Reimplementing the Cauchy Distribution in Fulgurance (C++ Statistics Library)

Introduction

Fulgurance is my C++ statistics library where I reimplemented several probability distributions from scratch, including the Cauchy distribution. The goal of Fulgurance is to provide flexible statistical tools in modern C++, without relying heavily on external libraries. It’s also a playground to explore how distributions work internally, both mathematically and computationally.

Why Reimplement Cauchy?

The Cauchy distribution is a well-known heavy-tailed distribution, often used in physics, Bayesian inference, and robustness testing. Unlike the normal distribution, its mean and variance are undefined — which makes it a good candidate to test statistical implementations. Rebuilding it from scratch in C++ was a great exercise in both numerical computation and random number generation.

The Four Building Blocks

1. dcauchy — Density Function

The probability density function (PDF) of the Cauchy distribution is:

f(x) = 1 / (scale · π · (1 + ((x - location)/scale)²))

In Fulgurance:

std::vector<double> dcauchy(std::vector<double> &x,
                            double location = 0,
                            double scale = 1);

This returns the probability density values for each element in the vector x.

2. pcauchy — Cumulative Distribution Function

The cumulative distribution function (CDF) of the Cauchy distribution can be written in closed form using the arctangent. I implemented it as:

std::vector<double> pcauchy(std::vector<double> &x,
                            double location = 0,
                            double scale = 1,
                            double step = 0.01);

The formula used is:

F(x) = (1/π) · atan((x − location)/scale) + 0.5

I also normalize relative to the first value of the input vector, so results can be interpreted as probabilities over the input set.

3. qcauchy — Quantile Function

The quantile (inverse CDF) is useful for generating samples or computing thresholds. It is defined as:

Q(p) = location + scale · tan(π(p − 0.5))

In Fulgurance:

std::vector<double> qcauchy(std::vector<double> &p,
                            double location = 0,
                            double scale = 1);

It takes a vector of probabilities p and returns the corresponding quantile values.

4. rcauchy — Random Number Generation

This one was more fun (and a little hacky 😅). I wanted to generate pseudo-random numbers following a Cauchy distribution without relying directly on the standard library’s random engines. So I improvised by using:

• System clock timestamps (std::chrono)
• Modulo arithmetic to get variability
• Short std::this_thread::sleep_for calls to add entropy
• The tan() transform to map uniform-like values into a Cauchy distribution

The implementation is:

std::vector<double> rcauchy(unsigned int n,
                            double location = 0,
                            double scale = 1);

It generates n values distributed approximately like Cauchy(location, scale).

Limitations

• rcauchy is not a cryptographically secure or statistically rigorous RNG — it’s a custom experiment to simulate randomness.
• For production use, you would normally rely on std::cauchy_distribution from <random>.
• The goal here was educational: to understand how each component of the distribution is built.

Conclusion

Reimplementing the Cauchy distribution in Fulgurance was both a math refresher and a programming challenge. I now have my own versions of:

• dcauchy — density function
• pcauchy — cumulative distribution function
• qcauchy — quantile function
• rcauchy — random number generator

And this is only one example: I’ve implemented dozens of other statistical laws in Fulgurance, all available here:
👉 GitHub Repository: Fulgurance

Even though C++ already provides some distributions in <random>, reimplementing them from scratch gave me a much deeper understanding of probability distributions, and was a fun addition to Fulgurance.