 A processor from 50 years ago, the historian Intel 4040managed to calculate 2035 decimal places of the value of pi in just under 70 hours of work.

Il pi greco (π) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. Regardless of the size of the circle, the ratio of the circumference to the diameter remains constant and approximately equal to 3.14159. The value of pi it is one of the most important and most used constants in mathematics and many other disciplines. His irrationality (cannot be expressed as an exact fraction) and its infinite, non-periodic sequence of decimals make calculating the value of pi a topic of great interest to mathematical scholars. Even if, in practice, few decimal places they are usually sufficient for most applications.

Google Cloud technicians managed to calculate 100,000 billion decimal places of pi in mid-2022 (in approximately 157 days of work) by relying on the company’s infrastructure and exploiting aarchitecture strongly parallelised. Without reaching the heights reached by Google, even the most modern processors used on our PCs and mobile devices are still able to calculate an ever-increasing number of decimal places of π in a relatively short time. Calculate millions of decimal places it can even be a matter of minutes.

What a leap forward information technology has made in the space of a few decades!

## Intel 4040 passes the Pi-2035 test in less than 70 hours

The microprocessor Intel 4040, introduced in 1974, has gone down in history as a key component in the evolution of technology. Designed by Federico FagginIntel 4040 was one chip a 4 bit, meaning it could process instructions and data on up to 4 bits at a time. He had one clock frequency maximum of about 740 kHz and could execute about 60,000 instructions per second.

Initially designed for use in calculators, the Intel 4040 had a much more widespread impact by pioneering the miniaturization of integrated circuits and playing a fundamental role in the emerging personal computers.

An independent researcher managed to speed up the Intel 4040 by 180,000%, making it surpass the Pi-2035 test, or arriving at the calculation of 2035 decimal places of pi. With a series of optimizations focused on the implementation of the algorithm, the use of the system’s working memory, the format of fixed precision numbers and the search for more effective paths, it was possible to reduce the time for calculating 2035 decimals of pi from 14 and a half years at “just” 70 hours of work.

A truly incredible result if you consider that theENIAC (Electronic Numerical Integrator and Computer), the first large digital electronic computer dating back to the Second World War (completed in 1945), passed the Pi-2035 test in 1950 in about 70 hours. But it was a computer that it took up an entire roomhad a structure composed of thermionic valves and could perform a wide range of calculations, including those relating to ballistic trajectories for military purposes.

## Calculating thousands of decimal places of pi is trivial today

Calculating pi with a high degree of accuracy is still a benchmark interesting for CPUs in 2023. However, we recommend using other software tools as a CPU stress test. Also because software like Simple Pi Calculator does not offer a sufficient degree of parallelization in order to evaluate the real potential of the CPU. A good approach could be to use GNU Parallel and a script that allows you to optimize the decimal calculation task.

Modern processors can in fact use advanced algorithms to calculate the decimal places of π. There are several computational methods although one of the most common methods is to use thealgorithm in Gauss-Legendre.

The algorithm is based on the method of iterative formulas that progressively refine the decimal places of π. It uses the rapid convergence of some mathematical series and takes advantage of the computing power of modern processors to perform multiple iterations, calculating the decimal places of π efficiently.