`symbolic_solve` -- can I replace unicode symbols in the answer?

[B-LIE]

I'm solving a set of 6 symbolic (algebraic) equations in 6 variables. The answer looks like this:

Is there a way to avoid the unicode square-root symbol, and instead get sqrt with the parentheses as needed?

[B-LIE]

OK -- I tried to ask Windows "Co-pilot"... First it suggested using some code_lowering function (does it exist?), and next it suggested to convert the answer to a string and use replace... So this seems to work:

Tp_sol[2][sx_2] |> string |> x-> replace(x, "√" => "sqrt") |> println

The answers (6 unknowns) are really, really ugly and lengthy. So I am wondering whether it would be simpler to get the solution presented as:

1. A monovariable polynomial for one of the variables (ideally, I'd like to choose which one; or one variable that Symbolics introduces).
2. Simple expressions for the remaining variables, typically linear mappings from the one found by solving the monovariable polynomial. I think the Groebner algorithm should be able to do that??

Anyways, great that symbolic_solve() handles my problem!

[ChrisRackauckas]

The print out shown there is Latex, not the symbolic representation. There's a MIME type for latex print outs that Pluto uses, since Pluto defaults to showing Latex when it can. If you display on the result IIRC it will show you the original representation.

[B-LIE]

If you display on the result IIRC it will show you the original representation.

Thanks! I will streamline my code tomorrow afternoon (after an exercise hike) and check. I will then try to show what I get from Symbolics.jl + what I get from Nemo.jl, and indicate what would (to me) be the ultimate response. I'll try to provide a complete description so that someone can use it as a test case, in case you find it useful. [It relates to so-called equilibrium flash calculations in Thermodynamics... water + hydrogen + oxygen, as an example. ]

[B-LIE]

OK... here is a report on how things work in Symbolics.jl vs. Nemo.jl, and also how I would have preferred it to work.

1: The problem

Assume I consider a thermodynamic "Tp flash" problem. The idea is that I know the mole fractions of total amounts of the components in the given volume. In addition, suppose I know the temperature T and the pressure, and that the components split into two phases, liquid and gas, for simplicity. I denote the mole fractions in the liquid by x, and the mole fractions in the gas by y.

In this case, I consider a mixture of 3 components: water H2O + two light gases (e.g., Oxygen, Hydrogen). I index water by H2O, and the two other un-specified components by 1 and 2. So z, x, and y contain 3 components each, and as is the case, the elements of each of z, x, and y must sum to unity.

The following 5 equations describe: (i) equilibrium conditions for each of the 3 components, and (ii) the fact that the elements of x and y must sum to unity. Here, p, T, and z are known quantities. In additions, f_H2O(T,p), H_1(T), and H_2(T) are known functions.

x_H2O * f_H2O(T,p) = y_H2O * p
x_1 * H_1(T) = y_1 * p
x_2 * H_2(T) = y_2 * p
x_H2O + x_1 + x_2 = 1
y_H2O + y_1 + y_1 = 1

In addition, there is a constraint related to the mole balances:

z_2 * (x_H2O * y_1-x_1 * y_H2O) = x_2 * (y_2 * z_H2O - y_H2O * z_1) + y_2 *(-x_1 * z_H2O + x_H2O * z_1)

So now, the problem is to find the 3 elements of x and the 3 elements of y, with T, p, and z known.

2: Using Symbolics

To avoid name collisions, I prepend the variables used with Symbolics by _s_.

using Symbolics
using Groebner

@variables _s_xH2O, _s_x1, _s_x2, _s_yH2O, _s_y1, _s_y2, _s_fH2O, _s_p, _s_H1, _s_H2, _s_zH2O, _s_z1, _s_z2

_s_eq1 = _s_xH2O*_s_fH2O - _s_yH2O*_s_p 
_s_eq2 = _s_x1*_s_H1-_s_y1*_s_p 
_s_eq3 = _s_x2*_s_H2 - _s_y2*_s_p 
_s_eq4 = 1 - _s_xH2O - _s_x1 - _s_x2 
_s_eq5 = 1 - _s_yH2O - _s_y1 - _s_y2 
_s_eq6 = _s_z2*(_s_xH2O*_s_y1 - _s_x1*_s_yH2O) - _s_x2*(_s_y2*_s_zH2O - _s_yH2O*_s_z1) -
                _s_y2*(-_s_x1*_s_zH2O + _s_xH2O*_s_z1)

_s_eqs_Tp_flash = [_s_eq1, _s_eq2, _s_eq3, _s_eq4, _s_eq5, _s_eq6]
_s_vars_Tp_flash = [_s_xH2O, _s_x1, _s_x2, _s_yH2O, _s_y1, _s_y2]

Tp_sol = symbolic_solve(_s_eqs_Tp_flash, _s_vars_Tp_flash)

The solution Tp_sol contains two possible solutions, which look like:

Here, I would like to get rid of the square-root symbol. Doing, e.g., display(Tp_sol[1][]) does not do this. At least not in a Jupyter notebook within VSCode:

The only way I have found to do it is as follows:

julia> Tp_sol[1][_s_x2] |> string |> x-> replace(x, "√" => "sqrt") |> println
(_s_p*sqrt(4(_s_H1^2)*(_s_H2^4)*(_s_p^2)*(_s_z1^2) + 8(_s_H1^2)*(_s_H2^4)*(_s_p^2)*_s_z1*_s_z2 + 4(_s_H1^2)*(_s_H2^4)*(_s_p^2)*(_s_z2^2) - 8(_s_H1^2)*(_s_H2^3)*_s_fH2O*(_s_p^2)*(_s_z1^2) - 8(_s_H1^2)*(_s_H2^3)*_s_fH2O*(_s_p^2)*_s_z1*_s_z2 - 16(_s_H1^2)*(_s_H2^3)*_s_fH2O*(_s_p^2)*_s_z2*_s_zH2O - 8(_s_H1^2)*(_s_H2^3)*(_s_p^3)*_s_z1*_s_z2 - 8(_s_H1^2)*(_s_H2^3)*(_s_p^3)*(_s_z2^2) + 16(_s_H1^2)*(_s_H2^3)*(_s_p^3)*_s_z2*_s_zH2O + 4(_s_H1^2)*(_s_H2^2)*(_s_fH2O^2)*(_s_p^2)*(_s_z1^2) + 8(_s_H1^2)*(_s_H2^2)*_s_fH2O*(_s_p^3)*_s_z1*_s_z2 + 4(_s_H1^2)*(_s_H2^2)*(_s_p^4)*(_s_z2^2) - 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*_s_z1*_s_z2 + 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*_s_z1*_s_zH2O - 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*(_s_z2^2) - 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*_s_z2*_s_zH2O - 8_s_H1*(_s_H2^4)*(_s_p^3)*(_s_z1^2) - 8_s_H1*(_s_H2^4)*(_s_p^3)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^4)*(_s_p^3)*_s_z1*_s_zH2O + 8_s_H1*(_s_H2^4)*(_s_p^3)*_s_z2*_s_zH2O + 8_s_H1*(_s_H2^3)*(_s_fH2O^2)*(_s_p^2)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^3)*(_s_fH2O^2)*(_s_p^2)*_s_z1*_s_zH2O + 16_s_H1*(_s_H2^3)*(_s_fH2O^2)*(_s_p^2)*_s_z2*_s_zH2O + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*(_s_z1^2) + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*_s_z1*_s_z2 + 8_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*_s_z1*_s_zH2O + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*(_s_z2^2) + 8_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*_s_z2*_s_zH2O + 8_s_H1*(_s_H2^3)*(_s_p^4)*_s_z1*_s_z2 - 24_s_H1*(_s_H2^3)*(_s_p^4)*_s_z2*_s_zH2O - 8_s_H1*(_s_H2^2)*(_s_fH2O^2)*(_s_p^3)*(_s_z1^2) - 8_s_H1*(_s_H2^2)*(_s_fH2O^2)*(_s_p^3)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^2)*_s_fH2O*(_s_p^4)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^2)*_s_fH2O*(_s_p^4)*(_s_z2^2) + 4(_s_H2^4)*(_s_fH2O^2)*(_s_p^2)*(_s_z2^2) + 8(_s_H2^4)*(_s_fH2O^2)*(_s_p^2)*_s_z2*_s_zH2O + 4(_s_H2^4)*(_s_fH2O^2)*(_s_p^2)*(_s_zH2O^2) + 8(_s_H2^4)*_s_fH2O*(_s_p^3)*_s_z1*_s_z2 - 8(_s_H2^4)*_s_fH2O*(_s_p^3)*_s_z1*_s_zH2O - 8(_s_H2^4)*_s_fH2O*(_s_p^3)*_s_z2*_s_zH2O - 8(_s_H2^4)*_s_fH2O*(_s_p^3)*(_s_zH2O^2) + 4(_s_H2^4)*(_s_p^4)*(_s_z1^2) + 8(_s_H2^4)*(_s_p^4)*_s_z1*_s_zH2O + 4(_s_H2^4)*(_s_p^4)*(_s_zH2O^2) - 8(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*_s_z1*_s_z2 + 8(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*_s_z1*_s_zH2O - 8(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*(_s_z2^2) - 24(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*_s_z2*_s_zH2O - 8(_s_H2^3)*_s_fH2O*(_s_p^4)*(_s_z1^2) - 8(_s_H2^3)*_s_fH2O*(_s_p^4)*_s_z1*_s_z2 - 8(_s_H2^3)*_s_fH2O*(_s_p^4)*_s_z1*_s_zH2O + 24(_s_H2^3)*_s_fH2O*(_s_p^4)*_s_z2*_s_zH2O + 4(_s_H2^2)*(_s_fH2O^2)*(_s_p^4)*(_s_z1^2) + 8(_s_H2^2)*(_s_fH2O^2)*(_s_p^4)*_s_z1*_s_z2 + 4(_s_H2^2)*(_s_fH2O^2)*(_s_p^4)*(_s_z2^2)) - 2_s_H1*(_s_H2^2)*(_s_p^2)*_s_z1 - 2_s_H1*(_s_H2^2)*(_s_p^2)*_s_z2 + 2_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z1 + 4_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z2 + 2_s_H1*_s_H2*(_s_p^3)*_s_z1 - 2_s_H1*_s_H2*(_s_p^3)*_s_zH2O - 2_s_H1*_s_fH2O*(_s_p^3)*_s_z1 - 2_s_H1*_s_fH2O*(_s_p^3)*_s_z2 - 2(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_z2 - 2(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_zH2O + 2(_s_H2^2)*(_s_p^3)*_s_z2 + 4_s_H2*_s_fH2O*(_s_p^3)*_s_zH2O) / (-4_s_H1*(_s_H2^2)*(_s_p^2)*_s_z1 - 4_s_H1*(_s_H2^2)*(_s_p^2)*_s_z2 + 4_s_H1*(_s_H2^2)*(_s_p^2)*_s_zH2O + 4_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z1 + 4_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z2 + 4(_s_H2^3)*(_s_p^2)*_s_z1 + 4(_s_H2^3)*(_s_p^2)*_s_z2 + 4(_s_H2^3)*(_s_p^2)*_s_zH2O - 4(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_z1 - 4(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_z2 - 8(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_zH2O) + ((1//2)*_s_p) / _s_H2

As seen, the solution is exceedingly complex. The solutions for the other unknowns are of similar complexity, and there are two solutions for each case because there is an underlying quadratic polynomial involved.

3: Using Nemo

For Nemo, I prepend the variable names by _n_.

using Nemo
using Groebner
using ParamPunPam

# 1. Define parameters and variables
# - known values
_r, (_n_fH2O, _n_p, _n_H1, _n_H2, _n_zH2O, _n_z1, _n_z2) = Nemo.QQ[:_n_fH2O, :_n_p, :_n_H1, :_n_H2, :_n_zH2O, :_n_z1, :_n_z2]
# - unknowns
_R, (_n_xH2O, _n_x1, _n_x2, _n_yH2O, _n_y1, _n_y2) = fraction_field(_r)[:_n_xH2O, :_n_x1, :_n_x2, :_n_yH2O, :_n_y1, :_n_y2]

_n_vars_Tp_flash = [_n_xH2O, _n_x1, _n_x2, _n_yH2O, _n_y1, _n_y2]

# 2. Define equations

_n_eq1 = _n_xH2O*_n_fH2O - _n_yH2O*_n_p 
_n_eq2 = _n_x1*_n_H1-_n_y1*_n_p 
_n_eq3 = _n_x2*_n_H2 - _n_y2*_n_p 
_n_eq4 = 1 - _n_xH2O - _n_x1 - _n_x2 
_n_eq5 = 1 - _n_yH2O - _n_y1 - _n_y2 
_n_eq6 = _n_z2*(_n_xH2O*_n_y1 - _n_x1*_n_yH2O) - _n_x2*(_n_y2*_n_zH2O - _n_yH2O*_n_z1) - _n_y2*(-_n_x1*_n_zH2O + _n_xH2O*_n_z1)

_n_eqs_Tp_flash = [_n_eq1, _n_eq2, _n_eq3, _n_eq4, _n_eq5, _n_eq6]

# 3. Make these variables the smallest
_n_interesting_vars = [_n_x2]

# 4. Compute a GB, allowing division by parameters

Tp_gb = paramgb(
    _n_eqs_Tp_flash, 
    # vars > interesting_vars > params
   ordering=DegRevLex(setdiff(_n_vars_Tp_flash, _n_interesting_vars)) * DegRevLex(_n_interesting_vars)
)

# 5. Inspect. Note: gb[1] is univariate
map(vars, Tp_gb)

The response of the above looks like:

Specifically, I can inspect the first element:

julia> Tp_gb[1]
_n_x2^2 + (_n_fH2O*_n_p*_n_z1 + _n_fH2O*_n_p*_n_z2 - _n_fH2O*_n_H1*_n_z1 - 2*_n_fH2O*_n_H1*_n_z2 + _n_fH2O*_n_H2*_n_zH2O + _n_fH2O*_n_H2*_n_z2 + _n_p*_n_H1*_n_z2 - _n_p*_n_H2*_n_zH2O - _n_p*_n_H2*_n_z1 - 2*_n_p*_n_H2*_n_z2 + _n_H1*_n_H2*_n_z1 + _n_H1*_n_H2*_n_z2)//(_n_fH2O*_n_H1*_n_z1 + _n_fH2O*_n_H1*_n_z2 - 2*_n_fH2O*_n_H2*_n_zH2O - _n_fH2O*_n_H2*_n_z1 - _n_fH2O*_n_H2*_n_z2 + _n_H1*_n_H2*_n_zH2O - _n_H1*_n_H2*_n_z1 - _n_H1*_n_H2*_n_z2 + _n_H2^2*_n_zH2O + _n_H2^2*_n_z1 + _n_H2^2*_n_z2)*_n_x2 + (-_n_fH2O*_n_p*_n_z2 + _n_fH2O*_n_H1*_n_z2 + _n_p^2*_n_z2 - _n_p*_n_H1*_n_z2)//(_n_fH2O*_n_H1*_n_z1 + _n_fH2O*_n_H1*_n_z2 - 2*_n_fH2O*_n_H2*_n_zH2O - _n_fH2O*_n_H2*_n_z1 - _n_fH2O*_n_H2*_n_z2 + _n_H1*_n_H2*_n_zH2O - _n_H1*_n_H2*_n_z1 - _n_H1*_n_H2*_n_z2 + _n_H2^2*_n_zH2O + _n_H2^2*_n_z1 + _n_H2^2*_n_z2)

This is a quadratic polynomial for x2, which -- although it looks ugly -- is trivial to solve.

If I look at Tb_gb[2], I get [_n_x2, _n_y2] which essentially means that I get the expression for y2 where now x2 is known. Etc.

So Nemo.jl gives me the task of finding the root of a quadratic polynomial for finding x2 (simple, although the polynomial parameters are relatively complex -- still much, much simpler than the full solution given by Symbolics.jl).

However, even the solution provided by Nemo.jl could be simpler.

4: What would be the ideal way to handle the problem?

The ideal response from Symbolics.jl (perhaps such a function should not be called symbolic_solve() because it would not provide the final solution) would be to spit back:

1. A monovariable polynomial, such as the polynomial for x2 via Nemo.jl. This leaves it to the user to use a polynomial root finding algorithm to find the numeric solution when inserting numeric values.

2. From the 5+1 original equations, observe that the 5 first equations are linear when x2 is assumed to be known. And the monovariable polynomial in point 1 of this list provides the expression to compute x2.

3. I would then like to be provided a matrix M and a vector v such that M*q = v, where q = [xH2O, x1, yH2O, y1, y2].

OK... I don't know whether the Groebner basis routine actually can do this. My understanding is that the Groebner basis routine "triangularizes" the multivariable polynomial problem into

• first, a monovariable polynomial for one unknown (possibly a fictitious new one, not among the original variables)
• second, a bivariate polynomial where one of the variables is the one we found from the previous basis (i.e., is known), effectively making this also into a monovariable polynomial that can be solved
• third, a trivariate polynomial where two of the variables are the ones found in the two points above, effectively making it into a monovariable polynomial
• etc.

5: Some final comments

I'm impressed that Symbolics.jl can find the solution. However, when solving "engineering problems", such complex solutions are not ideal. My "ideal" solution would be better, I think.

Nemo.jl currently gets me closer to what I want (thanks a lot to sumiya11 in the Julia community who set up the structure in Nemo.jl for me). Still, Nemo.jl is somewhat more complicated to use than Symbolics.jl.

I believe that for solving engineering problems, my indicated "ideal way" would be the best. One could always argue that it is sound to do some manual manipulation as a quality assurance in engineering problems, though... maybe I am too lazy.

For this particular problem, I have found Clapeyron.jl invaluable for testing out flash computations and checking my resulting algorithm (thanks to the developers there, in particular Andres).

---

Note: thermodynamics is only one out of many fields where I think Symbolics.jl and Nemo.jl can provide essential help in algorithm development.

[ChrisRackauckas]

@n0rbed I'm curious about your take on this.

[n0rbed]

For the initial issue (showing sqrt rather than √), this is much easier than what you're doing. Observe the code here:

Symbolics.jl/src/solver/solve_helpers.jl

Line 81 in da85a95

Base.show(io::IO, f::typeof(ssqrt)) = print(io, "√")

If this is causing widespread headache for users, we can remove this option. That being said i think it looks kinda nice in the REPL when using Symbolics.jl as a small and neat CAS. Its important to note that ssqrt behaves differently than sqrt, refer to its definition here:

Symbolics.jl/src/solver/solve_helpers.jl

Line 23 in da85a95

function ssqrt(n)

I would then like to be provided a matrix M and a vector v such that M*q = v, where q = [xH2O, x1, yH2O, y1, y2].

My problem with these complaints is that they're all referring to the formats we have, unless there is a good reason, the dict system we have is much more usable during scripts and its easier to read in the REPL as well. You can probably define this matrix on your own by writing a couple lines of code.

LMK if i missed something - and sorry if i did.

[B-LIE]

That being said i think it looks kinda nice in the REPL when using Symbolics.jl as a small and neat CAS

Yes, I do agree that the square root symbol looks nice in the REPL. But when I want to copy the result into other Julia code, it complicates matter. So I don't think it is necessary to remove it. Could an alternative be to have a "result_format" option that does not include such fancy symbols?

My problem with these complaints is that they're all referring to the formats we have

Please note: I am not complaining! Your code is very impressive and good. What I am suggesting is extra possibilities. Perhaps we are considering different uses? Also, I have tested more on my suggestion... What I have found is that:

1. My proposal of returning a linear sub-system in the form $M q = v$ is not that good, after all: I did this "manually" at first using Symbolics.jl. Then I also tested what would happen if I inserted the evaluated symbolic solution in my Julia code instead of using LinearAIgebra and v=M\q. The result of avoiding LinearAlgebra was a speed-up of the code of 4.7 times.

2. Also, symbolically solving the linear part in Symbolics.jl leads to un-necessarily "complex" expressions... In my example, Nemo.jl etc. returned (remember: _n_ prefix means that it is a symbolic variable in the Nemo subsystem... I do this to avoid messing up symbolic vs. standard variables):

This gives me a quadratic expression for mole fraction xH2O (I switched from x2 in my discussion above). This quadratic expression is simpler than the symbolic_solve response from Symbolics.jl, and more useful for further development in a Julia function. (I am not complaining about symbolic_solve, just stating that there may be other uses where an explicit solution is not the most useful). Nemo.jl also gives me similar quadratic expressions for all other variables. But that is unnecessarily complicated. If I use the quadratic expression for xH2O and compute its solution, all other equations are linear once xH2O is known.

So I used Symbolics.jl (this time, prefix _s_ for variables) to solve this linear set of equations, and get:

3. Isn't this fine? Yes, in a way. But still unnecessarily complicated. In this case, the linear equations can be expressed in the following, much simpler (triangular) way:

xH2O = (-_b + sqrt(_b^2 - 4*_c))/2
#
x1 = (p - _H2 + xH2O*(_H2-_fH2O))/(_H1-_H2)
x2 = 1 - xH2O - x1
yH2O = _fH2O*xH2O/p 
y1 = _H1*x1/p
y2 = 1 - yH2O - y

Summing up

a. symbolic_solve is fine! I don't complain about it.
b. Perhaps what I am "wishing for" is a new function that, for algebraic expressions ("multivariable polynomials"), returns a triangularized solution with (in my case):

• the second order polynomial/equation for xH2O which only depends on the given/known values [it would be nice to be able to use Symbolics.jl to collect terms proportional to _s_xH2O^2, _s_xH2O, and constants -- essentially finding the expressions for _b and _c in my Julia code]
• a linear equation relating x1 and xH2O and known terms
• a linear equation relating yH2O and possibly x1 and xH2O and known terms
• a linear equation relating y1 and possibly yH2O and x1 and xH2O and known terms
• etc.

In more generality, is this what the Groebner basis algorithm actually does? I.e., converting the system to a triangular form? The user's difficulty (ok: "mine") with Nemo.jl is a somewhat complicated way to define known symbolic values and unknown symbolic values, + the need to know that you need to use ParamPunPam.jl (I don't know exactly why that is needed).

As a kind of motivation for my use of Symbolics.jl and Nemo.jl for code development, consider the following comparisons of my simple "VT flash" calculation vs. what I get with Clapeyron.jl:

My "quick" algorithm is much faster than those in Clapeyron.jl (NB: I'm still testing things, so numbers are at best indicative). That does not mean that I am criticizing Clapeyron.jl - without Clapeyron.jl I have no idea about whether my results are good or not. Also, my "quick" algorithm is very limited in scope (if I add one more component, I need a new algorithm, etc.)

So -- without Symbolics.jl and Nemo.jl, it would have taken me much more time to develop my routines.

[n0rbed]

But when I want to copy the result into other Julia code, it complicates matter.

I see what you mean. Copying from the REPL into other julia code feels unnecessary though, cant you run everything into one script/notebook? In any case, i agree that we should have some result() function that prints out everything in functional form, in cases where the unicode root symbol is not looking good.

I am not complaining!

Complaints are good

just stating that there may be other uses where an explicit solution is not the most useful

Sure! what function are you running in Nemo.jl? As for the complex symbolic_solve output, this can eventually be fixed by adding more rewrite rules in here: https://github.com/JuliaSymbolics/Symbolics.jl/blob/da85a9573024d36ea29d20a3aca2f0a495e9e13c/src/solver/postprocess.jl

[n0rbed]

Also, for these quicker algorithms youve developed in Clapeyron.jl, I suggest showcasing your work there, it will definitely be appreciated.

[B-LIE]

I see what you mean. Copying from the REPL into other julia code feels unnecessary though, cant you run everything into one script/notebook?

I do run everything in one script/notebook. If I copy the output from Symbolics, say:

and paste it into another notebook cell where I define the function, I get what I show at the bottom of this response -- a horrible, unusable text.

If I convert the output to a string and do manual replacement of symbols, then it becomes manageable.

julia> Tp_sol[1][_s_x2] |> string |> x-> replace(x, "√" => "sqrt") |> println
((1//2)*_s_p) / _s_H2 + (_s_p*sqrt(4(_s_H1^2)*(_s_H2^4)*(_s_p^2)*(_s_z1^2) + 8(_s_H1^2)*(_s_H2^4)*(_s_p^2)*_s_z1*_s_z2 + 4(_s_H1^2)*(_s_H2^4)*(_s_p^2)*(_s_z2^2) - 8(_s_H1^2)*(_s_H2^3)*_s_fH2O*(_s_p^2)*(_s_z1^2) - 8(_s_H1^2)*(_s_H2^3)*_s_fH2O*(_s_p^2)*_s_z1*_s_z2 - 8(_s_H1^2)*(_s_H2^3)*_s_fH2O*(_s_p^2)*_s_z1*_s_zH2O + 8(_s_H1^2)*(_s_H2^3)*_s_fH2O*(_s_p^2)*_s_z2*_s_zH2O - 8(_s_H1^2)*(_s_H2^3)*(_s_p^3)*_s_z1*_s_z2 + 8(_s_H1^2)*(_s_H2^3)*(_s_p^3)*_s_z1*_s_zH2O - 8(_s_H1^2)*(_s_H2^3)*(_s_p^3)*(_s_z2^2) - 8(_s_H1^2)*(_s_H2^3)*(_s_p^3)*_s_z2*_s_zH2O + 4(_s_H1^2)*(_s_H2^2)*(_s_fH2O^2)*(_s_p^2)*(_s_z1^2) + 8(_s_H1^2)*(_s_H2^2)*(_s_fH2O^2)*(_s_p^2)*_s_z1*_s_zH2O + 4(_s_H1^2)*(_s_H2^2)*(_s_fH2O^2)*(_s_p^2)*(_s_zH2O^2) + 8(_s_H1^2)*(_s_H2^2)*_s_fH2O*(_s_p^3)*_s_z1*_s_z2 - 8(_s_H1^2)*(_s_H2^2)*_s_fH2O*(_s_p^3)*_s_z1*_s_zH2O - 8(_s_H1^2)*(_s_H2^2)*_s_fH2O*(_s_p^3)*_s_z2*_s_zH2O - 8(_s_H1^2)*(_s_H2^2)*_s_fH2O*(_s_p^3)*(_s_zH2O^2) + 4(_s_H1^2)*(_s_H2^2)*(_s_p^4)*(_s_z2^2) + 8(_s_H1^2)*(_s_H2^2)*(_s_p^4)*_s_z2*_s_zH2O + 4(_s_H1^2)*(_s_H2^2)*(_s_p^4)*(_s_zH2O^2) - 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*_s_z1*_s_z2 + 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*_s_z1*_s_zH2O - 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*(_s_z2^2) - 8_s_H1*(_s_H2^4)*_s_fH2O*(_s_p^2)*_s_z2*_s_zH2O - 8_s_H1*(_s_H2^4)*(_s_p^3)*(_s_z1^2) - 8_s_H1*(_s_H2^4)*(_s_p^3)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^4)*(_s_p^3)*_s_z1*_s_zH2O + 8_s_H1*(_s_H2^4)*(_s_p^3)*_s_z2*_s_zH2O + 8_s_H1*(_s_H2^3)*(_s_fH2O^2)*(_s_p^2)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^3)*(_s_fH2O^2)*(_s_p^2)*_s_z1*_s_zH2O - 8_s_H1*(_s_H2^3)*(_s_fH2O^2)*(_s_p^2)*_s_z2*_s_zH2O - 8_s_H1*(_s_H2^3)*(_s_fH2O^2)*(_s_p^2)*(_s_zH2O^2) + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*(_s_z1^2) + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*_s_z1*_s_z2 + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*_s_z1*_s_zH2O + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*(_s_z2^2) + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*_s_z2*_s_zH2O + 16_s_H1*(_s_H2^3)*_s_fH2O*(_s_p^3)*(_s_zH2O^2) + 8_s_H1*(_s_H2^3)*(_s_p^4)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^3)*(_s_p^4)*_s_z1*_s_zH2O - 8_s_H1*(_s_H2^3)*(_s_p^4)*_s_z2*_s_zH2O - 8_s_H1*(_s_H2^3)*(_s_p^4)*(_s_zH2O^2) - 8_s_H1*(_s_H2^2)*(_s_fH2O^2)*(_s_p^3)*(_s_z1^2) - 8_s_H1*(_s_H2^2)*(_s_fH2O^2)*(_s_p^3)*_s_z1*_s_z2 - 8_s_H1*(_s_H2^2)*(_s_fH2O^2)*(_s_p^3)*_s_z1*_s_zH2O + 8_s_H1*(_s_H2^2)*(_s_fH2O^2)*(_s_p^3)*_s_z2*_s_zH2O - 8_s_H1*(_s_H2^2)*_s_fH2O*(_s_p^4)*_s_z1*_s_z2 + 8_s_H1*(_s_H2^2)*_s_fH2O*(_s_p^4)*_s_z1*_s_zH2O - 8_s_H1*(_s_H2^2)*_s_fH2O*(_s_p^4)*(_s_z2^2) - 8_s_H1*(_s_H2^2)*_s_fH2O*(_s_p^4)*_s_z2*_s_zH2O + 4(_s_H2^4)*(_s_fH2O^2)*(_s_p^2)*(_s_z2^2) + 8(_s_H2^4)*(_s_fH2O^2)*(_s_p^2)*_s_z2*_s_zH2O + 4(_s_H2^4)*(_s_fH2O^2)*(_s_p^2)*(_s_zH2O^2) + 8(_s_H2^4)*_s_fH2O*(_s_p^3)*_s_z1*_s_z2 - 8(_s_H2^4)*_s_fH2O*(_s_p^3)*_s_z1*_s_zH2O - 8(_s_H2^4)*_s_fH2O*(_s_p^3)*_s_z2*_s_zH2O - 8(_s_H2^4)*_s_fH2O*(_s_p^3)*(_s_zH2O^2) + 4(_s_H2^4)*(_s_p^4)*(_s_z1^2) + 8(_s_H2^4)*(_s_p^4)*_s_z1*_s_zH2O + 4(_s_H2^4)*(_s_p^4)*(_s_zH2O^2) - 8(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*_s_z1*_s_z2 + 8(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*_s_z1*_s_zH2O - 8(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*(_s_z2^2) - 8(_s_H2^3)*(_s_fH2O^2)*(_s_p^3)*_s_z2*_s_zH2O - 8(_s_H2^3)*_s_fH2O*(_s_p^4)*(_s_z1^2) - 8(_s_H2^3)*_s_fH2O*(_s_p^4)*_s_z1*_s_z2 - 8(_s_H2^3)*_s_fH2O*(_s_p^4)*_s_z1*_s_zH2O + 8(_s_H2^3)*_s_fH2O*(_s_p^4)*_s_z2*_s_zH2O + 4(_s_H2^2)*(_s_fH2O^2)*(_s_p^4)*(_s_z1^2) + 8(_s_H2^2)*(_s_fH2O^2)*(_s_p^4)*_s_z1*_s_z2 + 4(_s_H2^2)*(_s_fH2O^2)*(_s_p^4)*(_s_z2^2)) - 2_s_H1*(_s_H2^2)*(_s_p^2)*_s_z1 - 2_s_H1*(_s_H2^2)*(_s_p^2)*_s_z2 + 2_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z1 + 4_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z2 + 2_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_zH2O + 2_s_H1*_s_H2*(_s_p^3)*_s_z1 - 2_s_H1*_s_fH2O*(_s_p^3)*_s_z1 - 2_s_H1*_s_fH2O*(_s_p^3)*_s_z2 - 2_s_H1*_s_fH2O*(_s_p^3)*_s_zH2O - 2(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_z2 - 2(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_zH2O + 2(_s_H2^2)*(_s_p^3)*_s_z2 + 2_s_H2*_s_fH2O*(_s_p^3)*_s_zH2O) / (-4_s_H1*(_s_H2^2)*(_s_p^2)*_s_z1 - 4_s_H1*(_s_H2^2)*(_s_p^2)*_s_z2 - 4_s_H1*(_s_H2^2)*(_s_p^2)*_s_zH2O + 4_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z1 + 4_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_z2 + 4_s_H1*_s_H2*_s_fH2O*(_s_p^2)*_s_zH2O + 4(_s_H2^3)*(_s_p^2)*_s_z1 + 4(_s_H2^3)*(_s_p^2)*_s_z2 + 4(_s_H2^3)*(_s_p^2)*_s_zH2O - 4(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_z1 - 4(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_z2 - 4(_s_H2^2)*_s_fH2O*(_s_p^2)*_s_zH2O)

But this is slightly cumbersome if I don't know in advance what the various symbols that Symbolics inserts are (e.g., other than the square root symbol, etc.).

Complaints are good

To some degree, but the word "complaint" may indicate some negativity/"bitching" (which is totally unfair, in particular since Julia with toolboxes is a free tool), and I think it is better to do constructive wishes.

Sure! what function are you running in Nemo.jl?

See Section 3 a few posts above this one -- I include the complete Nemo.jl ++ code.

Below, I show the unusable result of just copy+pasting from the original output from Symbolics:

s_H2
​
 
2
1
​
  
s_p
​
 
​
 + 
−4 
s_p
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
s_z1
​
 −4 
s_p
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
s_z2
​
 −4 
s_p
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
s_zH2O
​
 +4 
s_p
​
  
2
  
s_H1
​
  
s_H2
​
  
s_fH2O
​
  
s_z1
​
 +4 
s_p
​
  
2
  
s_H1
​
  
s_H2
​
  
s_fH2O
​
  
s_z2
​
 +4 
s_p
​
  
2
  
s_H1
​
  
s_H2
​
  
s_fH2O
​
  
s_zH2O
​
 +4 
s_p
​
  
2
  
s_H2
​
  
3
  
s_z1
​
 +4 
s_p
​
  
2
  
s_H2
​
  
3
  
s_z2
​
 +4 
s_p
​
  
2
  
s_H2
​
  
3
  
s_zH2O
​
 −4 
s_p
​
  
2
  
s_H2
​
  
2
  
s_fH2O
​
  
s_z1
​
 −4 
s_p
​
  
2
  
s_H2
​
  
2
  
s_fH2O
​
  
s_z2
​
 −4 
s_p
​
  
2
  
s_H2
​
  
2
  
s_fH2O
​
  
s_zH2O
​
 
s_p
​
  
4 
s_z1
​
  
2
  
s_p
​
  
2
  
s_H2
​
  
4
  
s_H1
​
  
2
 +8 
s_p
​
  
2
  
s_H2
​
  
4
  
s_H1
​
  
2
  
s_z1
​
  
s_z2
​
 +4 
s_z2
​
  
2
  
s_p
​
  
2
  
s_H2
​
  
4
  
s_H1
​
  
2
 −8 
s_z1
​
  
2
  
s_p
​
  
2
  
s_H2
​
  
3
  
s_H1
​
  
2
  
s_fH2O
​
 −8 
s_p
​
  
2
  
s_H2
​
  
3
  
s_H1
​
  
2
  
s_fH2O
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
2
  
s_H2
​
  
3
  
s_H1
​
  
2
  
s_fH2O
​
  
s_z1
​
  
s_zH2O
​
 +8 
s_p
​
  
2
  
s_H2
​
  
3
  
s_H1
​
  
2
  
s_fH2O
​
  
s_z2
​
  
s_zH2O
​
 −8 
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
2
  
s_z1
​
  
s_z2
​
 +8 
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
2
  
s_z1
​
  
s_zH2O
​
 −8 
s_z2
​
  
2
  
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
2
 −8 
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
2
  
s_z2
​
  
s_zH2O
​
 +4 
s_z1
​
  
2
  
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
2
 +8 
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
2
  
s_z1
​
  
s_zH2O
​
 +4 
s_zH2O
​
  
2
  
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
2
 +8 
s_p
​
  
3
  
s_H2
​
  
2
  
s_H1
​
  
2
  
s_fH2O
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
3
  
s_H2
​
  
2
  
s_H1
​
  
2
  
s_fH2O
​
  
s_z1
​
  
s_zH2O
​
 −8 
s_p
​
  
3
  
s_H2
​
  
2
  
s_H1
​
  
2
  
s_fH2O
​
  
s_z2
​
  
s_zH2O
​
 −8 
s_zH2O
​
  
2
  
s_p
​
  
3
  
s_H2
​
  
2
  
s_H1
​
  
2
  
s_fH2O
​
 +4 
s_z2
​
  
2
  
s_p
​
  
4
  
s_H2
​
  
2
  
s_H1
​
  
2
 +8 
s_p
​
  
4
  
s_H2
​
  
2
  
s_H1
​
  
2
  
s_z2
​
  
s_zH2O
​
 +4 
s_zH2O
​
  
2
  
s_p
​
  
4
  
s_H2
​
  
2
  
s_H1
​
  
2
 −8 
s_p
​
  
2
  
s_H2
​
  
4
  
s_H1
​
  
s_fH2O
​
  
s_z1
​
  
s_z2
​
 +8 
s_p
​
  
2
  
s_H2
​
  
4
  
s_H1
​
  
s_fH2O
​
  
s_z1
​
  
s_zH2O
​
 −8 
s_z2
​
  
2
  
s_p
​
  
2
  
s_H2
​
  
4
  
s_H1
​
  
s_fH2O
​
 −8 
s_p
​
  
2
  
s_H2
​
  
4
  
s_H1
​
  
s_fH2O
​
  
s_z2
​
  
s_zH2O
​
 −8 
s_z1
​
  
2
  
s_p
​
  
3
  
s_H2
​
  
4
  
s_H1
​
 −8 
s_p
​
  
3
  
s_H2
​
  
4
  
s_H1
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
3
  
s_H2
​
  
4
  
s_H1
​
  
s_z1
​
  
s_zH2O
​
 +8 
s_p
​
  
3
  
s_H2
​
  
4
  
s_H1
​
  
s_z2
​
  
s_zH2O
​
 +8 
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
3
  
s_H1
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
3
  
s_H1
​
  
s_z1
​
  
s_zH2O
​
 −8 
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
3
  
s_H1
​
  
s_z2
​
  
s_zH2O
​
 −8 
s_zH2O
​
  
2
  
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
3
  
s_H1
​
 +16 
s_z1
​
  
2
  
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
s_fH2O
​
 +16 
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
s_fH2O
​
  
s_z1
​
  
s_z2
​
 +16 
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
s_fH2O
​
  
s_z1
​
  
s_zH2O
​
 +16 
s_z2
​
  
2
  
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
s_fH2O
​
 +16 
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
s_fH2O
​
  
s_z2
​
  
s_zH2O
​
 +16 
s_zH2O
​
  
2
  
s_p
​
  
3
  
s_H2
​
  
3
  
s_H1
​
  
s_fH2O
​
 +8 
s_p
​
  
4
  
s_H2
​
  
3
  
s_H1
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
4
  
s_H2
​
  
3
  
s_H1
​
  
s_z1
​
  
s_zH2O
​
 −8 
s_p
​
  
4
  
s_H2
​
  
3
  
s_H1
​
  
s_z2
​
  
s_zH2O
​
 −8 
s_zH2O
​
  
2
  
s_p
​
  
4
  
s_H2
​
  
3
  
s_H1
​
 −8 
s_z1
​
  
2
  
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_H1
​
 −8 
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
s_z1
​
  
s_zH2O
​
 +8 
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_H1
​
  
s_z2
​
  
s_zH2O
​
 −8 
s_p
​
  
4
  
s_H2
​
  
2
  
s_H1
​
  
s_fH2O
​
  
s_z1
​
  
s_z2
​
 +8 
s_p
​
  
4
  
s_H2
​
  
2
  
s_H1
​
  
s_fH2O
​
  
s_z1
​
  
s_zH2O
​
 −8 
s_z2
​
  
2
  
s_p
​
  
4
  
s_H2
​
  
2
  
s_H1
​
  
s_fH2O
​
 −8 
s_p
​
  
4
  
s_H2
​
  
2
  
s_H1
​
  
s_fH2O
​
  
s_z2
​
  
s_zH2O
​
 +4 
s_z2
​
  
2
  
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
4
 +8 
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
4
  
s_z2
​
  
s_zH2O
​
 +4 
s_zH2O
​
  
2
  
s_p
​
  
2
  
s_fH2O
​
  
2
  
s_H2
​
  
4
 +8 
s_p
​
  
3
  
s_H2
​
  
4
  
s_fH2O
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
3
  
s_H2
​
  
4
  
s_fH2O
​
  
s_z1
​
  
s_zH2O
​
 −8 
s_p
​
  
3
  
s_H2
​
  
4
  
s_fH2O
​
  
s_z2
​
  
s_zH2O
​
 −8 
s_zH2O
​
  
2
  
s_p
​
  
3
  
s_H2
​
  
4
  
s_fH2O
​
 +4 
s_z1
​
  
2
  
s_p
​
  
4
  
s_H2
​
  
4
 +8 
s_p
​
  
4
  
s_H2
​
  
4
  
s_z1
​
  
s_zH2O
​
 +4 
s_zH2O
​
  
2
  
s_p
​
  
4
  
s_H2
​
  
4
 −8 
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
3
  
s_z1
​
  
s_z2
​
 +8 
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
3
  
s_z1
​
  
s_zH2O
​
 −8 
s_z2
​
  
2
  
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
3
 −8 
s_p
​
  
3
  
s_fH2O
​
  
2
  
s_H2
​
  
3
  
s_z2
​
  
s_zH2O
​
 −8 
s_z1
​
  
2
  
s_p
​
  
4
  
s_H2
​
  
3
  
s_fH2O
​
 −8 
s_p
​
  
4
  
s_H2
​
  
3
  
s_fH2O
​
  
s_z1
​
  
s_z2
​
 −8 
s_p
​
  
4
  
s_H2
​
  
3
  
s_fH2O
​
  
s_z1
​
  
s_zH2O
​
 +8 
s_p
​
  
4
  
s_H2
​
  
3
  
s_fH2O
​
  
s_z2
​
  
s_zH2O
​
 +4 
s_z1
​
  
2
  
s_p
​
  
4
  
s_fH2O
​
  
2
  
s_H2
​
  
2
 +8 
s_p
​
  
4
  
s_fH2O
​
  
2
  
s_H2
​
  
2
  
s_z1
​
  
s_z2
​
 +4 
s_z2
​
  
2
  
s_p
​
  
4
  
s_fH2O
​
  
2
  
s_H2
​
  
2
 
​