Implementing the Transcendental Functions in Ivy

Summary

Towards the end of 2014, in need of pleasant distraction, I began writing, in Go, my second pseudo-APL, called Ivy. Over the years, Ivy's capabilities expanded to the point that it's now being used to do things such as to design floating-point printing algorithms. That was unexpected, doubly so because among the set of array languages founded by APL, Ivy remains one of the weakest.Also unexpected was how the arrival of high-precision floating point in Go's libraries presented me with some mathematical challenges. Computing functions such as sine and cosine when there are more bits than the standard algorithms provide required polishing up some mathematics that had rusted over in my brain. In this article, I'll show the results and the sometimes surprising paths that led to them.First I need to talk a bit more about Ivy itself.For obvious reasons, Ivy avoids APL's idiosyncratic character set and uses ASCII tokens and words, just as in my previous, Lisp-written pseudo-APL back in 1979. To give it a different flavor and create a weak but valid justification for it, from the start Ivy used exact arithmetic, colloquially big ints and exact rationals. This made it unusual and a little bit interesting, and also meant it could be used to perform calculations that were clumsy or infeasible in traditional languages, particularly cryptographic work. The factorial of 100, in Ivy (input is indented): !100 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000Or perhaps more surprising: 0.51/2Ivy also became, due to the work of Hana Kim and David Crawshaw, the first mobile application written in Go. I take no credit for that. The iOS version is long gone and the Android version is very old, but that situation might change before long.For those who don't know the history and peculiarities of APL, it's worth doing your own research; that education is not the purpose of this artic...