Evaluating LLM Math and Coding Capabilities: A Systematic Framework

Evaluating LLM Capabilities: A Systematic Approach from Math Reasoning to Engineering Quotas

In model selection and day-to-day usage, developers commonly face two challenges: how to objectively assess a model’s true capability on complex reasoning tasks, and how to complete engineering work efficiently under limited quotas. This article takes a general technical perspective to abstract reusable evaluation methods and resource management strategies, without recommending or benchmarking any specific product.

1. Evaluation Dimensions for Mathematical Reasoning

Complex math problems—open-ended questions, multi-step proofs, competition-level problems—are a touchstone for testing a model’s reasoning depth. The following dimensions are worth monitoring during evaluation:

1. Adaptability to open-ended problems: Whether the model can handle problems without standard answers that require constructive proofs, rather than merely matching problem types within its training distribution. This capability is typically related to the model’s chain-of-thought length, search strategy, and self-consistency verification mechanisms.
2. Chain-of-thought quality: Longer is not always better. Concise chains of thought often indicate stronger abstraction ability and lower inference cost; verbose, convoluted chains may reflect repeated trial-and-error at intermediate steps. Evaluation should record both accuracy and average token consumption.
3. End-to-end stability: The consistency rate across multiple samples on the same difficult problem (pass@k and self-consistency) reflects true capability better than a single success.
4. Tool invocation coordination: In scenarios requiring symbolic computation or numerical verification, whether the model can reasonably orchestrate external computation tools is an important indicator for engineering deployment.

In practice, a layered evaluation set can be constructed: basic problems (textbook-level), advanced problems (competition-level), and open problems (research-level), with accuracy, average reasoning steps, and token cost tallied separately to form a three-dimensional radar chart, avoiding misleading conclusions from any single metric.

2. Capability Layering for Coding Engineering Scenarios

Coding tasks cannot be evaluated generically; they should be layered by task type:

• Bug localization and fixing: Examines the model’s breadth of context understanding and minimal-change capability. The “regression test pass rate after fix” is recommended as a hard metric.
• Code review: Focuses on false positive rate and actionability of suggestions. The model can be asked to output structured review comments (issue location, severity level, fix suggestion), followed by manual spot-checking.
• Incremental feature development: Examines adherence to existing architectural conventions. During evaluation, provide project specification documents as context and observe whether the model follows layering and naming conventions.
• Subtask orchestration capability: When the primary model handles planning and lightweight models handle execution, planning quality directly determines overall efficiency. Evaluation should focus on whether task decomposition granularity is reasonable, whether context passing is complete, and whether failure retry strategies are sound.

A reusable orchestration pattern is illustrated below (schematic pseudocode):

def orchestrate(task, planner, executor):
    plan = planner.decompose(task)          # Split into independently verifiable subtasks
    for step in plan.steps:
        ctx = collect_context(step)          # Pass only necessary context to control tokens
        result = executor.run(step, ctx)
        if not verify(result):               # Verify each step to avoid error accumulation
            plan = planner.replan(step, result)
    return plan.assemble()

Key principle: Concentrate high-cost reasoning in planning and verification stages, delegate repetitive execution to low-cost models, and codify clear division-of-labor instructions in project specification files to reduce wasted context consumption.

3. Quota Management and Cost Control Practices

Under usage models billed by periodic quotas, a common pain point is “quota consumption rate not matching task progress.” Optimization can be approached from three directions:

1. Task tiering: Classify tasks by reasoning density into high, medium, and low tiers. Schedule high-density tasks in the early part of the quota period for concentrated processing, and use lightweight models for batch execution of low-density tasks.
2. Session granularity control: Long sessions continuously accumulate context, causing per-turn token costs to rise. Split sessions by functional module, and proactively reset context after task completion to avoid repeated billing for historical information.
3. Quota budget system: Set periodic quota caps for each task category (e.g., planning no more than 40%, execution no more than 50%, verification no more than 10%), combined with monitoring alerts to prevent a single task from exhausting the entire period’s allowance.
4. Degradation strategy: When quotas approach thresholds, automatically switch to lightweight models for execution-type subtasks, reserving high-capability models only for critical planning steps.

It should be emphasized that quota rules are commercial policies of service providers; developers should refer to their official documentation. This article discusses only general resource planning methods.

4. Selection Recommendations and Rational Expectations

Model capability rankings are dynamic, and any judgment of “currently strongest” has a limited shelf life. The rational approach is:

• Build your own evaluation set covering your business’s real distribution, and run regression tests regularly;
• Focus on the ratio of capability to cost, not the absolute capability ceiling;
• Maintain a multi-model replaceable architecture, isolating specific model differences through a unified abstraction layer for rapid switching as capabilities evolve;
• Maintain cautious expectations for open-ended reasoning tasks—any model has a failure rate, and engineering should include fallback and human review mechanisms.

In summary, evaluating large models should return to the task itself: clarify task types, define quantifiable metrics, control context and quota costs, and build a replaceable architecture. This methodology does not depend on any specific product and can help teams maintain rationality and efficiency in a rapidly changing technology landscape.