Problem Statement

In Computer programming, there are often situations where finding the absolute value of an integer is required. The conventional approach involves using conditional statements or branching to handle positive and negative integers separately. However, this may not be efficient in some scenarios, especially when execution speed is crucial. The challenge is to find a way to calculate the absolute value of an integer without using conditional statements.

Problem Understanding

Consider the problem of finding the absolute value of an integer x. The absolute value of a number is its distance from zero on the number line, disregarding its sign. For positive integers, the absolute value is the number itself, and for negative integers, it is the negation of the number.

Example:

Input: x = -8
Output: Absolute value = 8

Input: x = 15
Output: Absolute value = 15

Solutions

One clever way to find the absolute value without branching is by using bit manipulation. The idea is to exploit the properties of two's complement representation of integers. For positive integers, their two's complement representation is the same as their binary representation. For negative integers, their two's complement is obtained by inverting all the bits and adding 1.

The solution involves two steps:

1. Mask out the sign bit.
2. Subtract the original value with the sign bit masked.

Example:

Let's take the example of x = -8:

1. Original binary representation of -8: 1111 1111 1111 1111 1111 1111 1111 1000.
2. Mask out the sign bit: 0000 0000 0000 0000 0000 0000 0000 1000 (mask = 1<< (sizeof(int)*8 -1))
3. Subtract original value with the sign bit masked: 1111 1111 1111 1111 1111 11111 1111 1000 - 0000 0000 0000 0000 0000 0000 0000 1000 = 1111 1111 1111 1111 1111 1111 1111 0000