How to combine and simplify the following PostScript operators?

Question

Background

Let

\def\RR#1{(2^(#1/10))}
\def\TT#1{(#1)}

be the radius and angle functions, respectively.

Christoph defined a PS operator AAA as follows.

/AAA {AlgParser cvx exec exch AlgParser cvx exec exch RadtoDeg PtoC} def 

It can be used to translate

\pscircle*[linecolor=red](!\RR{7*Pi/2} \TT{7*Pi/2} AAA){2pt}

from polar infix to polar postfix. However 7*Pi/2 are repeated, it breaks DRY (Don't Repeat Yourself) principle and cannot make use of t as explained shortly below.

I modified Herbert's operators as follows

/BBB {/TT ED TT AlgParser cvx /t ED  /RR ED RR AlgParser cvx /r ED r t RadtoDeg PtoC} def
/CCC { AlgParser cvx /t ED  AlgParser cvx exec t RadtoDeg PtoC} def

to allow me to write two styles as follows.

\pscircle*[linecolor=green](!{(2^(t/10))} {(3*Pi/2)} BBB){2pt}
\pscircle*[linecolor=blue](!\RR{t} \TT{5*Pi/2} CCC){2pt}

The first does not make use of \RR and \TT and the last does make use of \RR and \TT.

Complete code

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\pstVerb
{
    /AAA {AlgParser cvx exec exch AlgParser cvx exec exch RadtoDeg PtoC} def 
    /BBB {/TT ED TT AlgParser cvx /t ED  /RR ED RR AlgParser cvx /r ED r t RadtoDeg PtoC} def
    /CCC { AlgParser cvx /t ED  AlgParser cvx exec t RadtoDeg PtoC} def
}


\def\RR#1{(2^(#1/10))}
\def\TT#1{(#1)}

\begin{document}
\begin{pspicture}[showgrid](-3,-3)(3,3)
  \psplot[algebraic,polarplot]{TwoPi}{TwoPi 2 mul}{\RR{x}}
    \pscircle*[linecolor=red](!\RR{7*Pi/2} \TT{7*Pi/2} AAA){2pt}
    \pscircle*[linecolor=green](!{(2^(t/10))} {(3*Pi/2)} BBB){2pt}
    \pscircle*[linecolor=blue](!\RR{t} \TT{5*Pi/2} CCC){2pt}
\end{pspicture}

\end{document}

Question

Can I combine BBB and CCC to create DDD as simple as AAA but they must still work for the following?

\pscircle*[linecolor=green](!{(2^(t/10))} {(3*Pi/2)} DDD){2pt}
\pscircle*[linecolor=blue](!\RR{t} \TT{5*Pi/2} DDD){2pt}

Bonus question: I still don't understand why Herbert used only one exec while Christoph used 2 execs? What does exec do actually?

Answer 1

simplified

/AAA {AlgParser cvx exec exch AlgParser cvx exec exch RadtoDeg PtoC} def 
/BBB {exec AlgParser cvx /t ED  exec AlgParser cvx exec t RadtoDeg PtoC} def
/CCC {AlgParser cvx /t ED  AlgParser cvx exec t RadtoDeg PtoC} def

for /BBB: the expression {(2^(t/10))} is moved to PostScript with the parentheses {...} the reason why we first need an exec for removing it, or, of course, /anyvar exch def anyvar or /anyvar ED anyvar (saving it and pushing the saved data onto the stack) will do the same. Without the {...} it can't work on TeX side.

For your examples you need only /BBB, it works for all:

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\pstVerb{
  /A2P {exec AlgParser cvx /t ED  exec AlgParser cvx exec t RadtoDeg PtoC} def }

\def\RR#1{(2^(#1/10))}
\def\TT#1{(#1)}

\begin{document}
\begin{pspicture}[showgrid](-3,-3)(3,3)
  \psplot[algebraic,polarplot]{2 Pi mul neg}{4 Pi mul}{\RR{x}}
  \pscircle*[linecolor=red  ](!\RR{7*Pi/2} \TT{7*Pi/2} A2P){2pt}
  \pscircle*[linecolor=green](!{(2^(t/10))} {(3*Pi/2)} A2P){2pt}
  \pscircle*[linecolor=blue ](!\RR{t} \TT{5*Pi/2} A2P){2pt}
\end{pspicture}

\end{document}

Answer 2

Only the bonus answer regarding the exec:

AlgParser parses a string (...) and leaves an array [...] on the stack. The expression

(2^(Pi/10)) AlgParser

evaluates to the array

[tx@AddMathFunc begin 2.0 Pi 10.0 div exp end]

In order to evaluate this, one must turn this into an executable array (or procedure) with cvx, which gives

{tx@AddMathFunc begin 2.0 Pi 10.0 div exp end}

Now you can either call exec to execute this procedure which is on the stack (which is what I did twice), or you can save it and execute it by calling the variable, i.e. use /x exch def x to execute it (which is what Herbert did once).

So the following statements are equal

(2^(Pi/10)) AlgParser cvx exec

and

(2^(Pi/10)) AlgParser cvx /x exch def x

Answer 3

Thanks to Christoph's answer on exec, I managed to make a DDD working in both cases. Here is the MWE (Minimal Working Example).

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-plot}
\pstVerb
{
    /AAA {AlgParser cvx exec exch AlgParser cvx exec exch RadtoDeg PtoC} def 
    /DDD {exec AlgParser cvx /t ED exec AlgParser cvx exec t RadtoDeg PtoC} def
}    

\def\RR#1{(2^(#1/10))}
\def\TT#1{(#1)}
\def\RT#1{\RR{#1} \TT{#1}}

\begin{document}
\begin{pspicture}[showgrid](-3,-3)(3,3)
  \psplot[algebraic,polarplot]{2 Pi mul neg}{4 Pi mul}{\RR{x}}
  \pscircle*[linecolor=red](!\RT{7*Pi/2} AAA){2pt}
  \pscircle*[linecolor=green](!{(2^(t/10))} {(3*Pi/2)} DDD){2pt}
  \pscircle*[linecolor=blue](!\RR{t} \TT{5*Pi/2} DDD){2pt}
\end{pspicture}
\end{document}