Backward type inference in SBCL?

Hi! I am learning SBCL internals in order to teach the compiler to generate
more optimal code. I've noticed that a type of a function's result can be
derived from types of its arguments by writing a DERIVE-TYPE optimizer
using DEFOPTIMIZER. Is there a mechanism which can improve the compiler's
knowledge about the types of arguments based on a current knowledge about
the types of arguments and the result?

E.g. if we have NUMBER + SINGLE-FLOAT = DOUBLE-FLOAT, then the first arg of
+ must be DOUBLE-FLOAT

I am curious because I want to improve my cl-forward-diff library which
adds a new numeric type and provides its own math functions for automatic
differentiation. I want to go a bit further and make type inference to play
nicely with those functions.

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