Are there a lot of systems that define arctangent but not pi?

The cheap Sinclair Scientific calculator (1974) was rather primitive, so instead of storing constants internally, it printed constants such as e and pi on the case. The calculator was rather inaccurate, so you were much better off using 3.14159 from the case than computing 4×atan(1), which gave 3.1440.

It'd be interesting to see their implementation and the trade-offs its designers made.

I'm glad you asked :-) I reverse-engineered the Sinclair Scientific calculator, documented its code, and built a simulator: http://files.righto.com/calculator/sinclair_scientific_simul...

The short answer is that they used a 4-function calculator chip with just 320 words of ROM and managed to reprogram it into a cheap scientific calculator, a remarkable feat. The tradeoff was that the calculator was very slow and inaccurate.

That's amazing! Thank you for sharing!

C/C++ don't define PI but have an atan function. According to the TIOBE Index for September 2020 (https://www.tiobe.com/tiobe-index/) those languages are both in the top 5 of languages used by developers.

C++20 defines mathematical constants including Pi.

https://en.cppreference.com/w/cpp/numeric/constants

Thanks for the heads up! I haven't been paying any attention to C++20. You must be on the cutting edge of C++ to know that - I ended up having to work a bit on my Macbook Pro to get a version of gcc working that defines std::numbers::pi.

They don’t but POSIX requires M_PI.

Yes - Fortan doesn't have a built-in pi constant, but does have trigonometric functions. Usually programs define pi based on atan or some other trig relation.

bc calculator doesn't (or didn't) define pi, but has atan. I've used "4*atan(1) method" in bc to get pi.

This works, `echo 'scale=100;4*a(1)' | bc -lq`

I assume the later part of you comment refers to the (in)famous attempt by the Indiana state assembly to legalize Pi to be exactly equal to 3.

That thought had come across my mind, but no, I've seen engineering requirements stating something to the effect that the value of PI used in these calculations is to be 3.1415, for example.