I'm writing a paper which explores a new forecasting method. I compare the results with those of an existing algorithm as a baseline. Plenty of literature already exists on the existing algorithm, and the definitions for the new forecasting method already make the paper very math-heavy. Being in-depth mathematical definitions of the existing algorithm can easily be found, is it necessary to define it mathematically in the paper? Would it suffice to clearly describe what it does in English? Adding math for the existing algorithm on top of the math for the new one adds significant complexity for the reader. On the other hand, readers who want the full definition may have to research elsewhere which could lessen the appeal of the paper to peer reviewers. Any thoughts?
In my experience, most of the times, when someone tries to "clearly describe in English" what an algorithm does avoiding mathematical notation, the result is not as clear as intended. Ask yourself if that is the case also for your description.
Math exists precisely because it makes it clear and unambiguous to speak about combinatorial concepts.
You need to cite the earlier method, making it easy for readers to find the details if needed. But you also need to satisfy reviewers and editors before your paper can be published. Or, for course work, satisfy the professor that your work is complete.
But other than that, the structure and content of your work is up to you. Make a decision about how you think it should be written and then do that. If complaints follow (reviewers...) make the needed changes to satisfy their concerns.
But, for the headline question, no it probably isn't necessary to completely define the earlier work unless yours can't be understood without it.
If you are comparing to published results of an existing method, then you are fine to cite the paper and provide a brief description.
However, if you are producing the results yourself, then you need to describe the exact algorithm you ran. This is because you will probably claim better results than that method (else why would you be publishing?), and I in my role as the reviewer will be skeptical of your claim. My first question will be why you did not compare directly against published results and instead ran the algorithm yourself. I will assume that since you didn't describe the algorithm you ran, you likely don't understand it fully yourselves and have misimplemented it. Finally, as you said "Being in-depth mathematical definitions of the existing algorithm can easily be found" --- multiple definitions of many algorithms can be found, typically not all equivalent. Reviewing, I will assume that you'll have picked the one variation that performs most poorly on the example you are presenting.
And, frankly, if the algorithm description isn't even in the appendix of the document, and I do have to look up another paper to understand your baseline, you'll have annoyed me and I'll be less likely to give you a favourable review.
At least in my discipline (economics-econometrics) the solution is called "On-line Supplementary material". It is submitted together with the paper, as a separate file and not part of the page count of the paper, but subject to peer-review.
If the paper is accepted for publication, this "Supplementary material" is posted on-line only, on the journal's website. It won't appear in the printed version (apart from an alert that "additional on-line material exists for this article").
It usually includes long mathematical derivations (i.e things for a Technical Appendix that are too long to attach to the main text of the paper), but also the details of some subsection of the paper that it is only summarized in the main text. For example a bag of simulations whose detailed tables are included in the Supplement, and only the main conclusion is reported in the main text.
So you can include all this additional math of the competing algorithm to such a file.
You may want to check with the journal that they do this.