := # of set bits in x
14 	dec al          //  al := al - 1;  ZF := al = 0
15 	setz al         //  al := ZF
16 end;
17 {$else} 18 begin 19 isPowerOfTwo := popCnt(x) = 1; 20 end; 21 {$endIf}


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

 7 function isPowerOfTwo(const x: qWord): longBool; assembler;
8 asm
9 	mov rax, x                  // rax := rdi
10 	// Do not enter loop, because dividing zero will never emit a 1.
11 	test rax, rax               //  ZF := rax = 0
12 	jz @is_power_of_zero_done   // if ZF then goto done
13
14 	// Repeated division by two: The first bit we've carried out (CF)
15 	// must have also been the _only_ set bit in the entire word (ZF).
16 @is_power_of_two_divide:
17 	shr rax, 1                  // rax := rax div 2
18 	jnc @is_power_of_two_divide // if not CF goto divide
19
20 	setz al                     //  al := ZF
21 @is_power_of_zero_done:
22 end;