I am simulating a system with state that can be thought of as a set of scalar-valued signals that are combined by arithmetic operations (+, -, *, /). This system is a model of some (*very* vaguely specified) physical processes, so I want to limit the ranges of the signals to some finite interval, say [-1, +1] for all the signals for the sake of concreteness and simplicity. (Mister Physical Process is not at home to unbounded ranges of values. Think along the lines of an electronic analogue computer only being able use voltages in a fixed range.)
I want definitions of the arithmetic operators such that are defined only for operands in the interval [-1, +1] such that the result also lies in [-1, +1] and that otherwise these operators behave as much like the standard arithmetic operators as possible. (That's grossly underspecified as an objective.)
I could do something like apply the standard operators and linearly rescale the results so that the interval of possible results is scaled back to [-1, +1]. However, that would effectively be applying a uniform gain of less than 1 to all the signals, so the signals would fade as they flow through the system. I think it would be better to have something where the effective gain is one when the result is 0 and the effective gain decreases as the magnitude of the unscaled result gets larger - so that the absolute value of the scaled result can approach but never exceed 1.
I would greatly appreciate any pointers to work that might help defining such arithmetic operators. (Unfortunately I can't think of any sensible search terms other than the long-winded description above).
BONUS QUESTION: The above, but for complex values rather than reals. For concreteness, assume that the operators are closed within the unit circle on the complex plane.
