# Introduction to RPL

The original RPL (*Reverse Polish Lisp*) programming language was designed and
implemented by Hewlett Packard for their calculators from the mid-1980s until
2015 (the year the HP50g was discontinued). It is based on older calculators
that used RPN (*Reverse Polish Notation*). Whereas RPN had a limited stack size of
4, RPL has a stack size only limited by memory and also incorporates
programmatic concepts from the Lisp programming language.

The first implementation of RPL accessible by the user was on the HP28C, circa
1987, which had an HP Saturn processor. More recent implementations (e.g., HP49,
HP50g) run through a Saturn emulation layer on an ARM based processor. These
ARM-based HP calculators would be good targets for a long-term port of DB48X.

DB48X is a fresh implementation of RPL on ARM, initially targetting the
SwissMicros DM42 calculator. This has [implications on the design](#design-overview)
of this particular implementation of RPL.

## The RPL stack

The RPL stack can grow arbitrarily in size.

By convention, and following RPN usage, this document gives the names `X`, `Y`,
`Z` and `T` to the first four levels of the stack. This is used to describe the
operations on the stack with synthetic stack diagrams showing the state of the
stack before and after the operation.

For example, the addition of two objects in levels 1 and 2 with the result
deposited in stack level 1 can be described in synthetic form using the
following stack diagram:

`Y` `X` ▶ `Y+X`

The duplication operation `Duplicate` (`DUP`) can be described in synthetic form
using the following synthetic stack diagram:

`X` ▶ `X` `X`


## Algebraic mode

Unlike earlier RPN calculators from Hewlett-Packard, RPL calculators from HP
includes complete support for algebraic objects written using the standard
precedence rules in mathematics. This gives you the best of both worlds,
i.e. the keyboard efficiency of RPN, requiring less keystrokes for a given
operation, as well as the mathematical readability of the algebraic
notation. Better yet, it is possible and easy to build an algebraic expression
from RPN keystrokes. These nice properties are also true for DB48X.

In RPL, algebraic expressions are placed between ticks. For
example, `'2+3×5'` will evaluate as `17`: the multiplication `3×5`, giving `15`,
is performed before the addition `2+15`, which gives `17`. An algebraic
expression can also be symbolic and contain unevaluated variables. For example,
`2+x` is a valid algebraic operation. If, having this expression on the stack,
you type `3` and then hit the `×` key, you will end up with `(2+x)×3`, showing
how the algebraic expression was built from RPN keystrokes.

Algebraic expressions are not evaluated automatically. The _R/S_ key (bound to
the [Evaluate](#evaluate) function) will compute their value as needed. On the
DB48X keyboard overlay, this key is also marked as `=` for that reason.

## Rich data types

Since introducing the first scientific pocket calculator, the HP-35, in 1972,
and with it the reverse polish notation (RPN), Hewlett-Packard perfected its
line-up for decades. This led to such powerhouses pocket computers such as as
the HP-41C series, or tiny wonders of pocket efficiency such as the HP-15C. Many
of these calculators, including the models we just cited, were capable of
advanced mathematics, including dealing with complex numbers, matrix operations,
root finding or numeric integration.

Then in 1986, everything changed with the HP-28C, which introduced a new user
interface called RPL. While the most evidently visible change was an unlimited
stack, what instantly made it both more powerful and easier to use than all its
RPN predecessors was the introduction of [data types](#types). Every value
on the stack, instead of having to be a number, could be a text, a name or an
equation. This made operations completely uniform irrespective of the data being
operated on. The same `+` operation that adds numbers can also add complex
numbers, vectors, matrices, or concatenate text. The exact same logic applies in
all case. This solved a decade-long struggle to extend the capabilities of
pocket calculators.

For example, whereas the HP-41C had some support for text, with an "Alpha" mode
and an alpha register, text operations were following their own logic, with for
example `ARCL` and `ASTO` dealing with at most 6 characters at a time, because
they were artificially fitted in a register designed to hold a numerical value.
Dealing with complex numbers on the HP-41C was
[similarly clunky](https://coertvonk.com/sw/hp41/complex-arithmetic-xmem-4426).
Even the HP-15C, which had built-in support for complex numbers, remained a bit
awkward to use in "complex mode" because its display could only show one half of
a complex number, e.g. the real or imaginary part. Similarly, matrix or
statistic operations had non-obvious interactions with numbered data registers.

All this was solved with RPL, because now a complex number, a matrix or a text
would occupy a single entry on the stack. So whereas adding two integers would
require a sequence like `1 ENTER 2 +` like in RPN, a very similar sequence would
add two texts: `"ABC" ENTER "DEF" +`, and the exact same logic would also add
two vectors in `[1 2 3] ENTER [4 5 6] +`.

DB48X adopts this extremely powerful idea, with a focus on making it as
efficient as possible for interactive calculations as well as for custom
programmed solution.