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

Annotations already mark it as read-only, idempotent, and non-destructive, but the description adds important behavioral details: squares are rejected, pricing ($0.001 USDC per call), and an alternative HTTPS endpoint. It doesn't specify what error is returned for squares, but the added context is substantial.

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 definition, followed by illustrative examples, a constraint, range, and relevant pricing/endpoint info. Every sentence is purposeful and no redundancy is present.

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?

Given a simple one-parameter tool with no output schema and read-only annotations, the description covers the calculation, constraints, cost, and alternative access. It could explicitly state the return value is an integer period, but otherwise it provides enough for correct invocation.

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?

With 0% schema description coverage, the description compensates by stating the domain (d from 1 to 100000) and the key constraint (squares rejected). The parameter's role is implied by 'sqrt(d)', and the added constraint is not present in the schema.

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 computation: the period of the continued fraction of sqrt(d). Includes illustrative examples (sqrt(2)→1, sqrt(3)→2, etc.) and a constraint (squares rejected), making the tool's exact purpose unambiguous.

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?

No explicit when-to-use guidance or alternatives mentioned, despite many sibling tools like cf_sqrt, cf_expand, and sqrt_cf that could be confused with this one. The implied context is clear from the purpose, but the agent is left to infer when this tool is preferable.

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