Automated synthesis techniques generate systems—such as functions or circuits—that provably satisfy a formal specification. Traditional synthesis frameworks often adopt an all-or-nothing approach: either the system satisfies all constraints, or synthesis fails. However, in many practical settings, such strict completeness is either infeasible or too costly to achieve, especially in terms of resources like time, memory, or circuit area. This work addresses such scenarios by moving beyond the all-or-nothing paradigm. We propose a novel synthesis framework that distinguishes between hard constraints, which must be strictly satisfied, and soft constraints, which may be relaxed. The goal is to synthesize systems that provably satisfy all hard constraints while achieving a user-defined threshold of satisfiability on the soft constraints. We quantify this relaxation using a satisficing measure, such as accuracy—i.e., the proportion of inputs for which the system satisfies all constraints. Our approach integrates AI-based methods to generate candidate systems and automated reasoning techniques to ensure formal guarantees. Through extensive experiments, we show that our framework significantly reduces synthesis time compared to traditional approaches. Moreover, the synthesized systems (e.g., circuits) tend to be smaller, connecting our method naturally to the domain of approximate circuit synthesis. Unlike existing approximate synthesis techniques, our framework provides formal guarantees on both correctness (for hard constraints) and quality (for soft constraints).