Why Does My C Probability Calculation Incur Floating Point Errors?

Understanding Floating Point Error Through a Simple C Example

In the realm of programming, floating point variables can introduce errors due to their finite precision. This phenomenon, known as floating point error, can arise when performing mathematical operations involving such variables.

Consider the following C code snippet, which attempts to calculate the probability of exactly two successes in a sequence of 10 independent events, where each event has a probability 'p' of success:

double p_2x_success = pow(1-p, (double)8) * pow(p, (double)2) * (double)choose(8, 2);

The variables 'pow()' and 'choose()' represent mathematical functions.

Now, let's examine whether this code faces potential floating point errors. As the value of 'k' in the above equation increases, the magnitude of the terms 'pow(1-p, k)' and 'choose(k, 2)' will become very large. This can lead to an accumulation of floating point errors, as these operations are performed on increasingly large numbers.

To visualize this, let's graph the equation 'f(k)':

f(k) = pow(1-p, k) * pow(p, k) * choose(k, 2)

where both 'X' and 'Y' are in logarithmic scale.

For a computer with 32-bit floating point representation, we would expect 'f(k)' to be zero for all values of 'k'. However, due to floating point errors, the error increases significantly with larger 'k' values. This is evident from the graph shown below:

[Image of XY Graph with Logarithmic Scale]

In this graph, the X-axis represents 'k', and the Y-axis represents the absolute value of the error. As 'k' increases, the error accumulation becomes more pronounced.

Therefore, the code snippet provided is indeed susceptible to floating point errors due to the accumulation of round-off errors in the calculation of probabilities.

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