Booth's algorithm is a multiplication algorithm that multiplies two signed binary numbers in 2's complement 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 implementation 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 arithmetic shift by 1 bit.
2. If Qn Qn+1 = 01 do A= A + BR and perform arithmetic shift by 1 bit.
3. If Qn Qn+1 = 10 do A= A - BR and perform arithmetic shift by 1 bit.
// 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 to 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 operations
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];
}
// Java code to implement booth's algorithm
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];
x[0] = 1;
for (i = 0; i < n; i++)
{
a[i] = (a[i] + 1) % 2;
}
add(a, x, n);
}
// function ro perform right shift
static void rightShift(int ac[], int qr[],
int qn, int qrn)
{
int temp, i;
temp = ac[0];
qn = qr[0];
System.out.print("\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--)
{
System.out.print(ac[i]);
}
System.out.print("\t");
// multiplier content
for (i = qrn - 1; i >= 0; i--)
{
System.out.print(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];
int temp = 0;
System.out.print("qn\tq[n+1]\t\tBR\t\tAC\tQR\t\tsc\n");
System.out.print("\t\t\tinitial\t\t");
display(ac, qr, qrn);
System.out.print("\t\t" + sc + "\n");
while (sc != 0)
{
System.out.print(qr[0] + "\t" + qn);
// SECOND CONDITION
if ((qn + qr[0]) == 1)
{
if (temp == 0)
{
// subtract BR from accumulator
add(ac, mt, qrn);
System.out.print("\t\tA = A - BR\t");
for (int i = qrn - 1; i >= 0; i--)
{
System.out.print(ac[i]);
}
temp = 1;
}
// THIRD CONDITION
else if (temp == 1)
{
// add BR to accumulator
add(ac, br, qrn);
System.out.print("\t\tA = A + BR\t");
for (int i = qrn - 1; i >= 0; i--)
{
System.out.print(ac[i]);
}
temp = 0;
}
System.out.print("\n\t");
rightShift(ac, qr, qn, qrn);
}
// FIRST CONDITION
else if (qn - qr[0] == 0)
{
rightShift(ac, qr, qn, qrn);
}
display(ac, qr, qrn);
System.out.print("\t");
// decrement counter
sc--;
System.out.print("\t" + sc + "\n");
}
}
static void reverse(int a[])
{
int i, k, n = a.length;
int t;
for (i = 0; i < n / 2; i++)
{
t = a[i];
a[i] = a[n - i - 1];
a[n - i - 1] = t;
}
}
// Driver code
public static void main(String[] args)
{
int[] mt = new int[10];
int 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);
complement(mt, brn);
// No. of multiplier bit
qrn = 4;
// sequence counter
sc = qrn;
// multiplier
int qr[] = {1, 0, 1, 0};
reverse(qr);
boothAlgorithm(br, qr, mt, qrn, sc);
System.out.print("\n"
+ "Result = ");
for (int i = qrn - 1; i >= 0; i--)
{
System.out.print(qr[i]);
}
}
}
/* This code contributed by PrinciRaj1992 */
# Python3 code to implement booth's algorithm
# function to perform adding in the accumulator
def add(ac, x, qrn):
c = 0
for i in range(qrn):
# 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
def complement(a, n):
x = [0] * 8
x[0] = 1
for i in range(n):
a[i] = (a[i] + 1) % 2
add(a, x, n)
# function to perform right shift
def rightShift(ac, qr, qn, qrn):
temp = ac[0]
qn = qr[0]
print("\t\trightShift\t", end = "");
for i in range(qrn - 1):
ac[i] = ac[i + 1]
qr[i] = qr[i + 1]
qr[qrn - 1] = temp
# function to display operations
def display(ac, qr, qrn):
# accumulator content
for i in range(qrn - 1, -1, -1):
print(ac[i], end = '')
print("\t", end = '')
# multiplier content
for i in range(qrn - 1, -1, -1):
print(qr[i], end = "")
# Function to implement booth's algo
def boothAlgorithm(br, qr, mt, qrn, sc):
qn = 0
ac = [0] * 10
temp = 0
print("qn\tq[n+1]\t\tBR\t\tAC\tQR\t\tsc")
print("\t\t\tinitial\t\t", end = "")
display(ac, qr, qrn)
print("\t\t", sc, sep = "")
while (sc != 0):
print(qr[0], "\t", qn, sep = "", end = "")
# SECOND CONDITION
if ((qn + qr[0]) == 1):
if (temp == 0):
# subtract BR from accumulator
add(ac, mt, qrn)
print("\t\tA = A - BR\t", end = "")
for i in range(qrn - 1, -1, -1):
print(ac[i], end = "")
temp = 1
# THIRD CONDITION
elif (temp == 1):
# add BR to accumulator
add(ac, br, qrn)
print("\t\tA = A + BR\t", end = "")
for i in range(qrn - 1, -1, -1):
print(ac[i], end = "")
temp = 0
print("\n\t", end = "")
rightShift(ac, qr, qn, qrn)
# FIRST CONDITION
elif (qn - qr[0] == 0):
rightShift(ac, qr, qn, qrn)
display(ac, qr, qrn)
print("\t", end = "")
# decrement counter
sc -= 1
print("\t", sc, sep = "")
# driver code
def main():
mt = [0] * 10
# Number of multiplicand bit
brn = 4
# multiplicand
br = [ 0, 1, 1, 0 ]
# copy multiplier to temp array mt[]
for i in range(brn - 1, -1, -1):
mt[i] = br[i]
br.reverse()
complement(mt, brn)
# No. of multiplier bit
qrn = 4
# sequence counter
sc = qrn
# multiplier
qr = [ 1, 0, 1, 0 ]
qr.reverse()
boothAlgorithm(br, qr, mt, qrn, sc)
print("\nResult = ", end = "")
for i in range(qrn - 1, -1, -1):
print(qr[i], end = "")
print()
main()
#This code is contributed by phasing17
// 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)
//JavaScript code to implement booth's algorithm
// function to perform adding in the accumulator
function add(ac, x, qrn)
{
let c = 0;
for (let 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
function complement(a, n)
{
let x = new Array(8).fill(0);
x[0] = 1;
for (let i = 0; i < n; i++)
a[i] = (a[i] + 1) % 2;
add(a, x, n);
}
// function to perform right shift
function rightShift(ac, qr, qn, qrn)
{
let temp = ac[0];
qn = qr[0];
process.stdout.write("\t\trightShift\t");
for (let i = 0; i < qrn - 1; i++)
{
ac[i] = ac[i + 1];
qr[i] = qr[i + 1];
}
qr[qrn - 1] = temp;
}
// function to display operations
function display(ac, qr, qrn)
{
// accumulator content
for (let i = qrn - 1; i > -1; i--)
process.stdout.write(ac[i] + "");
process.stdout.write("\t");
// multiplier content
for (i = qrn - 1; i > -1; i--)
process.stdout.write(qr[i] + "");
}
// Function to implement booth's algo
function boothAlgorithm(br, qr, mt, qrn, sc)
{
let qn = 0;
let ac = new Array(10).fill(0);
let temp = 0;
process.stdout.write("qn\tq[n+1]\t\tBR\t\tAC\tQR\t\tsc\n");
process.stdout.write("\t\t\tinitial\t\t");
display(ac, qr, qrn);
process.stdout.write("\t\t" + sc + "\n");
while (sc != 0)
{
process.stdout.write(qr[0] + "\t" + qn);
// SECOND CONDITION
if ((qn + qr[0]) == 1)
{
if (temp == 0)
{
// subtract BR from accumulator
add(ac, mt, qrn);
process.stdout.write("\t\tA = A - BR\t");
for (let i = qrn - 1; i > -1; i--)
process.stdout.write(ac[i] + "");
temp = 1;
}
// THIRD CONDITION
else if (temp == 1)
{
// add BR to accumulator
add(ac, br, qrn);
process.stdout.write("\t\tA = A + BR\t");
for (i = qrn - 1; i > -1; i--)
process.stdout.write(ac[i] + "");
temp = 0;
}
process.stdout.write("\n\t");
rightShift(ac, qr, qn, qrn);
}
// FIRST CONDITION
else if (qn - qr[0] == 0)
rightShift(ac, qr, qn, qrn);
display(ac, qr, qrn);
process.stdout.write("\t");
// decrement counter
sc -= 1;
process.stdout.write("\t" + sc + "\n");
}
}
// driver code
let mt = new Array(10).fill(0);
// Number of multiplicand bit
let brn = 4;
// multiplicand
let br = [ 0, 1, 1, 0 ];
// copy multiplier to temp array mt[]
for (let i = brn - 1; i > -1; i--)
mt[i] = br[i];
br.reverse()
complement(mt, brn)
// No. of multiplier bit
qrn = 4;
// sequence counter
let sc = qrn;
// multiplier
let qr = [ 1, 0, 1, 0 ];
qr.reverse();
boothAlgorithm(br, qr, mt, qrn, sc)
process.stdout.write("\nResult = ");
for (let i = qrn - 1; i > -1; i--)
process.stdout.write(qr[i] + "");
process.stdout.write("\n");
//This code is contributed by phasing17
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
Time Complexity: O(n)
Auxiliary Space: O(1)