Leveraging large language models (LLMs) for designing reward functions demonstrates significant potential. However, achieving effective design and improvement of reward functions in reinforcement learning (RL) tasks with complex custom environments and multiple requirements presents considerable challenges. In this paper, we enable LLMs to be effective white-box searchers, highlighting their advanced semantic understanding capabilities. Specifically, we generate reward components for each explicit user requirement and employ the reward critic to identify the correct code form. Then, LLMs assign weights to the reward components to balance their values and iteratively search and optimize these weights based on the context provided by the training log analyzer, while adaptively determining the search step size. We applied the framework to an underwater information collection RL task without direct human feedback or reward examples (zero-shot). The reward critic successfully correct the reward code with only one feedback for each requirement, effectively preventing irreparable errors that can occur when reward function feedback is provided in aggregate. The effective initialization of weights enables the acquisition of different reward functions within the Pareto solution set without weight search. Even in the case where a weight is 100 times off, fewer than four iterations are needed to obtain solutions that meet user requirements. The framework also works well with most prompts utilizing GPT-3.5 Turbo, since it does not require advanced numerical understanding or calculation.