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

Annotations already declare readOnly, idempotent, non-destructive, so the safety profile is covered; the description adds real value beyond that with the exact return shape, the failure mode for perfect squares, the accepted domain, and the $0.001 x402 pricing plus an HTTPS fallback endpoint.

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 operation, then a worked example, then constraints, then access details — no wasted sentences. The example pair earns its place by showing the output format, though the pricing/URL sentence is somewhat peripheral to calling the tool.

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?

With no output schema, the description correctly explains the return value and one error case, which is the big gap it had to fill. What it leaves out is sibling routing — for an agent facing cf_sqrt, sqrt_period, sqrt_tail, and cf_expand, the lack of disambiguation is the missing piece.

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 burden, and it does: it names the domain (1 to 10000, matching schema min/max) and adds the rejection rule for perfect squares, which the schema cannot express. It stops short of explaining output length or period-size limits.

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?

Specific verb+resource: computes the continued fraction of sqrt(d) and returns a0 plus the repeating block, illustrated with sqrt(2) and sqrt(19). However, it never distinguishes itself from very close siblings like cf_sqrt, sqrt_period, sqrt_tail, or square_obstruction in a catalog of hundreds of number-theory tools, leaving the agent to guess which one to call.

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?

'A square is rejected' and 'd runs from 1 to 10000' give input preconditions, which implies when the call is valid, but there is no explicit when-to-use/when-not guidance and no named alternative among the cf_* and sqrt_* siblings.

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