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I want to define a named constant standing for a more complicated symbolic expression, for example, PlasticConstant for the plastic constant:

((9 - √69)^(1/3) + (9 + √69)^(1/3))/(2^(1/3) 3^(2/3))

and I want it to have evaluation rules similar to GoldenRatio:

  • it remains in its named form during normal evaluation (so, simple Set does not solve my problem);
  • it can be numerically evaluated to an arbitrary precision using N;
  • it can be converted to the underlying expression in radicals by FunctionExpand, can be converted to a Root object by RootReduce;
  • FullSimplify is able to replace it with its underlying expression when it helps reducing overall complexity of a containing expression, but leaves it in its named form otherwise;

Equality and inequality operators between the named constant and its underlying expression should automatically evaluate to Boolean values:

PlasticConstant == ((9 - √69)^(1/3) + (9 + √69)^(1/3))/(2^(1/3) 3^(2/3))
(* True *)

similar to how it happens here:

GoldenRatio == (1 + √5)/2
(* True *)

How can we do it?

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    $\begingroup$ Related (128444). It doesn't cover FunctionExpand, RootReduce or FullSimplify. $\endgroup$
    – Carl Woll
    Jun 4, 2018 at 23:37
  • $\begingroup$ I'd try it with upvalues, but I'm not sure which cases will suffice. $\endgroup$
    – John Doty
    Jun 4, 2018 at 23:47

1 Answer 1

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Rather long for a comment:

I think the OP's goals are going to be hard to achieve. GoldenRatio is embedded in the system in ways I don't know. Here is some examples of the embedding that come from an internal Trace of FunctionExpand@GoldenRation:

GoldenRatio /. SimplifyDump`PositiveRules
SimplifyDump`$FSTab /@ %
(*
  {901}
  {GoldenRatio :> 1/2 (1 + Sqrt[5])}
*)

Some of the OP's goals can be handled programmatically. For instance:

ClearAll[OneThird];
SetAttributes[OneThird, {Constant}];
N[OneThird, p_: MachinePrecision] := N[1/3, p];

OneThird // N[#, 22] &
(*  0.3333333333333333333333  *)

However, automatic conversion to its numerical value does not happen for OneThird as it does for GoldenRatio:

3. OneThird
3. GoldenRatio
(*
  3. OneThird
  4.8541
*)

Maybe someone else will know how to do it. An internal trace reveals nothing.

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    $\begingroup$ You need NumericQ[OneThird] = True as in @Szabolcs' answer to the related question. $\endgroup$
    – Carl Woll
    Jun 4, 2018 at 23:49
  • $\begingroup$ @CarlWoll Thanks. $\endgroup$
    – Michael E2
    Jun 4, 2018 at 23:51
  • $\begingroup$ Also, I usually do N[OneThird, _] = 1/3 instead, but I don't think it makes a difference. $\endgroup$
    – Carl Woll
    Jun 4, 2018 at 23:51
  • $\begingroup$ I think rules for FunctionExpand and FullSimplify can be added somehow via SimplifyDump`PositiveRules and SimplifyDump`$FSTab. What do numeric indexes in the latter mean? What index do I use to add my own entry? $\endgroup$ Jun 5, 2018 at 17:34
  • $\begingroup$ @VladimirReshetnikov Yes, I think you're right. The numbers are indices for $FSTab. You can look at them with ?SimplifyDump`$FSTab. Obviously you would use an index that is not present in Cases[DownValues[SimplifyDump`$FSTab], HoldPattern@SimplifyDump`$FSTab[n_Integer] :> n, Infinity]. (I don't know for sure that any missing one is safe.) $\endgroup$
    – Michael E2
    Jun 5, 2018 at 17:41

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