Reverse-Engineering Real Subscription Quotas from Floating-Point Usage Ratios

Why the “Cheapest” Plan Is Often Difficult to Evaluate

When enterprises purchase AI coding tools or large language model access, the monthly fee is only the first layer of information. The actual unit cost depends on how quotas are defined, how different models and input/output tokens are weighted, and how rolling limits—such as short burst quotas and weekly caps—work together. Marketing claims such as “N times the usage” are often promotional figures and do not necessarily map directly to the credits recorded by the backend.

A less obvious issue is that usage APIs may return “credits used / total credits” as an IEEE 754 double-precision floating-point value so that the frontend can render a progress bar. If the server does not round the result, the client may display a suspiciously long decimal. Such values are often not random noise. They may be binary approximations of a rational number. By recovering that fraction and comparing multiple observations, it is possible to estimate the actual quotas and multipliers behind different subscription tiers.

This article presents a reusable accounting method. It does not depend on any specific vendor’s proprietary API and can be applied to self-hosted gateways, cloud-based LLM workloads, and internal credit systems.

A General Structure for Credit-Based Usage

Many usage systems introduce an intermediate unit such as credits or compute points instead of exposing raw token counts to customers. Common reasons include:

• Different model sizes have different unit costs and need a common accounting unit.
• Output is often more expensive than input because of asymmetric inference costs, KV-cache usage, sampling overhead, and related factors.
• Systems need to enforce both short-term burst limits and long-period totals to prevent workloads from saturating a cluster.

A practical conversion model can be written as:

cost = tokens_in * w_in * m_model + tokens_out * w_out * m_model

Here, m_model is the model’s multiplier relative to a baseline model, while w_out / w_in is often several times greater than one. If internal pricing and the public API price list maintain the same proportions, subscription plans and pay-as-you-go billing remain consistent.

To evaluate whether a subscription is worthwhile, convert its quota into credits available for an equivalent baseline model and an equivalent input/output workload, then divide by the monthly fee.

Short-window quotas determine interactive performance: whether users are throttled during peak activity. Weekly or monthly caps determine whether the plan can support sustained development. Looking at only one of these constraints leads to an inaccurate assessment of value.

Why Floating-Point Usage Can Reveal Underlying Fractions

A double-precision floating-point number uses 53 significant bits to represent a real number. For a fraction such as p/q, where q is not too large, the decimal representation may contain a long repeating pattern or an irregular-looking tail. For example, a usage ratio might appear as:

0.16327272727272726

The final digits reflect binary rounding, while the middle of the value may reveal a recognizable repeating cycle.

The goal is not to guess. Instead, search for a reduced fraction in the rational-number space:

1. Treat the observed value as an interval, [x - eps, x + eps], where eps is one ULP or the precision used by the frontend.
2. Search the Stern–Brocot tree or a continued-fraction expansion for a fraction with a relatively small denominator that falls within the interval.
3. Use multiple independent observations—across accounts, timestamps, and before-and-after quota consumption—to verify whether the same denominator explains the results.

The Stern–Brocot tree enumerates all positive rational numbers in reduced form and is useful when the denominator is relatively small and the numerator represents a quota count. Continued-fraction convergents and semiconvergents can produce candidates more quickly. In practice, it is useful to retain several leading candidates and check which one can explain both short-window ratios and weekly-limit ratios.

This approach interprets data that has already been returned to the client. It is intended to clarify billing and quota definitions, not to attack a service or bypass limits. Authorized usage dashboards, exports, invoices, and official usage APIs remain the appropriate sources for compliant accounting.

Converting Recovered Fractions into Plan Quotas

Assume the entry-level plan provides a short-window quota of B_short and a periodic quota of B_period. A higher tier is advertised as offering k times the usage, but its actual short-window quota may be α * B_short, while its periodic cap is β * B_period. Neither α nor β must equal k, and they do not necessarily equal each other.

A practical accounting process is:

• Use the recovered fraction to estimate “current usage / total quota.” If the ratio is close to the complement of 1.0 immediately after a window reset, the absolute quota may be estimated by combining it with a known consumption amount, such as a fixed-size probe request.
• Convert the probe request into credits using the official or publicly documented token pricing. This establishes a scale such as credits per unit of currency or credits per request.
• Estimate the short-window quota and the periodic cap separately, then calculate α and β.
• Calculate unit cost as monthly fee divided by effective available credits. Effective credits are determined by the constraint reached first under the real workload: burst-oriented usage is usually limited by the short window, while batch processing is more likely to hit the periodic cap.

Typical findings may include:

• A mid-tier plan may provide more than its advertised multiplier within the short window but use a different multiplier for its periodic cap.
• A higher-tier plan’s advertised multiplier may align with only one of the two constraints.
• Output-heavy workloads can greatly increase actual consumption, causing a seemingly inexpensive plan to reach its limit quickly.

Any concrete quota or price must come from your own account’s billing records and usage API. Limits can change across environments and time periods, so a single observation should not be treated as a permanent price list.

Comparing Plans Without Relying on Marketing Copy

It helps to standardize the comparison in a table:

Dimension	What to record	Why it matters
Workload structure	Input/output ratio and model-tier distribution	Strongly affects credit weights
Short window	Reset period, quota, and whether requests can queue	Determines interactive responsiveness
Long period	Weekly or monthly cap and whether unused capacity carries over	Determines sustained capacity
Overage behavior	Throttling, degradation, or pay-as-you-go purchase	Determines worst-case cost
Collaboration	Whether seats share a common quota pool	Can significantly affect team economics

When calculating an effective unit price, do not use the advertised multiplier as the denominator. Use the observed effective credits instead.

If a higher tier has a substantially greater multiplier for its periodic cap than for its short-window quota, it may be better suited to stable automation pipelines. If the opposite is true, it may be more suitable for intermittent interactive coding.

For budgeting teams, unused quota should be treated as sunk cost. When long-term utilization remains below a certain threshold, moving to a lower tier and supplementing it with pay-as-you-go usage may be cheaper. This decision should be based on at least two to four complete billing cycles to avoid being misled by a single high-usage week.

Integrating the Accounting Process into Engineering Systems

A gateway or client can add a read-only observation layer:

# Example: record usage ratios without attempting to bypass limits
def record_usage(raw_ratio: float, window: str):
    candidates = rational_approximants(raw_ratio, max_den=10_000)
    emit_metric(window=window, ratio=raw_ratio, approx=candidates[0])

Combined with logging and monitoring alerts, this can warn when the short-window consumption rate exceeds the pace sustainable by the remaining periodic quota. Teams can then avoid discovering that the weekly allowance was exhausted only at the end of the week.

Archive the token statistics for each probe request in object storage so that the conversion coefficients can be audited later if pricing or weighting changes.

When evaluation jobs run on a container platform, apply the same credit budget at the Job level to prevent benchmark workloads from consuming the quota needed for interactive work. A message queue can defer low-priority batch processing until the usage window resets. These are general architecture capabilities and do not depend on a particular model provider.

For cloud-based workloads, connect four accounting dimensions on the billing side: request count, tokens, credits, and currency. Aggregate them using consistent labels such as project, environment, and model tier. Monitoring and logging services provided by cloud platforms, including Baidu AI Cloud where applicable, are sufficient to support this type of reconciliation without building a separate usage-analysis platform.

Practical Recommendations for Purchasing and Usage

• Define representative workload samples first, including real repositories and realistic review-to-generation ratios. Compare plans using those samples rather than vendor demonstrations.
• Stress-test short-window and long-period limits separately to identify which constraint is actually limiting your workload.
• Include output length and tool-call rounds in the cost model. These factors can affect the bill more than moving to the next subscription tier.
• Treat floating-point leakage as a technical clue, but use contracts, invoices, and official quota documentation as the final authority. Observations can help formulate questions but cannot replace commercial confirmation.
• Review quota policies regularly. Model updates and service changes can alter the accounting rules, so a one-time calculation should not be assumed to remain valid for a year.

Conclusion

Finding the cheapest subscription is not primarily about discovering hidden discounts. It requires reconstructing the credit definition, window constraints, and workload structure, then calculating the cost of effective capacity.

Unrounded floating-point usage ratios can make rational quotas more visible. The method relies on fraction approximation within an error interval and cross-validation across multiple usage windows. Embedding this analysis into monitoring and budgeting workflows is more reliable than focusing on a single advertised multiplier—and provides a more stable way to control AI infrastructure costs.