The number 10.3500574150076 is a numeric approximation of $\log(2)^2+\pi^2$. If we do not know its closed form and want to find it using computer, then there are a few things we may tre:
identify function.By asking my previous question, I also learned a new one.
Is there any other software that may be helpful for this kind of task?
This is not an answer to the question, but a comment too long to be put in the comments section.
I don't know if they are other public softwares than those in the large list already given.
For the record, it will be noted that one of the first software of this kind was developed by Simon Plouffe , Laboratoire de combinatoire et d'informatique mathématique de l'UQAM, Québec University, Montréal(Canada) and in the IUT Informatique de Nantes (France). Unfortunately, the public access to the so called "Plouffe Inverter" is no longer available.
I would like to give another viewpoint about these kind of softwares. This concerns the computer students and more precisely mathematical software development.
Developing a basic form of software of this kind is not so complicated that one think at first sight. This could be an excellent training for students learning data processing and computational science : That is why I pick that point up here.
Even with a small computer rather slow and of low capacity, already remarkable results can be observed. For example, with a personal computer of old generation, the example proposed by ablmf (10.3500574150076 ) is solved in an acceptable time. The solution $(\ln(2))^2+\pi^2$ appears in top place on the list of approximate formulas found :
The symbols are : p=$\pi$ , c=Catalan constant, g=Euler-Mascheroni constant, sh=$\sinh$, ch=$\cosh$, ...
The general principle to write such a software is roughly described in the paper (in French) : https://fr.scribd.com/doc/14161596/Mathematiques-experimentales
