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see if this calculator agrees with the math you are using for the production chain
https://kirkmcdonald.github.io/calc.html#data=1-1-19&rate=s&min=2&items=production-science-pack:r:1
If you have one assembly machine II directly feeding iron sticks into 2 of your rail machines the inserter swing time may be the issue.
Thank you for the calculator link. If I understand it correctly, this calculator gives the same result.
You do need 10 rails per second to make 1 purple science per second. And 4 machines are enough because each production cycle produces 2 rails rather than just 1.
Direct insertion is indeed faster. You need two fast inserters to keep up with the 2.5 stick per second output you need, so you should be fine with the output of your stick maker if one maker is feeding two rail makers.
The bigger problem is the rail makers, which also need to be producing 2.5 items per second, and so you still need two fast inserters to keep up.
But you don't need to calculate this; once you solve your belt problem, you can recognize this problem from the fact that the assembler will briefly stop because its output slot is full.
One fast inserter will be fine here if you get enough inserter capacity research so they have a stack size of 2.
2 rail produced in 0.5 sec (or 4 rail/sec) at 1 crafting time.
Assembling Machine 2 : 0,75 crafting time.
So you're producing for 1 assembling machine 3 rails / sec.
With 4 assembling machine, it's 12 rails / sec.
1 yellow belt is 15 items / sec FOR BOTH LINE.
So if you're outputing on only 1 yellow lane, you can't have more than 7,5 items / sec and that's where is your bottleneck.
Upgrading to red belt will solve this without redoing anything. (15 items / sec PER LANE).