@acer 

Your worksheet gives

(is(abs(x) = max(x, -x)) assuming positive);
                              true

(is(abs(x) = -min(-x, x)) assuming positive);
                              true

(is(abs(x) = -min(-x, x)) assuming negative);
                              true

@dharr 

I have used the above worksheet to compute solutions for x, y and y2 for a given z with fsolve similar to the original call

A first result is about 10 times faster

Substituting this results into the original expression for eqm gives not really equality

I attribute this to the large values for y and y2. So maybe that is not a real solution to the origianal problem with exp().

On the other hand if I give a solution from my worksheet as inital values, the agreement is better

All this without procedures. I think that a further gain with codgen,makeproc and codegen,optimized is possible. We might see a gain in speed of about x25.

Before going further in this direction, the following should be discussed:

Technically we could run x as a parameter (which means varying T) and with that solve for y, y2 and z.
z will give us d. I have tried this with initial values starting from here:

This can only be done in tiny increments of x. At each increment the former solution is used as start value. Very soon the solution jumps to negative values for z. So either there is a bug in the xyz approach, which I do not think, or we are dealing with a singularity.

@dharr : Thank you for the background. I was realy thinking about something else.

@Gabriel Barcellos

I have tried modified equations with a reduced number of arithmetic operations (removing 1/T in the sums, see eqm_eqm2_eqTPO.mw). I did this by chanceling beta in m0. To my surprise fsolve took longer with the simplified equations. Now there is only one simplifcation left which is removing redundancies in the 3 equations (see update 3). This is in itself a new question where I have no good answer for and my time to work on it is limited. However, worksheets with a reduced number of terms in the sums are a big help to work on this problem.

Concerning physics: It is the first time that I see such expressions (big sums which are not series) in a physical context. Since it is new to me I share @one man doubts and I cannot interprete results. Any bockgound would be of help. I was surprise to see that only S0 has to be modifed for a lower spin number. I did expect Z0 also to have a reduced number of additions. Mathematically we can always plug in results in the set of equations that fsolve solves to verify Maples numerical results. This is less of a worry to me.

Concerning solving the problem: You have to tell us what you are finally looking for. Is it a T-d diagram, as we see it at the end of your last attachment for 3/2 spin? Depending on that answer a grid of data point might be ennough for you or is a restructured problem with less redundancies (where we do not know how much fsolve will get faster) needed.

@Gabriel Barcellos

As I see it at the moment:

• The equations eqm, eqm2 and eqTPO can be simplified to the below equations. But that's it pretty much. As @dharr said the problem is the large number of exponentials and this has not changed

• I could not optimize your original code further than the procedure p_O2 (see  my answer to @acer). A gain in speed by a factor of 2 to 3 is possible and on my fast machine I could compute solutions. On my old PC it simply took too long. The code should be faster with the above simplifications. I can try if you can provide simplified equations.
• I do not think that approximating the exponentials by series is a good idea. I cannot see now you will get exact results.
• I also think that using computer algebra is less prone to errors.
• I still think that a worksheet for spin 3/2 would be helpfull that @one man  can have a look at the equations to apply a different root finding method. If that works, it will be quite easy to scale up to 7/2. A worksheet with the big equations is simply not managable because of long computation delays and the responsiveness of the GUI. I can hardly edit code on my machine. What is possible is replacing S0 and Z0 in a worksheet for spin 3/2 and then executing the worksheet for a 7/2.

Update:

I forgot to upload

eqm_eqm2_eqTPO.mw (again corrected)

Update 2:

I had to correct eqTPO in the above.

Update 3:

Missed to add m2 in the argument of d__i in eqm2 and eqTPO. Now corrected.

Redundancies:
2 of the 6 sums in the above equations are doubled.

Also in the sums of eqm and eqm2 is redundancy. Computing all the 50238 exponentials, for example, in the denominator of

covers all the exponentials in the numerator. Here is code to verify this

exp_n:=indets(numer(lhs(eqm)),function):nops(%);
exp_d:=indets(denom(lhs(eqm)),function):nops(%);
exp_d minus exp_n;nops(%)

(note: the difference are the exponentials that do not depend on m. This is the reason why I had to correct a second time)

The number of non redundant exp calls is less than 50% of the exp calls in the above equations. So far I have not found a smart way to remove redundancies using Maple's strength. It could be that fsolve is already recognising some of the redundancies and there is not much to gain. Maybe someone can tell.

@acer 

I could make hfloat work togehter with the fsolve call. evalhf does not work when calling fsolve.
Making an optimized procedures out of the expression showed the biggest gain in computation time (real time).

It seems to me that for this particular problem there is not much to gain with hardware floats. I am surprised.

Download Campo_Médio_spin_7_2_-_Forum_optimize_03_b.mw

@Carl Love

How to map what you demonstrated above to a list of procedures. Here is an example

P:= [proc(x) sin(x) end proc,proc(x) sin(x)^2 end proc,proc(x) sin(x)^3 end proc]:

P(3.);

         [0.1411200081, 0.01991485669, 0.002810384737]

For the time beeing, I work separately listelemet by listelement

@acer 

There was an example in my question that has disappeared. I didn't have time to rewrite it.
I've progressed enough (with the support I got from this question) in the other question with the 100k exp calls that I feel confident in giving an overview of some ways to improve computing power. What I can see so far gain by hardware floats is limited.
I will post it there once presentable.

@acer 

dharr answered one part. His answer can be used for the second part of my question.

@dharr 

Excelent. Is there a way to do this with the body?

subs(sin = evalhf@sin, sin(x))
                        (evalhf@sin)(x)

@Gabriel Barcellos 

I the latest file i uploaded I tried Compiler:-Compile. This does not finish in a reasonable time which is probably due to the large expressions. It takes much more time than generating a procedure internally (the help page indicates that this can be a problem).

For the evalhf I will ask separately. I have tried a few things but always got error messages. Edit: This does not work for example:
It could be that evalhf is limited to procedures with not more than 500 function calls. In this case it is not an option.

A key to test this and other new ways would be a test file with less exponential expression (e.g. spin 3/2). See my reply to @one man. If we get new methods to work we can scale up to spin 7/2.

@dharr 

Not always: As if the GUI does some house keeping from time to time.

I am wondering if the context panel or the pallet can play a role (or something else) since they are updated depending on the kernel state and the cursor position.

Maybe there are GUI functions that can be turned off.

@one man 

My computer works but becomes unresponsive and I cannot work efficiently.

It should be possible to reduce the number of spins to work on algorithms and to better analyse the equations. 

If the OP is interested he should provide S0 and Z0 for Spin 3/2. This should make the problem managable.

Let's wait for his feedback.

@one man 

The lhs of eqm and eqm2 is basically a fraction of two big sums of exponential functions with the parameters d, m, m2 and T.

The lhs and rhs of eqTPO contain sums. One summand is a ln function with a big sum of exponential functions

I have updated my response with new procedures to evaluate the equations.

You can analyse this way.

aaa := op(3, lhs(eqm2));
bbb := op(4, lhs(eqm2));
whattype(aaa);
whattype(bbb);
                              
nops(aaa);
nops(bbb);
                             
op(1 .. 4, aaa);
                                  
ccc := op(1, bbb);
nops(ccc);
                           
op(1 .. 4, ccc);

@Gabriel Barcellos

The Time command I used in my worksheet is the system time. The codetools:-usage output show cpu time and the real time which should match the system time difference before and after the fsolve call.

@acer suggested something I have no experience with. If I interprete him correctly he thinks of converting the equations for fsolve into procedures. In doing so evalhf can be applied to the procedure and functions inside the procedure as well as code can be generated with the Compiler:-Compile command.

Attched is a way that uses the codegen library to generate procedures. Its about 2 times faster. Since it takes 5 minutes time to optimize the procedures it is advisable to define and optimize the procedures only once by including d as a fourth parameter. I have not done this because of lack of time.

I am curious how much my attempt can be outperformed by other methods.

Campo_Médio_spin_7_2_-_Forum_optimize.mw

P.S.:

I have seen in your code that n and h are not used. I was further wondering whether in the exponent of Z0 there is not an m missing (since further down in the document a is replaced by J*m*something)

P.P.S.:

The root you want to encircle by several fsolve calls could perhaps be found with Draghilev’s method.

Looks at this example. Similarily to your problem a projection from R^4 (in your case d, m, m2 and T) to R^3 (m, m2, T) could be performed. Varing d would walk you along the pink line toward the root your are looking for.

@one man could tell if that is an option for your problem. As for acers suggestion, I have no practical experience with that method.

Update: This file contains procedures for the equations with the four parameters d, m, m2 and T

Campo_Médio_spin_7_2_-_Forum_optimize_02.mw

fsolve is a notch faster when using these. An attempt with Compiler:-Compile did not succeed.