Re: 2button, 1op RPN calculator, again

Hi Richard:

Re accuracy:

While an undergraduate at MIT in the late 1960's, I often used the Wang
calculators which were sprinkled about some of the MIT libraries.

As pointed out below, these machines *multiplied* by using logs,
addition and exponentiation. In essence, they were digital slide
rules!

These Wang calculators were likely the *last* calculators ever made
without using sophisticated integrated circuits.

https://www.oldcalculatormuseum.com/wang360.html

"The 300-series machines calculate to a full 14 digits of accuracy, but
only display 10 digits. This makes the machine quite accurate. ***Since
the Wang calculators used logarithms to perform multiplication and
division,*** and even though Dr. Wang's log-generating circuit produced
very accurate results, many logarithms are transcendental numbers
which can never be represented with 100% accuracy no matter how many
digits they are calculated out to. Because of the use of logarithms
for multiplication and division operations, the previous-generation
LOCI-2 could come up with results which would be slightly off. An
example would be multiplying 6 by 8. The LOCI-2 would give an answer
of 47.99999999. While 47.99999999 is technically useful as a
reasonably accurate answer, such a result was generally considered
incorrect by most calculator users. To make the 300-series more
intuitive, these machines got the benefit of a special circuit that
provided a round off function, which would cause such 6 x 8 to yield
48.0000000, which made the 300-series machine much more useful to
non-technical (such as business and financial) users. This round-off
function operated based on the non-displayed 11th digit (remember, the
machines calculate to 14 digits of accuracy internally) of results."

-----Original Message-----
From: Richard Fateman <[email protected]>
Sent: May 5, 2026 10:29 AM
To: Stavros Macrakis <[email protected]>
Cc: Henry Baker <[email protected]>, Barton Willis via Maxima-discuss <[email protected]>
Subject: Re: [Maxima-discuss] 2button, 1op RPN calculator, again

This is just a curiosity.  I thought I'd point out its uselessness
for encoding computation of scientific use. 
 
What is the precision and accuracy of eml?
 
If it is not specified, it should be clear that it is useless. 
 
 If it is specified, then examples for which the result
is arbitrarily incorrect can be devised.
 
If eml were used (say) for plotting, the picture could be arbitrarily wrong.
 
Maybe the original author addresses this?
RJF


On Tue, May 5, 2026 at 1:12 PM Stavros Macrakis <[email protected] (mailto:[email protected])> wrote:
It has to have some way of entering other inputs if you want to denote anything but constants. Or maybe you pre-push the inputs onto the stack, in general multiple instances of them so that they'll be available at the right place.


On Tue, May 5, 2026 at 12:14 PM Henry Baker <[email protected] (mailto:[email protected])> wrote:
I just realized that this calculator doesn't have any 'regular' buttons at all; the only 2 buttons are the parentheses '(' and ')'.

The '(' button pushes '1' onto the stack; the ')' replaces the top 2 elts of the stack with the single object eml(x,y), where x,y were the top two elements of the stack.

Clearly '(' increments the stack length by 1, while ')' decrements the stack length by 1, so we have a *parenthesis language*, and only properly paired parentheses are allowed.

The stack comes initialized with a single object '1', because *every* legal computation has to start with a push, and the only constant that can be pushed onto the stack is a '1', so we needn't bother with that first push instruction at all !

The number of legal instruction sequences of length 2n is Catalan(n) = binomial(2*n,n)/(n+1).

This means that the fraction of legal instruction sequences approaches zero as the length of the sequence grows, so the bit efficiency of ANY RPN-type representation isn't going to be particularly good.



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