Avoiding Floating Point Errors in Python

In the realm of programming, it's essential to understand the intricacies of floating point calculations, as they can introduce unexpected errors if not handled properly. This article explores a practical example that highlights the pitfalls of floating point arithmetic.

The Square Root Problem

Consider a Python function designed to approximate square roots:

def sqrt(num):
    root = 0.0
    while root * root 
Using this function, we encounter surprising results:
>>> sqrt(4)
2.0000000000000013
>>> sqrt(9)
3.00999999999998
Floating point arithmetic explains these inaccuracies. When representing fractional numbers, computers store them as a combination of integer and exponent. Due to limitations in this representation, certain decimal values cannot be represented exactly, leading to approximations.
Understanding the Error
In the code above, the issue lies in the increment used to increase the root value. While we intend to add a value of 0.01, the actual value stored in the floating point register is slightly different and greater than 0.01.
Addressing the Error
To avoid floating point errors, various approaches can be employed:
Using the Decimal Module:
The Decimal module in Python provides more precise decimal arithmetic. By replacing float with Decimal in the code, we get more accurate results:
from decimal import Decimal as D

def sqrt(num):
    root = D(0)
    while root * root 
Now, the function returns precise results, such as:

```
>>> sqrt(4)
Decimal('2.00')
>>> sqrt(9)
Decimal('3.00')
```
Using Non-Floating Point Increments:
If using the Decimal module is not an option, another approach is to use floating point values that are exactly representable in binary, such as 1/2**J. This involves more complex code, but it ensures accurate increments.
Employing Numerical Analysis Techniques:
Numerical analysis offers specialized methods for handling complex mathematical calculations. Techniques like Newton's method can provide precise square root approximations.