When an ALU performs a floating point division operation using the non-restoring or the SRT algorithm, it maintains the current value of the "remainder" (in quotes, because it is not a true non-negative remainder) in a scratch register. After the operation, the quotient is presented in a user-visible location, and the contents of the scratch register are of no consequence.
However, if a user-visible register is used as the scratch, specifying its contents in all cases is too bothersome. For example, in the BESM-6, the instruction set manual leaves the contents of the accumulator extension register, normally holding the least significant bits of the mantissa after the FP additions or multiplications, unspecified after a division operation (another source explicitly says that it is undefined). I have not heard of attempts to reverse-engineer the value of that register after a division operation and to use it to speed up computation of the remainder.
A narrow-scope question: Is it possible to use the scratch register of a non-restoring or SRT division to figure out the remainder?
Update: As a matter of fact, it is the "positive" bits of the quotient which are left in a user-visible register. Unfortunately, there does not seem to be anything useful to be gleaned from comparing those bits with the final result.
My main question is, has there been another case of an unspecified data register contents after any arithmetic operation in another CPU architecture and were there any attempts to reverse-engineer them for a good use?