Write a function subtract(x, y) that returns x-y where x and y are integers. The function should not use any of the arithmetic operators (+, ++, –, -, … etc).

The idea is to use bitwise operators. Like addition, Write a function Add() that returns sum of two integers. The function should not use any of the arithmetic operators (+, ++, –, -, … etc). Sum of two bits can be obtained by performing XOR (^) of the two bits. Carry bit can be obtained by performing AND (&) of two bits. We can extend this logic for integers. If x and y don’t have set bits at same position(s), then bitwise XOR (^) of x and y gives the sum of x and y. To incorporate common set bits also, bitwise AND (&) is used. Bitwise AND of x and y gives all carry bits. We calculate (x & y) << 1 and add it to x ^ y to get the required result. The idea is to use subtractor logic.

The truth table for the half subtractor is given below.

X Y Diff Borrow

0 0 0 0

0 1 1 1

1 0 1 0

1 1 0 0

From the above table one can draw the Karnaugh map for “difference” and “borrow”.

So, Logic equations are:

```
Diff = y ⊕ x
Borrow = x' . y
```

Following is implementation based on above equations.

`<?php`

`// PHP Program to subtract two Number`

`// without using arithetic operater`

`function`

`subtract(`

`$x`

`, `

`$y`

`)`

`{`

` `

` `

`// Iterate till there is no carry`

` `

`while`

`(`

`$y`

`!= 0)`

` `

`{`

` `

` `

`// borrow contains common set`

` `

`// bits of y and unset`

` `

`// bits of x`

` `

`$borrow`

`= (~`

`$x`

`) & `

`$y`

`;`

` `

`// Subtraction of bits of x`

` `

`// and y where at least`

` `

`// one of the bits is not set`

` `

` `

`$x`

`= `

`$x`

`^ `

`$y`

`;`

` `

`// Borrow is shifted by one so`

` `

`// that subtracting it from`

` `

`// x gives the required sum`

` `

` `

`$y`

`= `

`$borrow`

`<< 1;`

` `

`}`

` `

`return`

`$x`

`;`

`}`

` `

`// Driver Code`

` `

`$x`

`= 29; `

`$y`

`= 13;`

` `

`echo`

`"x - y is "`

`, subtract(`

`$x`

`,`

`$y`

`);`

`?>`

**Output :**

x - y is 16