Booth’s Multiplication Algorithm

Booth’s algorithm is a multiplication algorithm that multiplies two signed binary numbers in 2’s compliment notation.
Booth used desk calculators that were faster at shifting than adding and created the algorithm to increase their speed. Booth’s algorithm is of interest in the study of computer architecture. Here’s the implememtation of the algorithm.

Examples:

Input : 0110, 0010
Output :  qn      q[n+1]                  AC      QR     sc(step count)
                          initial         0000   0010        4
          0       0       rightShift      0000   0001        3
          1       0       A = A - BR      1010
                          rightShift      1101   0000        2
          0       1       A = A + BR      0011
                          rightShift      0001   1000        1
          0       0       rightShift      0000   1100        0

Result=1100



Algorithm :

Put multiplicand in BR and multiplier in QR
and then the algorithm works as per the following conditions :
1. If Qn and Qn+1 are same i.e. 00 or 11 perform arithematic shift by 1 bit.
2. If Qn Qn+1 = 10 do A= A + BR and perform arithematic shift by 1 bit.
3. If Qn Qn+1 = 01 do A= A – BR and perform arithematic shift by 1 bit.

C++

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// CPP code to implement booth's algorithm
#include <bits/stdc++.h>
  
using namespace std;
  
// function to perform adding in the accumulator
void add(int ac[], int x[], int qrn)
{
    int i, c = 0;
      
    for (i = 0; i < qrn; i++) {
          
        // updating accumulator with A = A + BR
        ac[i] = ac[i] + x[i] + c;
          
        if (ac[i] > 1) {
            ac[i] = ac[i] % 2;
            c = 1;
        }
        else
            c = 0;
    }
}
  
// function to find the number's complement
void complement(int a[], int n)
{
    int i;
    int x[8] = { 0};
    x[0] = 1;
      
    for (i = 0; i < n; i++) {
        a[i] = (a[i] + 1) % 2;
    }
    add(a, x, n);
}
  
// function ro perform right shift
void rightShift(int ac[], int qr[], int& qn, int qrn)
{
    int temp, i;
    temp = ac[0];
    qn = qr[0];
      
    cout << "\t\trightShift\t";
      
    for (i = 0; i < qrn - 1; i++) {
        ac[i] = ac[i + 1];
        qr[i] = qr[i + 1];
    }
    qr[qrn - 1] = temp;
}
  
// function to display oprations
void display(int ac[], int qr[], int qrn)
{
    int i;
      
    // accumulator content
    for (i = qrn - 1; i >= 0; i--)
        cout << ac[i];
    cout << "\t";
      
    // multiplier content
    for (i = qrn - 1; i >= 0; i--)
        cout << qr[i];
}
  
// Function to implement booth's algo
void boothAlgorithm(int br[], int qr[], int mt[], int qrn, int sc)
{
  
    int qn = 0, ac[10] = { 0 };
    int temp = 0;
    cout << "qn\tq[n+1]\t\tBR\t\tAC\tQR\t\tsc\n";
    cout << "\t\t\tinitial\t\t";
      
    display(ac, qr, qrn);
    cout << "\t\t" << sc << "\n";
      
    while (sc != 0) {
        cout << qr[0] << "\t" << qn;
          
        // SECOND CONDITION
        if ((qn + qr[0]) == 1)
        {
            if (temp == 0) {
                  
                // subtract BR from accumulator
                add(ac, mt, qrn);
                cout << "\t\tA = A - BR\t";
                  
                for (int i = qrn - 1; i >= 0; i--)
                    cout << ac[i];
                temp = 1;
            }
              
            // THIRD CONDITION
            else if (temp == 1)
            {
                // add BR to accumulator
                add(ac, br, qrn);
                cout << "\t\tA = A + BR\t";
                  
                for (int i = qrn - 1; i >= 0; i--)
                    cout << ac[i];
                temp = 0;
            }
            cout << "\n\t";
            rightShift(ac, qr, qn, qrn);
        }
          
        // FIRST CONDITION
        else if (qn - qr[0] == 0)
            rightShift(ac, qr, qn, qrn);
         
        display(ac, qr, qrn);
         
        cout << "\t";
          
        // decrement counter
        sc--;
        cout << "\t" << sc << "\n";
    }
}
  
// driver code
int main(int argc, char** arg)
{
  
    int mt[10], sc;
    int brn, qrn;
      
    // Number of multiplicand bit
    brn = 4;
      
    // multiplicand
    int br[] = { 0, 1, 1, 0 };
      
    // copy multiplier to temp array mt[]
    for (int i = brn - 1; i >= 0; i--)
        mt[i] = br[i]; 
          
    reverse(br, br + brn);
  
    complement(mt, brn);
  
    // No. of multiplier bit
    qrn = 4;
      
    // sequence counter
    sc = qrn;
      
    // multiplier
    int qr[] = { 1, 0, 1, 0 };
    reverse(qr, qr + qrn);
  
    boothAlgorithm(br, qr, mt, qrn, sc);
  
    cout << endl
         << "Result = ";
           
    for (int i = qrn - 1; i >= 0; i--)
        cout << qr[i];
}

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C#

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// C# code to implement 
// booth's algorithm
using System;
  
class GFG
{
    // function to perform 
    // adding in the accumulator
    static void add(int []ac, 
                    int []x, 
                    int qrn)
    {
        int i, c = 0;
          
        for (i = 0; i < qrn; i++) 
        {
              
            // updating accumulator
            // with A = A + BR
            ac[i] = ac[i] + x[i] + c;
              
            if (ac[i] > 1) 
            {
                ac[i] = ac[i] % 2;
                c = 1;
            }
            else
                c = 0;
        }
    }
      
    // function to find 
    // the number's complement
    static void complement(int []a, int n)
    {
        int i;
        int []x = new int[8];
        Array.Clear(x, 0, 8);
  
        x[0] = 1;
          
        for (i = 0; i < n; i++)
        {
            a[i] = (a[i] + 1) % 2;
        }
        add(a, x, n);
    }
      
    // function to perform 
    // right shift
    static void rightShift(int []ac, int []qr,
                           ref int qn, int qrn)
    {
        int temp, i;
        temp = ac[0];
        qn = qr[0];
          
        Console.Write("\t\trightShift\t");
          
        for (i = 0; i < qrn - 1; i++)
        {
            ac[i] = ac[i + 1];
            qr[i] = qr[i + 1];
        }
        qr[qrn - 1] = temp;
    }
      
    // function to display 
    // operations
    static void display(int []ac, 
                        int []qr, 
                        int qrn)
    {
        int i;
          
        // accumulator content
        for (i = qrn - 1; i >= 0; i--)
            Console.Write(ac[i]);
        Console.Write("\t");
          
        // multiplier content
        for (i = qrn - 1; i >= 0; i--)
            Console.Write(qr[i]);
    }
      
    // Function to implement
    // booth's algo
    static void boothAlgorithm(int []br, int []qr, 
                               int []mt, int qrn, 
                               int sc)
    {
      
        int qn = 0;
        int []ac = new int[10];
        Array.Clear(ac, 0, 10);
          
        int temp = 0;
        Console.Write("qn\tq[n + 1]\tBR\t" +
                       "\tAC\tQR\t\tsc\n");
        Console.Write("\t\t\tinitial\t\t");
          
        display(ac, qr, qrn);
        Console.Write("\t\t" + sc + "\n");
          
        while (sc != 0) 
        {
            Console.Write(qr[0] + "\t" + qn);
              
            // SECOND CONDITION
            if ((qn + qr[0]) == 1)
            {
                if (temp == 0)
                {
                      
                    // subtract BR 
                    // from accumulator
                    add(ac, mt, qrn);
                    Console.Write("\t\tA = A - BR\t");
                      
                    for (int i = qrn - 1; i >= 0; i--)
                        Console.Write(ac[i]);
                    temp = 1;
                }
                  
                // THIRD CONDITION
                else if (temp == 1)
                {
                    // add BR to accumulator
                    add(ac, br, qrn);
                    Console.Write("\t\tA = A + BR\t");
                      
                    for (int i = qrn - 1; i >= 0; i--)
                        Console.Write(ac[i]);
                    temp = 0;
                }
                Console.Write("\n\t");
                rightShift(ac, qr, ref qn, qrn);
            }
              
            // FIRST CONDITION
            else if (qn - qr[0] == 0)
                rightShift(ac, qr, 
                           ref qn, qrn);
              
            display(ac, qr, qrn);
              
            Console.Write("\t");
              
            // decrement counter
            sc--;
            Console.Write("\t" + sc + "\n");
        }
    }
      
    // Driver code
    static void Main()
    {
      
        int []mt = new int[10]; 
        int sc, brn, qrn;
          
        // Number of 
        // multiplicand bit
        brn = 4;
          
        // multiplicand
        int []br = new int[]{ 0, 1, 1, 0 };
          
        // copy multiplier 
        // to temp array mt[]
        for (int i = brn - 1; i >= 0; i--)
            mt[i] = br[i]; 
              
        Array.Reverse(br);
      
        complement(mt, brn);
      
        // No. of 
        // multiplier bit
        qrn = 4;
          
        // sequence 
        // counter
        sc = qrn;
          
        // multiplier
        int []qr = new int[]{ 1, 0, 1, 0 };
        Array.Reverse(qr);
      
        boothAlgorithm(br, qr, 
                       mt, qrn, sc);
          
        Console.WriteLine();
        Console.Write("Result = ");
              
        for (int i = qrn - 1; i >= 0; i--)
            Console.Write(qr[i]);
    }
}
  
// This code is contributed by 
// Manish Shaw(manishshaw1)

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Output :

qn    q[n + 1]    BR        AC    QR        sc
            initial        0000    1010        4
0    0        rightShift    0000    0101        3
1    0        A = A - BR    1010
            rightShift    1101    0010        2
0    1        A = A + BR    0011
            rightShift    0001    1001        1
1    0        A = A - BR    1011
            rightShift    1101    1100        0

Result = 1100


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