Question

Write a short \(\mathrm{C}++\) function that takes a positive double value \(x\) and returns the number of times we can divide \(x\) by 2 before we get a number less than 2.

Short answer

Define `countDivisionsByTwo` function, initialize `count` to `0`, use a while loop to divide `x` by `2`, and return `count`.

Step-by-step solution

- Function Definition

Start by defining a C++ function named `countDivisionsByTwo`. The function should take a single parameter `x` of type `double`.

- Initialization

Inside the function, declare an integer variable `count` and initialize it to `0`. This variable will keep track of the number of times we divide `x` by `2`.

- While Loop

Use a while loop to continuously divide `x` by `2` as long as `x` is greater than or equal to `2`. On each iteration of the loop, divide `x` by `2` and increment the `count` variable by `1`.

- Return the Count

After the loop exits (when `x` is less than `2`), return the `count` variable, which now contains the number of times `x` was divided by `2`.

- Complete Function

Here is the complete function:```cppint countDivisionsByTwo(double x) { int count = 0; while (x >= 2) { x /= 2; count++; } return count;}```

Key concepts

C++ Functions

In C++, functions allow you to break your code into smaller, reusable pieces. Each function in C++ has a specific purpose and can take input parameters and return a value.
For example, in the provided code, the function `countDivisionsByTwo` takes a double value `x` and returns an integer indicating how many times `x` can be divided by 2 before it becomes less than 2.
Here's why functions are useful:

• Reusability: You can call the same function multiple times with different inputs.
• Modularity: Functions help you organize your code into logical sections.
• Ease of Testing: Smaller functions are easier to test and debug compared to a monolithic block of code.

In our example function, we see how defining the function and separating tasks into meaningful steps makes the code more readable and maintainable.

While Loop in C++

A while loop in C++ repeatedly executes a block of code as long as a specified condition is true. This type of loop is useful when the number of iterations is not known beforehand and depends on a certain condition being met.
An example is inside our `countDivisionsByTwo` function. The loop continues to divide `x` by 2 until `x` is less than 2:
```cppwhile (x >= 2) { x /= 2; count++;}```
Here's how the while loop works in this context:

• **Initialization:** Before the loop starts, the counter `count` is set to `0`.
• **Condition Checking:** The loop checks if `x` is still greater than or equal to 2.
• **Iteration:** If the condition is met, `x` is divided by `2` and the counter `count` is incremented by `1`.
• **Repeat:** The loop continues until `x` is no longer greater than or equal to 2.

The while loop ensures that we keep track of how many times `x` is divided by 2, fulfilling the purpose of our function.

Integer Division

Integer division in C++ is the division of one integer by another, resulting in an integer quotient. For example, `5 / 2` yields `2` as the fractional part is discarded. This is different from floating-point division which retains the decimal part.
In our function `countDivisionsByTwo`, we use floating-point division (`x /= 2`) because `x` is a double:
```cpp x /= 2;```
Here’s why understanding numerical types and division is important:

• **Precision:** In operations involving floating-point numbers, precision might be lost, which could affect the result.
• **Behavior Differences:** Integer division and floating-point division behave differently, so knowing the type of your variables is crucial.
• **Casting:** Sometimes you need to cast between types to ensure the correct type of division.
  

Remember that in the context of our function, since `x` is a double, each division by 2 is accurate to the floating-point precision defined by the `double` type.