Bitwise Multiply and Add in Java

Question

I have the methods that do both the multiplication and addition, but I'm just not able to get my head around them. Both of them are from external websites and not my own:

public static void bitwiseMultiply(int n1, int n2) {
    int a = n1, b = n2, result=0;
    while (b != 0) // Iterate the loop till b==0
    {
        if ((b & 01) != 0) // Logical ANDing of the value of b with 01
        {
            result = result + a; // Update the result with the new value of a.
        }
        a <<= 1;              // Left shifting the value contained in 'a' by 1.
        b >>= 1;             // Right shifting the value contained in 'b' by 1.
    }
    System.out.println(result);
}

public static void bitwiseAdd(int n1, int n2) {
    int x = n1, y = n2;
    int xor, and, temp;
    and = x & y;
    xor = x ^ y;

    while (and != 0) {
        and <<= 1;
        temp = xor ^ and;
        and &= xor;
        xor = temp;
    }
    System.out.println(xor);
}

I tried doing a step-by-step debug, but it really didn't make much sense to me, though it works.

What I'm possibly looking for is to try and understand how this works (the mathematical basis perhaps?).

Edit: This is not homework, I'm just trying to learn bitwise operations in Java.

Answer 1

Let's begin by looking the multiplication code. The idea is actually pretty clever. Suppose that you have n₁ and n₂ written in binary. Then you can think of n1 as a sum of powers of two: n2 = c₃₀ 2³⁰ + c₂₉ 2²⁹ + ... + c₁ 2¹ + c₀ 2⁰, where each c_i is either 0 or 1. Then you can think of the product n₁ n₂ as

n₁ n₂ =

n₁ (c₃₀ 2³⁰ + c₂₉ 2²⁹ + ... + c₁ 2¹ + c₀ 2⁰) =

n₁ c₃₀ 2³⁰ + n₁ c₂₉ 2²⁹ + ... + n₁ c₁ 2¹ + n₁ c₀ 2⁰

This is a bit dense, but the idea is that the product of the two numbers is given by the first number multiplied by the powers of two making up the second number, times the value of the binary digits of the second number.

The question now is whether we can compute the terms of this sum without doing any actual multiplications. In order to do so, we're going to need to be able to read the binary digits of n₂. Fortunately, we can do this using shifts. In particular, suppose we start off with n₂ and then just look at the last bit. That's c₀. If we then shift the value down one position, then the last bit is c₀, etc. More generally, after shifting the value of n₂ down by i positions, the lowest bit will be c_i. To read the very last bit, we can just bitwise AND the value with the number 1. This has a binary representation that's zero everywhere except the last digit. Since 0 AND n = 0 for any n, this clears all the topmost bits. Moreover, since 0 AND 1 = 0 and 1 AND 1 = 1, this operation preserves the last bit of the number.

Okay - we now know that we can read the values of c_i; so what? Well, the good news is that we also can compute the values of the series n₁ 2ⁱ in a similar fashion. In particular, consider the sequence of values n₁ << 0, n₁ << 1, etc. Any time you do a left bit-shift, it's equivalent to multiplying by a power of two. This means that we now have all the components we need to compute the above sum. Here's your original source code, commented with what's going on:

public static void bitwiseMultiply(int n1, int n2) {
    /* This value will hold n1 * 2^i for varying values of i.  It will
     * start off holding n1 * 2^0 = n1, and after each iteration will 
     * be updated to hold the next term in the sequence.
     */
    int a = n1;

    /* This value will be used to read the individual bits out of n2.
     * We'll use the shifting trick to read the bits and will maintain
     * the invariant that after i iterations, b is equal to n2 >> i.
     */
    int b = n2;

    /* This value will hold the sum of the terms so far. */
    int result = 0;

    /* Continuously loop over more and more bits of n2 until we've
     * consumed the last of them.  Since after i iterations of the
     * loop b = n2 >> i, this only reaches zero once we've used up
     * all the bits of the original value of n2.
     */
    while (b != 0)
    {
        /* Using the bitwise AND trick, determine whether the ith 
         * bit of b is a zero or one.  If it's a zero, then the
         * current term in our sum is zero and we don't do anything.
         * Otherwise, then we should add n1 * 2^i.
         */
        if ((b & 1) != 0)
        {
            /* Recall that a = n1 * 2^i at this point, so we're adding
             * in the next term in the sum.
             */
            result = result + a;
        }

        /* To maintain that a = n1 * 2^i after i iterations, scale it
         * by a factor of two by left shifting one position.
         */
        a <<= 1;

        /* To maintain that b = n2 >> i after i iterations, shift it
         * one spot over.
         */
        b >>>= 1;
    }

    System.out.println(result);
}

Hope this helps!

Answer 2

It looks like your problem is not java, but just calculating with binary numbers. Start of simple: (all numbers binary:)

0 + 0 = 0   # 0 xor 0 = 0
0 + 1 = 1   # 0 xor 1 = 1
1 + 0 = 1   # 1 xor 0 = 1
1 + 1 = 10  # 1 xor 1 = 0 ( read 1 + 1 = 10 as 1 + 1 = 0 and 1 carry)

Ok... You see that you can add two one digit numbers using the xor operation. With an and you can now find out whether you have a "carry" bit, which is very similar to adding numbers with pen&paper. (Up to this point you have something called a Half-Adder). When you add the next two bits, then you also need to add the carry bit to those two digits. Taking this into account you can get a Full-Adder. You can read about the concepts of Half-Adders and Full-Adders on Wikipedia: http://en.wikipedia.org/wiki/Adder_(electronics) And many more places on the web. I hope that gives you a start.

With multiplication it is very similar by the way. Just remember how you did multiplying with pen&paper in elementary school. Thats what is happening here. Just that it's happening with binary numbers and not with decimal numbers.

Answer 3

EXPLANATION OF THE bitwiseAdd METHOD:

I know this question was asked a while back but since no complete answer has been given regarding how the bitwiseAdd method works here is one.

The key to understanding the logic encapsulated in bitwiseAdd is found in the relationship between addition operations and xor and and bitwise operations. That relationship is defined by the following equation (see appendix 1 for a numeric example of this equation):

x + y = 2 * (x&y)+(x^y)     (1.1)

Or more simply:

x + y = 2 * and + xor       (1.2)

with
    and = x & y
    xor = x ^ y

You might have noticed something familiar in this equation: the and and xor variables are the same as those defined at the beginning of bitwiseAdd. There is also a multiplication by two, which in bitwiseAdd is done at the beginning of the while loop. But I will come back to that later.

Let me also make a quick side note about the '&' bitwise operator before we proceed further. This operator basically "captures" the intersection of the bit sequences against which it is applied. For example, 9 & 13 = 1001 & 1101 = 1001 (=9). You can see from this result that only those bits common to both bit sequences are copied to the result. It derives from this that when two bit sequences have no common bit, the result of applying '&' on them yields 0. This has an important consequence on the addition-bitwise relationship which shall become clear soon

Now the problem we have is that equation 1.2 uses the '+' operator whereas bitwiseAdd doesn't (it only uses '^', '&' and '<<'). So how do we make the '+' in equation 1.2 somehow disappear? Answer: by 'forcing' the and expression to return 0. And the way we do that is by using recursion.

To demonstrate this I am going to recurse equation 1.2 one time (this step might be a bit challenging at first but if needed there's a detailed step by step result in appendix 2):

x + y = 2*(2*and & xor) + (2*and ^ xor)     (1.3)

Or more simply:

x + y = 2 * and[1] + xor[1]     (1.4)

with
    and[1] = 2*and & xor,
    xor[1] = 2*and ^ xor,
    [1] meaning 'recursed one time'

There's a couple of interesting things to note here. First you noticed how the concept of recursion sounds close to that of a loop, like the one found in bitwiseAdd in fact. This connection becomes even more obvious when you consider what and[1] and xor[1] are: they are the same expressions as the and and xor expressions defined INSIDE the while loop in bitwiseAdd. We also note that a pattern emerges: equation 1.4 looks exactly like equation 1.2!

As a result of this, doing further recursions is a breeze, if one keeps the recursive notation. Here we recurse equation 1.2 two more times:

x + y = 2 * and[2] + xor[2]
x + y = 2 * and[3] + xor[3]

This should now highlight the role of the 'temp' variable found in bitwiseAdd: temp allows to pass from one recursion level to the next.

We also notice the multiplication by two in all those equations. As mentioned earlier this multiplication is done at the begin of the while loop in bitwiseAdd using the and <<= 1 statement. This multiplication has a consequence on the next recursion stage since the bits in and[i] are different from those in the and[i] of the previous stage (and if you recall the little side note I made earlier about the '&' operator you probably see where this is going now).

The general form of equation 1.4 now becomes:

x + y = 2 * and[x] + xor[x]     (1.5)
with x the nth recursion

FINALY:

So when does this recursion business end exactly?

Answer: it ends when the intersection between the two bit sequences in the and[x] expression of equation 1.5 returns 0. The equivalent of this in bitwiseAdd happens when the while loop condition becomes false. At this point equation 1.5 becomes:

    x + y = xor[x]      (1.6)

And that explains why in bitwiseAdd we only return xor at the end!

And we are done! A pretty clever piece of code this bitwiseAdd I must say :)

I hope this helped

APPENDIX:

1) A numeric example of equation 1.1

equation 1.1 says:

    x + y = 2(x&y)+(x^y)        (1.1)

To verify this statement one can take a simple example, say adding 9 and 13 together. The steps are shown below (the bitwise representations are in parenthesis):

We have

    x = 9 (1001)
    y = 13  (1101)

And

    x + y = 9 + 13 = 22
    x & y = 9 & 13 = 9 (1001 & 1101 = 1001)
    x ^ y = 9^13 = 4 (1001 ^ 1101 = 0100)

pluging that back into equation 1.1 we find:

    9 + 13 = 2 * 9 + 4 = 22 et voila!

2) Demonstrating the first recursion step

The first recursion equation in the presentation (equation 1.3) says that

if

x + y = 2 * and + xor           (equation 1.2)

then

x + y = 2*(2*and & xor) + (2*and ^ xor)     (equation 1.3)

To get to this result, we simply took the 2* and + xor part of equation 1.2 above and applied the addition/bitwise operands relationship given by equation 1.1 to it. This is demonstrated as follow:

if

    x + y = 2(x&y) + (x^y)      (equation 1.1)

then

     [2(x&y)] + (x^y)     =      2 ([2(x&y)] & (x^y)) + ([2(x&y)] ^ (x^y))
(left side of equation 1.1)  (after applying the addition/bitwise operands relationship)

Simplifying this with the definitions of the and and xor variables of equation 1.2 gives equation 1.3's result:

[2(x&y)] + (x^y) = 2*(2*and & xor) + (2*and ^ xor)

with
    and = x&y
    xor = x^y

And using that same simplification gives equation 1.4's result:

2*(2*and & xor) + (2*and ^ xor) = 2*and[1] + xor[1]

with
    and[1] = 2*and & xor
    xor[1] = 2*and ^ xor
    [1] meaning 'recursed one time'