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

Annotations already cover the safety profile (readOnly, idempotent, non-destructive). The description adds genuinely useful behavior that annotations cannot express: the 0–1000000 input range and the switch to a decimal string past 15 digits, so the caller knows the return type changes. Pricing and HTTPS endpoint are also disclosed up front.

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?

Three compact sentences front-load the definition, examples, and range, then append the payout and endpoint facts. Every sentence adds information an agent would otherwise have to guess.

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 and no annotations covering return format, the description correctly supplies the return-shape caveat (decimal string past 15 digits) and the accepted input range. It lacks only sibling routing, which context can partly infer from the highly specific name.

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 description coverage is 0%, so the description must carry the load, and it does: it defines n as the index of the tetrahedral number and restates the domain 0–1000000, matching the schema bounds. It stops short of describing behavior for edge values like n=0 beyond the formula itself.

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 mathematical operation with its exact closed-form formula and worked examples (5th=35, 8th=120), making the computation unambiguous. The name plus formula distinguishes it from siblings like polygonal or centered_polygonal.

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 makes the domain of use clear (compute the nth tetrahedral number) but never contrasts with alternatives such as polygonal, octahedral, or tetranacci, nor states when this is the right sibling instead. In a family of hundreds of near-identical number tools, that omission matters more than usual.

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