how many bits of precision are lost in a computer when we carry out the subtraction x-sinx for x=1/2?

To analyze how many bits of precision are lost in the subtraction \( x - \sin(x) \) for \( x = \frac{1}{2} \), we need to understand **catastrophic cancellation** — a phenomenon where subtracting two nearly equal numbers results in a significant loss of precision.

Here's the step-by-step process:

### 1. Compute \( x \) and \( \sin(x) \):
Given \( x = \frac{1}{2} \), compute \( \sin\left(\frac{1}{2}\right) \).

\[
\sin\left(\frac{1}{2}\right) \approx 0.4794255386
\]

So the subtraction becomes:

\[
x - \sin(x) = 0.5 - 0.4794255386 \approx 0.0205744614
\]

### 2. Relative difference:
The result \( 0.0205744614 \) is much smaller than the original value of \( x = 0.5 \), meaning the two values \( x \) and \( \sin(x) \) are very close to each other. This proximity suggests that precision will be lost during the subtraction due to limited floating-point accuracy.

### 3. Estimate precision loss:
To estimate the number of bits lost, we can use the following formula related to precision loss in subtraction:

\[
\text{Precision loss (bits)} \approx \log_2\left(\frac{x}{x - \sin(x)}\right)
\]

Plugging in the values:

\[
\frac{x}{x - \sin(x)} = \frac{0.5}{0.0205744614} \approx 24.30
\]

Now, applying the logarithm base 2:

\[
\log_2(24.30) \approx 4.60
\]

### 4. Conclusion:
Approximately **4.6 bits of precision** are lost during the subtraction \( x - \sin(x) \) for \( x = \frac{1}{2} \). 

Would you like more details or have any questions?

Here are 5 related questions:
1. What is catastrophic cancellation in numerical analysis?
2. How does floating-point arithmetic work in computers?
3. Why does subtraction of close numbers lead to precision loss?
4. How is precision represented in floating-point formats like IEEE 754?
5. How can one avoid or minimize precision loss in numerical computations?

**Tip**: When dealing with numerical subtraction of nearly equal values, consider using reformulations or higher precision data types to reduce the impact of catastrophic cancellation.

How many bits of precision are lost in a computer when we carry out the subtraction x - sin(x) for x = 1/2?

Explore how many bits of precision are lost in the subtraction of x - sin(x) for x = 1/2, with a detailed analysis on catastrophic cancellation and floating-point arithmetic. Suitable for undergraduate students in mathematics or computer science.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation