Understanding Floating Point Error through a Simple Example
The concept of floating point error arises when using floating point variables to represent numerical values due to their limited precision. Let's delve into a simple example to illustrate this error.
Example in C++
Consider the following scenario: An event has a probability 'p' of success. We perform the event 10 times independently, and we want to calculate the probability of exactly 2 successful trials. The calculation is expressed as:
double p_2x_success = pow(1-p, (double)8) * pow(p, (double)2) * (double)choose(8, 2);
Floating Point Error
The aforementioned calculation involves operations that can potentially introduce floating point error. When performing mathematical operations with floating point numbers, the computer may truncate or round the results to fit within the limited range of floating point representation.
Accumulation of Error
In this example, the probability of exactly 2 successful trials is computed using a product of terms involving exponentiation and the binomial coefficient. Each operation can introduce a small error due to the limited precision of floating point numbers. As these operations are multiplied, the errors can accumulate, leading to a deviation from the exact result.
Visualization of Error
To visualize the accumulation of floating point error, we can plot a graph of the function f(k):
f(k) = (1 - p)^k * p^k
where k is the number of trials. Using logarithmic scales, we can observe that the error increases as k becomes larger. This indicates that floating point error becomes more significant with repeated operations, especially for large values of k.
Practical Implications
Understanding floating point error is essential in scenarios where precision is crucial. In financial calculations, scientific simulations, or any application that involves complex numerical operations, the impact of floating point error must be considered to ensure the accuracy of the results.
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