Question

Is there a way to perform addition(or arithmetic operations) by using ONLY bitwise operators?

Answer 1

Addition of bits a,b and c

carry_out = b&c|a&b|a&c;  // there's a carry, if at least 2 bits are set
sum = a^b^c;              // the sum is the number of set bits modulo 2

One has to perform this for all bits in the word – first for bit 0 using carry_in = c = 0, and iterating to carry_in(next_bit) = carry_out(previous_result).

Subtraction happens with bit inverting b from (a-b) and setting the initial carry to 1.

However, if one has to add e.g 32 numbers in parallel, one can fit ‘a’ with 32 lsbs of all those numbers and perform the binary operations in parallel. This is a technique called bit slicing.

For multiplication CSA (carry save adder is definitely the best software approach — it has the smallest “area”)

As an exercise, here’s an algorithm that calculates a+b(+c) in parallel:

int a = 13113;
int b = 43334;
int c =     1;

int main()
{
   int sum=a^b^c,c2;
   c=((c&a)|(a&b)|(c&b))<<1;
   while (c) {
     c2=(c&sum)<<1;
     sum^=c;
    c=c2;
   }
   printf("%d\n",sum);
}