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.
3uses
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*)
14function isPowerOfTwo(const x: qword): longbool;
15begin
16 isPowerOfTwo := popCnt(x) = 1;
17end;
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.
19function isPowerOfTwo(const x: int64): longbool;
20begin
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
32end;
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
.
34function isPowerOfTwo(const x: float): longbool;
35begin
36 {$push}
37 {$boolEval off}
38 if (x <= 0) or isInfinite(x) or isNan(x) then
39 {$pop}
40 begin
41 isPowerOfTwo := false;
42 end
43 else
44 begin
45 isPowerOfTwo := frac(log2(x)) = 0;
46 end;
47end;
Alternatively one could use the following implementation, utilizing math.frExp
.
It is even neater, since it does not really calculate the logarithm but just extracts the numbers (fxtract
instruction on x86 platforms).
However, (without writing inline assembly blocks) one needs to allocate two variables, where only one is actually needed.
34function isPowerOfTwo(const x: float): longbool;
35var
36 mantissa: extended;
37 exponent: longint;
38begin
39 {$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;
51end;