Computing The Error Bound Of Floating-point Expression

by ADMIN 55 views

Introduction

Floating-point arithmetic is a fundamental component of modern computing, used extensively in various fields such as scientific simulations, engineering, and data analysis. However, due to the inherent limitations of floating-point representations, errors can occur during calculations. In this article, we will discuss how to compute the maximum absolute and relative error of a given IEEE-754 floating-point expression.

Understanding Floating-Point Errors

Floating-point errors arise from the finite precision of floating-point numbers, which can lead to rounding errors during arithmetic operations. These errors can be categorized into two types: absolute error and relative error.

  • Absolute Error: The absolute error is the difference between the exact result and the computed result. It represents the maximum amount by which the computed result can differ from the exact result.
  • Relative Error: The relative error is the ratio of the absolute error to the exact result. It represents the percentage by which the computed result can differ from the exact result.

Computing Error Bounds

To compute the error bounds of a floating-point expression, we need to analyze the expression and identify the sources of error. In this case, the given expression is:

a.y + (x - a.x) * ((b.y - a.y) / (b.x - a.x))

We will use the following assumptions:

  • The optimizer has already optimized the expression for performance.
  • The expression is evaluated using IEEE-754 floating-point arithmetic.

Step 1: Identify the Sources of Error

The expression contains several sources of error:

  • Rounding Errors: The expression involves division, multiplication, and addition operations, which can introduce rounding errors.
  • Cancelling Errors: The expression contains terms that can cancel each other out, leading to a loss of precision.

Step 2: Analyze the Expression

To analyze the expression, we need to identify the critical paths and the sources of error along those paths. The critical paths are the paths that contribute the most to the overall error.

  • Critical Path 1: The first critical path is the path from a.y to the final result. This path involves a single addition operation, which can introduce a rounding error.
  • Critical Path 2: The second critical path is the path from (x - a.x) to the final result. This path involves a multiplication operation, which can introduce a rounding error.

Step 3: Compute the Error Bounds

To compute the error bounds, we need to estimate the maximum absolute and relative errors along each critical path.

  • Maximum Absolute Error: The maximum absolute error along each critical path is the maximum of the rounding errors introduced by each operation.
  • Maximum Relative Error: The maximum relative error along each critical path is the ratio of the maximum absolute error to the exact result.

Computing the Error Bounds for Critical Path 1

For Critical Path 1, the expression is:

a.y + (x - a.x)

The maximum absolute error along this path is the maximum of the rounding errors introduced by the addition operation. Assuming a rounding error of ε, the maximum absolute error is:

|a + (x - a.x) - (a.y + (x - a.x))| ≤ ε

The maximum relative error along this path is the ratio of the maximum absolute error to the exact result:

|(a.y + (x - a.x)) / (a.y + (x - a.x)) - 1| ≤ ε / (|a.y + (x - a.x)|)

Computing the Error Bounds for Critical Path 2

For Critical Path 2, the expression is:

(x - a.x) * ((b.y - a.y) / (b.x - a.x))

The maximum absolute error along this path is the maximum of the rounding errors introduced by the multiplication and division operations. Assuming rounding errors of ε1 and ε2, respectively, the maximum absolute error is:

|(x - a.x) * ((b.y - a.y) / (b.x - a.x)) - ((x - a.x) * ((b.y - a.y) / (b.x - a.x)))| ≤ |x - a.x| * |(b.y - a.y) / (b.x - a.x)| * (ε1 + ε2)

The maximum relative error along this path is the ratio of the maximum absolute error to the exact result:

|((x - a.x) * ((b.y - a.y) / (b.x - a.x))) / ((x - a.x) * ((b.y - a.y) / (b.x - a.x))) - 1| ≤ |x - a.x| * |(b.y - a.y) / (b.x - a.x)| * (ε1 + ε2) / (|(x - a.x) * ((b.y - a.y) / (b.x - a.x))|)

Conclusion

In this article, we discussed how to compute the maximum absolute and relative error of a given IEEE-754 floating-point expression. We analyzed the expression and identified the sources of error, computed the error bounds for each critical path, and estimated the maximum absolute and relative errors. The results can be used to optimize the expression for performance and to ensure the accuracy of the results.

References

  • [1] IEEE 754-2008 Standard for Floating-Point Arithmetic
  • [2] Goldberg, D. (1991). What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys, 23(1), 5-48.

Code

The following code snippet demonstrates how to compute the error bounds for the given expression using the techniques discussed in this article:

#include <stdio.h>
#include <math.h>

// Function to compute the error bounds
void compute_error_bounds(double a_x, double a_y, double b_x, double b_y, double x, double *error_abs, double *error_rel) {
// Compute the maximum absolute error for Critical Path 1
double error_abs_path1 = fabs(a_y + (x - a_x) - (a_y + (x - a_x)));


// Compute the maximum relative error for Critical Path 1
double error_rel_path1 = error_abs_path1 / (fabs(a_y + (x - a_x)));

// Compute the maximum absolute error for Critical Path 2
double error_abs_path2 = fabs((x - a_x) * ((b_y - a_y) / (b_x - a_x)) - ((x - a_x) * ((_y - a_y) / (b_x - a_x))));

// Compute the maximum relative error for Critical Path 2
double error_rel_path2 = error_abs_path2 / (fabs((x - a_x) * ((b_y - a_y) / (b_x - a_x))));

// Compute the overall maximum absolute error
*error_abs = fmax(error_abs_path1, error_abs_path2);

// Compute the overall maximum relative error
*error_rel = fmax(error_rel_path1, error_rel_path2);



}


int main() {
double a_x = 1.0;
double a_y = 2.0;
double b_x = 3.0;
double b_y = 4.0;
double x = 5.0;


double error_abs, error_rel;

compute_error_bounds(a_x, a_y, b_x, b_y, x, &amp;error_abs, &amp;error_rel);

printf(&quot;Maximum Absolute Error: %f\n&quot;, error_abs);
printf(&quot;Maximum Relative Error: %f\n&quot;, error_rel);

return 0;



}

Introduction

In our previous article, we discussed how to compute the maximum absolute and relative error of a given IEEE-754 floating-point expression. In this article, we will answer some frequently asked questions related to computing error bounds for floating-point expressions.

Q: What is the purpose of computing error bounds for floating-point expressions?

A: Computing error bounds for floating-point expressions is essential to ensure the accuracy of the results. By estimating the maximum absolute and relative errors, we can determine the reliability of the results and make informed decisions about the precision of the calculations.

Q: How do I identify the sources of error in a floating-point expression?

A: To identify the sources of error, you need to analyze the expression and identify the operations that can introduce rounding errors. These operations include division, multiplication, and addition. You should also consider the critical paths in the expression, which are the paths that contribute the most to the overall error.

Q: What is the difference between absolute error and relative error?

A: The absolute error is the difference between the exact result and the computed result. It represents the maximum amount by which the computed result can differ from the exact result. The relative error, on the other hand, is the ratio of the absolute error to the exact result. It represents the percentage by which the computed result can differ from the exact result.

Q: How do I compute the error bounds for a floating-point expression?

A: To compute the error bounds, you need to analyze the expression and identify the sources of error. You should then estimate the maximum absolute and relative errors along each critical path. The maximum absolute error is the maximum of the rounding errors introduced by each operation, while the maximum relative error is the ratio of the maximum absolute error to the exact result.

Q: What are the assumptions made when computing error bounds for floating-point expressions?

A: When computing error bounds for floating-point expressions, we assume that the optimizer has already optimized the expression for performance. We also assume that the expression is evaluated using IEEE-754 floating-point arithmetic.

Q: Can I use the same method to compute error bounds for other types of expressions?

A: No, the method described in this article is specific to floating-point expressions. Other types of expressions, such as integer expressions, may require different methods to compute error bounds.

Q: How do I implement the error bound computation in code?

A: The implementation of the error bound computation in code depends on the programming language and the specific requirements of the application. However, the basic steps remain the same: analyze the expression, identify the sources of error, estimate the maximum absolute and relative errors, and compute the overall error bounds.

Q: What are the limitations of the error bound computation method described in this article?

A: The error bound computation method described in this article assumes that the expression is evaluated using IEEE-754 floating-point arithmetic. It also assumes that the optimizer has already optimized the expression for. In practice, these assumptions may not always hold, and the actual error bounds may differ from the estimated values.

Q: Can I use the error bound computation method to optimize the expression for performance?

A: Yes, the error bound computation method can be used to optimize the expression for performance. By analyzing the expression and identifying the sources of error, you can determine which operations are most critical to the overall error and optimize those operations for performance.

Conclusion

In this article, we answered some frequently asked questions related to computing error bounds for floating-point expressions. We discussed the purpose of computing error bounds, how to identify the sources of error, and how to compute the error bounds. We also addressed some common limitations and assumptions made when computing error bounds. By understanding these concepts, you can ensure the accuracy of your results and make informed decisions about the precision of your calculations.

References

  • [1] IEEE 754-2008 Standard for Floating-Point Arithmetic
  • [2] Goldberg, D. (1991). What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys, 23(1), 5-48.

Code

The following code snippet demonstrates how to compute the error bounds for a floating-point expression using the techniques described in this article:

#include <stdio.h>
#include <math.h>

// Function to compute the error bounds
void compute_error_bounds(double a_x, double a_y, double b_x, double b_y, double x, double *error_abs, double *error_rel) {
// Compute the maximum absolute error for Critical Path 1
double error_abs_path1 = fabs(a_y + (x - a_x) - (a_y + (x - a_x)));


// Compute the maximum relative error for Critical Path 1
double error_rel_path1 = error_abs_path1 / (fabs(a_y + (x - a_x)));

// Compute the maximum absolute error for Critical Path 2
double error_abs_path2 = fabs((x - a_x) * ((b_y - a_y) / (b_x - a_x)) - ((x - a_x) * ((b_y - a_y) / (b_x - a_x))));

// Compute the maximum relative error for Critical Path 2
double error_rel_path2 = error_abs_path2 / (fabs((x - a_x) * ((b_y - a_y) / (b_x - a_x))));

// Compute the overall maximum absolute error
*error_abs = fmax(error_abs_path1, error_abs_path2);

// Compute the overall maximum relative error
*error_rel = fmax(error_rel_path1, error_rel_path2);



}


int main() {
double a_x = 1.0;
double a_y = 2.0;
double b_x = 3.0;
double b_y = 4.0;
double x = 5.0;


double error_abs, error_rel;

compute_error_bounds(a_x, a_y, b_x, b_y, x, &amp;error_abs, &amp;error_rel);

printf(&quot;Maximum Absolute Error: %f\n&quot;, error_abs);
printf(&quot;Maximum Relative Error: %f\n&quot;, error_rel);

return 0;



}

Note that this code snippet is simplified example and may not accurately represent the actual error bounds for the given expression. The error bounds should be computed using a more sophisticated method, such as the one described in the article.