Ok one specific example. I'd say the 10x part isn't about productivity but insight into the problem. Most people solve the problem as given, but the times when I've been 10x I managed to change the definition of the problem to something simpler, and easier to solve.
Here's an example: At one company they had a 'financial calculator,' i.e. a program to help visualise how some financial math played out over time and it was giving incorrect results. When I looked at the code I saw that it had been written by composing a lot of those functions you see on actual financial calculators, (1/V, PV, PMT, etc...).
This may have seemed simpler to approach it this way, but because there were so many compositions and there was a root finding algorithm involved, it quickly got out of hand. I started from the problem definition, defined a recurrence relation for what we are trying to calculate, found a closed form formula for the recurrence relation. This part was the hard part.
Now those 3000 lines of FV1P(XYZ(ABC())) turned into like 20 lines of simple math, _and_ the formula could now be solved for the other variables, so no more root finding. I just solved for the other variables, wrote functions for them and that was it. It ended up being faster, correct, numerically more stable, and the code was smaller.
I wrote up a white-paper on how the math worked, included a proof and that was it. It took two weeks.
The value to the customer was that this bug in this one part of their 'suite' of applications was holding up a sale to major prospective customer of their's. So fixing this had an outsized valued compared to the severity of the bug.