I have code as follows:
`if (a <= 10){
z = 5;
} else {
z = -1;
}`
I figured out that when s(10 - a) = |10 - a| / (10 - a) where it outputs a 1 or -1. It outputs 1 if a < 10 and -1 if a > 10.
Then, I just solve the linear equation z = s(10 - a) * m + b, to find constants m and b.
5 = 1 * m + b and -1 = -1 * m + b
Which outputs b = 2, m = 3.
Then this can be modeled as z = 3 * s(10 - a) + 2.
Now the question becomes more tricky. What if I have two variables in nested if statements? Such as:
`if (a <= 10){
if(b <= 3){
z = 3;
} else {
z = 1;
}
} else {
if(b <= -5){
z = -11;
} else {
z = 4;
}
}`
I tried to solve this using another series of linear equations.
3 = A * s(10 - a) + B * s(3 - b) + C
1 = A * s(10 - a) + B * s(3 - b) + C
-11 = A * s(10 - a) + D * s(-5 - b) + C
4 = A * s(10 - a) + D * s(-5 - b) + C
with A, B, C, D as constants. However, this isn't giving me the right answer. What am I doing wrong?
An if statement can be transformed into a formula by using the following trick: we need to find a formula that's 1 if the if statement is true and 0 otherwise. We can use the signum function for this:
f(x, y) = (sign(y - x) + 1) / 2
f(x, y) is 1 if x < y and 0 if x > y. The inverse g(x, y) = 1 - f(x, y).
So with those two formulas we can easily put together the whole thing:
f(a, 10) * (f(b, 3) * 3 + g(b, 3) * 1) + g(a, 10) * (f(b, -5) * -11 + g(b, -5) * 4)
A general equation of the form:
((z2+z1)/2) + (|z2-z1|/2)*f(a,b)
where f(a,b) = |a-b|/(a-b)
In english:
(midpoint between 2 given z values) +
(distance from midpoint to either z value)*|a-b|/(a-b)
trying this on the original example:
if (a <= 10){
z = 5;
} else {
z = -1;
}
you get:
z1=5 z2=-1
f(a,b)=f(10,a)=|10-a|/(10-a)
plugging these in...
((5-1)/2) + (|5-(-1)|/2)*|10-a|/(10-a)
simplifying to your original z = 3 * s(10 - a) + 2
When applying this to nested conditional:
if (a <= 10) {
... // z1
} else {
... // z2
}
for z1 i get z1 = 2 + |3-b|/(3-b)
for z2 i get -3.5 + 7.5*(|-5-b|/(-5-b)). z1 seems ok but z2 doesn't seem to work since if you tried b=0 you have z2 = -3.5 - 7.5*(1) but since 0>-5 you would expect z2 = 4 since:
if (b <= -5) {
z = -11;
} else {
z = 4;
}
to get the correct expression i swapped the definition of f(a,b) = |a-b|/(a-b) to f(a,b) = |b-a|/(b-a) the new result being z2 = -3.5 + 7.5*(|b+5|/(b+5)) and testing b=0 gives the correct result of 4. This reduces the nested conditional to look like the simpler problem
if (a <= 10) z = 2 + |3-b|/(3-b)
else z = -3.5 + 7.5*(|b+5|/(b+5))
which assuming you know b you can apply the same method above used for the simple case.
I have two if statements in my code, but when both are true, it only only makes the second if. How should I write it?
String s1 = editTextl.getText().toString();
double d = 0;
if (s1.contains("H")) {
d = + H; //H is double with value of 1
}
if (s1.contains("O")) {
d = + O; //O is double with value of 16
}
TextView.setText(" " + getString(R.string.vysledek) + " " + d);
When I type HO in EditText, TextView should return value of 17, but it returns only 16. Why?
You need to write it as follows, replacing = + with +=:
if (s1.contains("H")) {
d += H;
}
if (s1.contains("O")) {
d += O;
}
+= is a so-called compound assignment operator. It adds the right operand to the left operand.
The inverse, =+, just assigns the right operand to the left operand and is equivalent to a simple assignment using =. The unary + operator is ignored.
Replace = + with +=.
= is an assignment operator, + is an unary plus operator that basically does nothing. So you're assigning a new value to the variable.
+= is "add to the current value of left hand side".
double d = 0;
if (s1.contains("H")) {
d += 1; // H is double with value of 1
}
if (s1.contains("O")) {
d += 16; // O is double with value of 16
}
Change the = + assignments to += assignments - d = + H is really just interpreted by the compiler as d = +H. You're not actually adding H to d, you're just setting d = H.
Replace d = + H with d += H and d = + O with d += O
I couldn't figure out how the decrement operator (e--)
works in code below, so i wrote the other class below it
to get the same result. I want to know how the decrement operator
achieves that result in the Power class. - Newbie.
int result, e;
for(int i=0; i < 10; i++) {
result = 1;
e = i;
while(e > 0) {
result *= 2;
e--;
}
System.out.println("2 to the " + i +
" power is " + result);
}
Code written to achieve same result
int result = 1;
for(int i=0; i < 10; i++) {
if (i > 0) {
result*=2;
}
System.out.println("2 to the " + i +
" power is " + result);
}
So the first example is resetting result for each iteration of the main for loop, so it needs to recalculate from scratch each time, where as the second example is keeping the previous computed value. The if in the second example is not needed is it.
The decrement operator modifies the variable on which it's called. So e-- is effectively e = e - 1 (except the overall result of the expression is different, see below).
This code:
result = 1;
e = i;
while(e > 0) {
result *= 2;
e--;
}
starts with result = 1 and then loops for i iterations doubling the value in result. Equivalent code using for which you seem more comfortable with:
result = 1;
for (e = 0; e < i; e++) {
result *= 2;
}
There are two forms of the decrement (and increment) operator: Prefix and postfix, depending on whether the operator is before (prefix) or after (postfix) its operand. Either could be used in the code you were asking about, because the only difference is the result of the expression.
Prefix: Suppose we have x = 5. The expression --x has the value 4: First we decrement x, then we take its new value as the result of the expression.
Postfix: Suppose we had x = 5 (again). The expression x-- has the value 5, with x ending up containing 4: First we grab the current value of x as the result of the expression, then we decrement it (because the -- is after x).
int x, r;
x = 5;
r = --x; // Prefix
System.out.println("r = " + r + ", x = " + x); // "r = 4, x = 4"
x = 5;
r = x--; // Postfix
System.out.println("r = " + r + ", x = " + x); // "r = 5, x = 4"
i figure out that by placing a System.out.println(e) i could "see" the variable "e" behavior in order to make sense of the decrement.
class Power {
public static void main(String args[]) {
int e;
int result;
for(int i=0; i < 10; i++) {
result =1 ;
e = i;
while(e > 0) {
System.out.println(e); // not part of the original program
result *= 2 ;
e--;
System.out.println(e); // not part of the original program
}
//System.out.println("2 to the " + i +
//" power is " + result);
}
This is the output:
C:\Users\enrique\Desktop\Hello.java>java Power: 1, 0, 2, 1, 1, 0, 3
e = 1(iteration 1), 2^1, e (1) decremented to 0, e = 2 (iteration 2), 2^2, e(2) decremented to 1, e = 1 re-enter The while but is ignored as 2^1 is already registered, e (1) decremented to 0, e = 3 (iteration 3), 2^3…
I was just going through some basic stuff as I am learning C. I came upon a question to multiply a number by 7 without using the * operator. Basically it's like this
(x << 3) - x;
Now I know about basic bit manipulation operations, but I can't get how do you multiply a number by any other odd number without using the * operator? Is there a general algorithm for this?
Think about how you multiply in decimal using pencil and paper:
12
x 26
----
72
24
----
312
What does multiplication look like in binary?
0111
x 0101
-------
0111
0000
0111
-------
100011
Notice anything? Unlike multiplication in decimal, where you need to memorize the "times table," when multiplying in binary, you are always multiplying one of the terms by either 0 or 1 before writing it down in the list addends. There's no times table needed. If the digit of the second term is 1, you add in the first term. If it's 0, you don't. Also note how the addends are progressively shifted over to the left.
If you're unsure of this, do a few binary multiplications on paper. When you're done, convert the result back to decimal and see if it's correct. After you've done a few, I think you'll get the idea how binary multiplication can be implemented using shifts and adds.
Everyone is overlooking the obvious. No multiplication is involved:
10^(log10(A) + log10(B))
The question says:
multiply a number by 7 without using * operator
This doesn't use *:
number / (1 / 7)
Edit:
This compiles and works fine in C:
int number,result;
number = 8;
result = number / (1. / 7);
printf("result is %d\n",result);
An integer left shift is multiplying by 2, provided it doesn't overflow. Just add or subtract as appropriate once you get close.
int multiply(int multiplicand, int factor)
{
if (factor == 0) return 0;
int product = multiplicand;
for (int ii = 1; ii < abs(factor); ++ii) {
product += multiplicand;
}
return factor >= 0 ? product : -product;
}
You wanted multiplication without *, you got it, pal!
It's easy to avoid the '*' operator:
mov eax, 1234h
mov edx, 5678h
imul edx
No '*' in sight. Of course, if you wanted to get into the spirit of it, you could also use the trusty old shift and add algorithm:
mult proc
; Multiplies eax by ebx and places result in edx:ecx
xor ecx, ecx
xor edx, edx
mul1:
test ebx, 1
jz mul2
add ecx, eax
adc edx, 0
mul2:
shr ebx, 1
shl eax, 1
test ebx, ebx
jnz mul1
done:
ret
mult endp
Of course, with modern processors, all (?) have multiplication instructions, but back when the PDP-11 was shiny and new, code like this saw real use.
Mathematically speaking, multiplication distributes over addition. Essentially, this means:
x * (a + b + c ...) = (x * a) + (x * b) + (x * c) ...
Any real number (in your case 7), can be presented as a series of additions (such as 8 + (-1), since subtraction is really just addition going the wrong way). This allows you to represent any single multiplication statement as an equivalent series of multiplication statements, which will come up with the same result:
x * 7
= x * (8 + (-1))
= (x * 8) + (x * (-1))
= (x * 8) - (x * 1)
= (x * 8) - x
The bitwise shift operator essentially just multiplies or divides a number by a power of 2. So long as your equation is only dealing with such values, bit shifting can be used to replace all occurrence of the multiplication operator.
(x * 8) - x = (x * 23) - x = (x << 3) - x
A similar strategy can be used on any other integer, and it makes no difference whether it's odd or even.
It is the same as x*8-x = x*(8-1) = x*7
Any number, odd or even, can be expressed as a sum of powers of two. For example,
1 2 4 8
------------------
1 = 1
2 = 0 + 2
3 = 1 + 2
4 = 0 + 0 + 4
5 = 1 + 0 + 4
6 = 0 + 2 + 4
7 = 1 + 2 + 4
8 = 0 + 0 + 0 + 8
11 = 1 + 2 + 0 + 8
So, you can multiply x by any number by performing the right set of shifts and adds.
1x = x
2x = 0 + x<<1
3x = x + x<<1
4x = 0 + 0 + x<<2
5x = x + 0 + x<<2
11x = x + x<<1 + 0 + x<<3
When it comes down to it, multiplication by a positive integer can be done like this:
int multiply(int a, int b) {
int ret = 0;
for (int i=0; i<b; i++) {
ret += b;
}
return ret;
}
Efficient? Hardly. But it's correct (factoring in limits on ints and so forth).
So using a left-shift is just a shortcut for multiplying by 2. But once you get to the highest power-of-2 under b you just add a the necessary number of times, so:
int multiply(int a, int b) {
int ret = a;
int mult = 1;
while (mult <= b) {
ret <<= 1;
mult <<= 1;
}
while (mult < b) {
ret += a;
}
return ret;
}
or something close to that.
To put it another way, to multiply by 7.
Left shift by 2 (times 4). Left shift 3 is 8 which is >7;
Add b 3 times.
One evening, I found that I was extremely bored, and cooked this up:
#include <iostream>
typedef unsigned int uint32;
uint32 add(uint32 a, uint32 b) {
do {
uint32 s = a ^ b;
uint32 c = a & b;
a = s;
b = c << 1;
} while (a & b)
return (a | b)
}
uint32 mul(uint32 a, uint32 b) {
uint32 total = 0;
do {
uint32 s1 = a & (-(b & 1))
b >>= 1; a <<= 1;
total = add(s1, total)
} while (b)
return total;
}
int main(void) {
using namespace std;
uint32 a, b;
cout << "Enter two numbers to be multiplied: ";
cin >> a >> b;
cout << "Total: " << mul(a,b) << endl;
return 0;
}
The code above should be quite self-explanatory, as I tried to keep it as simple as possible. It should work, more or less, the way a CPU might perform these operations. The only bug I'm aware of is that a is not permitted to be greater than 32,767 and b is not permitted to be large enough to overflow a (that is, multiply overflow is not handled, so 64-bit results are not possible). It should even work with negative numbers, provided the inputs are appropriately reinterpret_cast<>.
O(log(b)) method
public int multiply_optimal(int a, int b) {
if (a == 0 || b == 0)
return 0;
if (b == 1)
return a;
if ((b & 1) == 0)
return multiply_optimal(a + a, b >> 1);
else
return a + multiply_optimal(a + a, b >> 1);
}
The resursive code works as follows:
Base case:
if either of the number is 0 ,product is 0.
if b=1, product =a.
If b is even:
ab can be written as 2a(b/2)
2a(b/2)=(a+a)(b/2)=(a+a)(b>>1) where'>>' arithematic right shift operator in java.
If b is odd:
ab can be written as a+a(b-1)
a+a(b-1)=a+2a(b-1)/2=a+(a+a)(b-1)/2=a+(a+a)((b-1)>>1)
Since b is odd (b-1)/2=b/2=b>>1
So ab=a+(2a*(b>>1))
NOTE:each recursive call b is halved => O(log(b))
unsigned int Multiply(unsigned int m1, unsigned int m2)
{
unsigned int numBits = sizeof(unsigned int) * 8; // Not part of the core algorithm
unsigned int product = 0;
unsigned int mask = 1;
for(int i =0; i < numBits; ++i, mask = mask << 1)
{
if(m1 & mask)
{
product += (m2 << i);
}
}
return product;
}
#Wang, that's a good generalization. But here is a slightly faster version. But it assumes no overflow and a is non-negative.
int mult(int a, int b){
int p=1;
int rv=0;
for(int i=0; a >= p && i < 31; i++){
if(a & p){
rv += b;
}
p = p << 1;
b = b << 1;
}
return rv;
}
It will loop at most 1+log_2(a) times. Could be faster if you swap a and b when a > b.
import java.math.BigInteger;
public class MultiplyTest {
public static void main(String[] args) {
BigInteger bigInt1 = new BigInteger("5");
BigInteger bigInt2 = new BigInteger("8");
System.out.println(bigInt1.multiply(bigInt2));
}
}
Shift and add doesn't work (even with sign extension) when the multiplicand is negative. Signed multiplication has to be done using Booth encoding:
Starting from the LSB, a change from 0 to 1 is -1; a change from 1 to 0 is 1, otherwise 0. There is also an implicit extra bit 0 below the LSB.
For example, the number 5 (0101) will be encoded as: (1)(-1)(1)(-1). You can verify this is correct:
5 = 2^3 - 2^2 + 2 -1
This algorithm also works with negative numbers in 2's complement form:
-1 in 4-bit 2's complement is 1111. Using the Booth algorithm: (1)(0)(0)(0)(-1), where there is no space for the leftmost bit 1 so we get: (0)(0)(0)(-1) which is -1.
/* Multiply two signed integers using the Booth algorithm */
int booth(int x, int y)
{
int prev_bit = 0;
int result = 0;
while (x != 0) {
int current_bit = x & 0x1;
if (prev_bit & ~current_bit) {
result += y;
} else if (~prev_bit & current_bit) {
result -= y;
}
prev_bit = current_bit;
x = static_cast<unsigned>(x) >> 1;
y <<= 1;
}
if (prev_bit)
result += y;
return result;
}
The above code does not check for overflow. Below is a slightly modified version that multiplies two 16 bit numbers and returns a 32 bit number so it never overflows:
/* Multiply two 16-bit signed integers using the Booth algorithm */
/* Returns a 32-bit signed integer */
int32_t booth(int16_t x, int16_t y)
{
int16_t prev_bit = 0;
int16_t sign_bit = (x >> 16) & 0x1;
int32_t result = 0;
int32_t y1 = static_cast<int32_t>(y);
while (x != 0) {
int16_t current_bit = x & 0x1;
if (prev_bit & ~current_bit) {
result += y1;
} else if (~prev_bit & current_bit) {
result -= y1;
}
prev_bit = current_bit;
x = static_cast<uint16_t>(x) >> 1;
y1 <<= 1;
}
if (prev_bit & ~sign_bit)
result += y1;
return result;
}
unsigned int Multiply( unsigned int a, unsigned int b )
{
int ret = 0;
// For each bit in b
for (int i=0; i<32; i++) {
// If that bit is not equal to zero
if (( b & (1 << i)) != 0) {
// Add it to our return value
ret += a << i;
}
}
return ret;
}
I avoided the sign bit, because it's kind of not the subject of the post. This is an implementation of what Wayne Conrad said basically. Here is another problem is you want to try more low level math operations. Project Euler is cool!
If you can use the log function:
public static final long multiplyUsingShift(int a, int b) {
int absA = Math.abs(a);
int absB = Math.abs(b);
//Find the 2^b which is larger than "a" which turns out to be the
//ceiling of (Log base 2 of b) == numbers of digits to shift
double logBase2 = Math.log(absB) / Math.log(2);
long bits = (long)Math.ceil(logBase2);
//Get the value of 2^bits
long biggerInteger = (int)Math.pow(2, bits);
//Find the difference of the bigger integer and "b"
long difference = biggerInteger - absB;
//Shift "bits" places to the left
long result = absA<<bits;
//Subtract the "difference" "a" times
int diffLoop = Math.abs(a);
while (diffLoop>0) {
result -= difference;
diffLoop--;
}
return (a>0&&b>0 || a<0&&b<0)?result:-result;
}
If you cannot use the log function:
public static final long multiplyUsingShift(int a, int b) {
int absA = Math.abs(a);
int absB = Math.abs(b);
//Get the number of bits for a 2^(b+1) larger number
int bits = 0;
int bitInteger = absB;
while (bitInteger>0) {
bitInteger /= 2;
bits++;
}
//Get the value of 2^bit
long biggerInteger = (int)Math.pow(2, bits);
//Find the difference of the bigger integer and "b"
long difference = biggerInteger - absB;
//Shift "bits" places to the left
long result = absA<<bits;
//Subtract the "difference" "a" times
int diffLoop = absA;
while (diffLoop>0) {
result -= difference;
diffLoop--;
}
return (a>0&&b>0 || a<0&&b<0)?result:-result;
}
I found this to be more efficient:
public static final long multiplyUsingShift(int a, int b) {
int absA = Math.abs(a);
int absB = Math.abs(b);
long result = 0L;
while (absA>0) {
if ((absA&1)>0) result += absB; //Is odd
absA >>= 1;
absB <<= 1;
}
return (a>0&&b>0 || a<0&&b<0)?result:-result;
}
and yet another way.
public static final long multiplyUsingLogs(int a, int b) {
int absA = Math.abs(a);
int absB = Math.abs(b);
long result = Math.round(Math.pow(10, (Math.log10(absA)+Math.log10(absB))));
return (a>0&&b>0 || a<0&&b<0)?result:-result;
}
In C#:
private static string Multi(int a, int b)
{
if (a == 0 || b == 0)
return "0";
bool isnegative = false;
if (a < 0 || b < 0)
{
isnegative = true;
a = Math.Abs(a);
b = Math.Abs(b);
}
int sum = 0;
if (a > b)
{
for (int i = 1; i <= b; i++)
{
sum += a;
}
}
else
{
for (int i = 1; i <= a; i++)
{
sum += b;
}
}
if (isnegative == true)
return "-" + sum.ToString();
else
return sum.ToString();
}
JAVA:Considering the fact, that every number can be splitted into powers of two:
1 = 2 ^ 0
2 = 2 ^ 1
3 = 2 ^ 1 + 2 ^ 0
...
We want to get x where:
x = n * m
So we can achieve that by doing following steps:
1. while m is greater or equal to 2^pow:
1.1 get the biggest number pow, such as 2^pow is lower or equal to m
1.2 multiply n*2^pow and decrease m to m-2^pow
2. sum the results
Sample implementation using recursion:
long multiply(int n, int m) {
int pow = 0;
while (m >= (1 << ++pow)) ;
pow--;
if (m == 1 << pow) return (n << pow);
return (n << pow) + multiply(n, m - (1 << pow));
}
I got this question in last job interview and this answer was accepted.
EDIT: solution for positive numbers
This is the simplest C99/C11 solution for positive numbers:
unsigned multiply(unsigned x, unsigned y) { return sizeof(char[x][y]); }
Another thinking-outside-the-box answer:
BigDecimal a = new BigDecimal(123);
BigDecimal b = new BigDecimal(2);
BigDecimal result = a.multiply(b);
System.out.println(result.intValue());
public static int multiply(int a, int b)
{
int temp = 0;
if (b == 0) return 0;
for (int ii = 0; ii < abs(b); ++ii) {
temp = temp + a;
}
return b >= 0 ? temp : -temp;
}
public static int abs(int val) {
return val>=0 ? val : -val;
}
public static void main(String[] args) {
System.out.print("Enter value of A -> ");
Scanner s=new Scanner(System.in);
double j=s.nextInt();
System.out.print("Enter value of B -> ");
Scanner p=new Scanner(System.in);
double k=p.nextInt();
double m=(1/k);
double l=(j/m);
System.out.print("Multiplication of A & B=> "+l);
}
package com.amit.string;
// Here I am passing two values, 7 and 3 and method getResult() will
// return 21 without use of any operator except the increment operator, ++.
//
public class MultiplyTwoNumber {
public static void main(String[] args) {
int a = 7;
int b = 3;
System.out.println(new MultiplyTwoNumber().getResult(a, b));
}
public int getResult(int i, int j) {
int result = 0;
// Check for loop logic it is key thing it will go 21 times
for (int k = 0; k < i; k++) {
for (int p = 0; p < j; p++) {
result++;
}
}
return result;
}
}
Loop it. Run a loop seven times and iterate by the number you are multiplying with seven.
Pseudocode:
total = 0
multiply = 34
loop while i < 7
total = total + multiply
endloop
A JavaScript approach for positive numbers
function recursiveMultiply(num1, num2){
const bigger = num1 > num2 ? num1 : num2;
const smaller = num1 <= num2 ? num1 : num2;
const indexIncrement = 1;
const resultIncrement = bigger;
return recursiveMultiplyHelper(bigger, smaller, 0, indexIncrement, resultIncrement)
}
function recursiveMultiplyHelper(num1, num2, index, indexIncrement, resultIncrement){
let result = 0;
if (index === num2){
return result;
}
if ((index+indexIncrement+indexIncrement) >= num2){
indexIncrement = 1;
resultIncrement = num1;
} else{
indexIncrement += indexIncrement;
resultIncrement += resultIncrement;
}
result = recursiveMultiplyHelper(num1, num2, (index+indexIncrement), indexIncrement, resultIncrement);
result += resultIncrement;
console.log(num1, num2, index, result);
return result;
}
Think about the normal multiplication method we use
1101 x =>13
0101 =>5
---------------------
1101
0000
1101
0000
===================
1000001 . => 65
Writing the same above in the code
#include<stdio.h>
int multiply(int a, int b){
int res = 0,count =0;
while(b>0) {
if(b & 0x1)
res = res + (a << count);
b = b>>1;
count++;
}
return res;
}
int main() {
printf("Sum of x+y = %d", multiply(5,10));
return 0;
}
Very simple, pal... Each time when you left shift a number it means you are multiplying the number by 2 which means the answer is (x<<3)-x.
To multiply of two numbers without * operator:
int mul(int a,int b) {
int result = 0;
if(b > 0) {
for(int i=1;i<=b;i++){
result += a;
}
}
return result;
}