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Asked: 2026-06-01T20:13:07+00:00

I need to write an expression of one byte Hamming weight in terms of

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I need to write an expression of one byte Hamming weight in terms of binary operations only (&, ^, >>); without any loop, just a formula.

I know that there are plenty algorithms, that allow computing Hamming weight, but all of them use arithmetical operations or looping.

If we take an algorithm from http://en.wikipedia.org/wiki/Hamming_weight, then the first sum D=B+C can be written as D = B^C^(B&C << 1), but two following sums are more complicated.

Does anyone have a hint?

UPDATE:
Thank you for help guys. Actually, I needed something like following:

int popcount_1(unsigned char in){

unsigned char m1  = 0x55;
unsigned char m2  = 0x33;
unsigned char m4  = 0x0f;
unsigned char B,C = 0;
unsigned char x = in;

x = (x & (x << 1) & (m1 << 1)) | (m1 & (x ^ (x >> 1)));

B = x & m2;
C = (x >>  2) & m2;
x = B ^ C ^ ((B & C) << 1);

B = (x & m4 ) ^ ((x >>  4) & m4);
C = (x & ((x >>  4) & m4)) << 1;
x = B ^ C ^ ((B & C) << 1);
return x;
}

This code will result in Hamming weight of variable in. It does not contain any +, -, or comparison instructions and it can work on 8bits microcontrollers.
Nevertheless, it takes more operations than most of other solutions. Now, I am trying to simplify it.

UPDATE2: Another solution, based on 64 bits registers, is proposed by @Evgeny Kluev

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1 Answer

  1. Editorial Team

    I’m not sure if this is what you search for, but here is just a formula using only shifts and bitwise and:

    int weight(unsigned char x)
{
  return ((0x876543210 >>
    (((0x4332322132212110 >> ((x & 0xF) << 2)) & 0xF) << 2)) >>
    ((0x4332322132212110 >> (((x & 0xF0) >> 2)) & 0xF) << 2))
    & 0xf;
}

    Here shift operation is twice used as a substitute for array indexing (to find 4-bit hamming weights). And one more shift operation uses array indexing to perform addition.

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