Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

Annotations already declare read-only, idempotent, non-destructive, open-world behavior, so the bar is lower. The description still adds real context beyond them: deterministic output format (reduced fraction), the zero-value rule for odd indices, the $0.001 USDC x402 charge per successful call, and the equivalent HTTPS endpoint. That is substantial transparency about cost and result shape.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Is the description appropriately sized, front-loaded, and free of redundancy?

Front-loaded with the core result, then examples, then the domain rule, then cost and transport. Every sentence carries distinct information (semantics, edge case, constraint, price, endpoint) with no repetition or filler.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

There is no output schema, so the description carries the return-shape burden and does so concretely via reduced-fraction examples and the odd-index zero rule. Combined with the disclosed price and alternate HTTPS endpoint, an agent has everything needed to call this correctly and understand the cost.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema coverage is 0%, so the description must carry the parameter meaning, and it partially does: it fixes n as the Bernoulli index, restates the 0-40 domain already in the schema, and describes the value at special indices. What it does not add is anything about behavior at invalid/out-of-range input or precision/format nuances beyond the examples, leaving a gap against the low coverage baseline.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Does the description clearly state what the tool does and how it differs from similar tools?

States a specific result (the nth Bernoulli number as a reduced fraction) and even names the algorithm used (Akiyama-Tanigawa). It anchors the output with concrete examples (B0=1, B1=1/2, B2=1/6, B4=-1/30), so an agent knows exactly what it gets, distinct from numeric siblings like bernoulli_sum or bernoulli_denom.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Does the description explain when to use this tool, when not to, or what alternatives exist?

The description establishes operating scope (n from 0 to 40) and a useful edge-case rule (odd indices after 1 are 0), which guides correct invocation. However, it never says when to prefer this over siblings bernoulli_denom or bernoulli_sum, nor does it state prerequisites for payment beyond the price line. Usage is implied by the domain, not explicitly routed.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.