Why assuming floating point rounding errors are random is bad

# Julia code goes here

Option 1: Using the Random package

One way to solve the problem of assuming floating point rounding errors are random in Julia is by using the Random package. This package provides functions for generating random numbers, which can be used to simulate floating point rounding errors.

Here is an example code snippet that demonstrates how to use the Random package to generate random floating point rounding errors:

using Random

function simulate_rounding_errors(n::Int)
    errors = Float64[]
    for i in 1:n
        x = rand()
        error = x - round(x)
        push!(errors, error)
    end
    return errors
end

n = 1000
errors = simulate_rounding_errors(n)

In this code, we define a function simulate_rounding_errors that takes an integer n as input and returns an array of n random floating point rounding errors. We use the rand function from the Random package to generate random numbers between 0 and 1, and then calculate the rounding error by subtracting the rounded value from the original value. The push! function is used to add each error to the errors array.

Option 2: Analyzing the distribution of rounding errors

Another approach to solving the problem is by analyzing the distribution of rounding errors. By studying the characteristics of rounding errors, we can gain insights into their behavior and make informed decisions about how to handle them.

Here is a code snippet that demonstrates how to analyze the distribution of rounding errors:

function analyze_rounding_errors(n::Int)
    errors = Float64[]
    for i in 1:n
        x = rand()
        error = x - round(x)
        push!(errors, error)
    end
    return errors
end

n = 1000
errors = analyze_rounding_errors(n)

# Calculate mean and standard deviation of rounding errors
mean_error = mean(errors)
std_error = std(errors)

In this code, we define a function analyze_rounding_errors that takes an integer n as input and returns an array of n random floating point rounding errors. We use the same approach as in the previous option to calculate the rounding errors. After generating the errors, we calculate the mean and standard deviation of the errors using the mean and std functions, respectively.

Option 3: Using a statistical test

A third option is to use a statistical test to determine whether the assumption of random floating point rounding errors holds. This approach involves comparing the observed rounding errors to a theoretical distribution and performing a hypothesis test to assess the randomness of the errors.

Here is a code snippet that demonstrates how to use a statistical test to assess the randomness of rounding errors:

using HypothesisTests

function test_rounding_errors(n::Int)
    errors = Float64[]
    for i in 1:n
        x = rand()
        error = x - round(x)
        push!(errors, error)
    end
    return errors
end

n = 1000
errors = test_rounding_errors(n)

# Perform a hypothesis test for randomness
test_result = israndom(errors)

In this code, we define a function test_rounding_errors that takes an integer n as input and returns an array of n random floating point rounding errors. We use the same approach as in the previous options to calculate the rounding errors. After generating the errors, we use the israndom function from the HypothesisTests package to perform a hypothesis test for randomness. The israndom function returns a test result indicating whether the observed errors are consistent with a random distribution.

After considering the three options, it is difficult to determine which one is better without more context about the specific problem and requirements. Option 1 provides a straightforward way to simulate rounding errors, while Option 2 allows for deeper analysis of the errors’ distribution. Option 3 offers a statistical approach to assess the randomness of the errors. The best option depends on the specific needs and goals of the problem at hand.