1. What is Floating Point Error?

Floating point error refers to the difference between an approximate numerical value and its exact mathematical value. This is particularly important in numerical computations where finite precision can lead to significant errors.

2. How Does the Calculator Work?

The calculator uses the relative error formula:

Where:

• \( \text{Approx} \) — The approximate or computed value
• \( \text{Exact} \) — The exact or theoretical value

Explanation: The formula calculates the relative error as a percentage, which shows how large the error is compared to the exact value.

3. Importance of Error Calculation

Details: Understanding floating point errors is crucial in numerical analysis, scientific computing, and any field where precise calculations are needed. It helps assess the accuracy of computational methods.

4. Using the Calculator

Tips: Enter both the approximate and exact values. The exact value cannot be zero (division by zero is undefined). The calculator will compute the relative error percentage.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between absolute and relative error?
A: Absolute error is the simple difference (Approx - Exact), while relative error shows this difference as a percentage of the exact value.

Q2: What are acceptable error ranges?
A: This depends on the application. Scientific computing often aims for errors less than 0.1%, while engineering might tolerate 1-5%.

Q3: Why can't the exact value be zero?
A: The relative error formula divides by the exact value, making zero undefined. For values near zero, consider using absolute error instead.

Q4: How does floating point representation affect errors?
A: Computers store numbers with finite precision, leading to rounding errors that accumulate in calculations.

Q5: Can this calculator handle very small numbers?
A: Yes, the calculator uses PHP's float type which can handle numbers as small as about 1.8e-308, though precision decreases with very small numbers.