A tourist visa/no visa allows you to meet your work colleagues, have meetings, lunches, to basically "do business" with them. It does not allow you to "work".
Imagine your job were moving boxes. You can visit the company, discuss further box moving, meet your boss in person, have lunch with your coworkers, take a tour of the new warehouse they opened, maybe spent half a shift learning all the things that are new in this specific warehouse about box moving, speak with the people moving boxes there, all those things. You cannot get into your dirty work pants and do a week of box moving for money.
The line is very blurry for knowledge workers. Where is the line between a business meeting and pair programming? And who would know? If you don't push your programming results to the code repository until after you are back in the US, did you actually work, officially? Somewhere between the falling tree in the woods and schrödingers Programming.
If you have a degree (lets say a BSc from any college in the US that vaguely matches your job description) and a minimum gross salary of 58,400 Euros (A reduced minimum annual gross salary of 45,552 Euros applies to employment in the occupational fields of mathematics, computer science, the natural sciences, engineering and human medicine (not including dentistry)), then you can apply for a blue card. As a US citizen you do not need a visa to come to Germany for a limited time, and you can apply for the blue card in country, while you are on your "free" visit.
Please note that Germany is very buerocratic, slow, and does not have an option to have your processes expedited, even for money. So the idea that you could just swing by and get your blue card in the 90 days you are allowed to be here "free" without any visa requirements is... theoretically and legally sound, but practically, you may not get your appointment to even tell them what you want until after 6 weeks. German "Amts" (civil administration offices) are no drive-ins. They are slow and overcrowded. A bit of planning ahead and a lot of patience goes a long way in that regard.