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Ultimath

Name: Ultimath
Author: Flupke68

by Flupke68

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Ultimath MCP

数字のハルシネーションを止めよう。 あらゆる数式を8つの独立したエンジンで並列に実行するMCPサーバーです。正確記号演算、ボール演算、区間演算、3種類の異なる任意精度浮動小数点数、遅延記号演算、IEEE 754 — それぞれが独自の故障モードを持ちます — そして8つの結果をすべて並べて返します。それらが食い違うとき、答えが数値的に信頼できないことが分かります。

エンジン	動作内容
Exact symbolic	正確な算術: 近似するのではなく値を決定する
Multiprecision	ボール演算。厳密な半径を計算全体で保持
Interval	厳密な下限/上限(保証された封入)
Binary float	任意精度のバイナリ浮動小数点数。内部にGMPを使わない
PARI/GP	GMPなしで構築: 独自の整数カーネルと超越関数カーネル
NTL RR	正確な整数仮数× 2^e; 丸めが発生するのは演算のみ
Deferred symbolic	評価の前に書き換え: キャンセルは決して丸められない
IEEE-754 double	標準的なハードウェア浮動小数点数

各エンジンが保証するもの

8つの数値を読むことは答えの半分にすぎません。各エンジンは、自分自身の結果をどこまで保証するかも示します。そして、その3つの回答は互いに程度の違いではありません:

フィールド	エンジンが示していること
`"exact": true`	値を保持しています。表示された数字はその値の切り詰めです。要求したさらなる桁はすべて、より近い近似ではなくその数そのものの桁です。
`"digits": N`	それ自身の誤差限界が先頭`N`桁の有効数字を保証します。`0`は実際の答えです。それが壊滅的な桁落ちが値に対して行うことです。
どちらでもない	誤差限界を持ちません。要求した桁数だけを出力し、そのどの桁についても何も言えません。

最後の行は0ではありません。そしてそれが、他のエンジンが存在する理由です。8つのうち5つは設計上そうなっています。

exp(pi*sqrt(163)) - 262537412640768744では、1つのエンジンは63桁を出力し、そのうち34桁が正しいのですが、それについて何も言いません。他の2つは33桁を証明して停止します。1つは値を正確に保持します。8つの数値を読むだけでは、どれがどれだか分からないでしょう。

どのエンジンも、他のエンジンについて尋ねられることはありません。各記述は、それ自体についてのものです。

Related MCP server: mcp-abacus

正確エンジン

calciumは桁を比較するのではなく等号を確定し、他のエンジンが近づくことしかできない極に到達します(tan(pi/2)はuinf)。また、出力された桁が単なる切り詰めである場合、閉形式も返します — sqrt(2)+1はプレーン形式とLaTeX形式の両方でAdd(Sqrt(2), 1)として返され、acosh(0.5)はDiv(Mul(NumberI, Pi), 3)として返されます。これは保存されている体の元ではなく、名前の付いた値です。

閉形式は、それを証明するのに桁を読むよりもコストがかかるため、HTTP APIではオプトインです("closed_form": true)。このクライアントは常にそれを要求します: 正確な値をそのまま返すことが、出力スキーマの約束だからです。

ツール

evaluate — 式を8つの全エンジンで評価し、比較します。算術演算、三角関数(sin, cos, tan)、指数/対数(exp, ln, log)、根とべき乗(sqrt, x^y)、階乗、複素数(3+2i)、異なる基数(0xFF, 0b1010)、定数(pi, e, 黄金比PHI)に対応しています。
list_functions — エンジンが公開するすべての関数を一覧表示します (名前、引数の数、カテゴリ、説明)。カテゴリによるフィルタリングも可能です。

比較は、すべてのエンジンで同じ意味を持つわけではありません。封入エンジンでは、!=、<、>は証明された場合にのみ真になります。==、<=、>=は反証されない限り真です。そこでの真の==は等号の証明ではありません — それを正確に決定するのはcalciumです — しかし、真の!=は差異の証明です。

乗算は明示的でなければなりません: 2*pi、2*sin(x)、(a+b)*(c+d)のように書いてください。隣接は積ではありません(2piはエラーです)。式は純粋に数学的なものです — 型キャストやコンストラクタはありません。複素数は1+2iのように書いてください。

セットアップ

https://ultimath.ai で無料のAPIキーを取得してください。
サーバーをMCPクライアント設定に追加します(Claude Desktopの例):

{
  "mcpServers": {
    "ultimath": {
      "command": "npx",
      "args": ["-y", "ultimath-mcp"],
      "env": {
        "ULTIMATH_API_KEY": "your_api_key_here"
      }
    }
  }
}

これだけです — npxが必要に応じてサーバーを取得して実行します。

要件

Node.js 18以上
UltimathのAPIキー(ULTIMATH_API_KEY)

リンク

ウェブサイト: https://ultimath.ai
ソースコード: https://github.com/Flupke68/ultimath-mcp

内部構造

UltimathのエンジンはFLINT/Arb、Calcium、GMP/MPFR、MPFIを基盤としています。クレジットの詳細は https://ultimath.ai/credits をご覧ください。

プライバシー

Ultimathは、サービスの運営と保護のために、APIキーのハッシュと基本的な利用メトリクスのみを保存します。式は評価されるだけで、トレーニング用に保持されることはありません。ポリシー全文: https://ultimath.ai/privacy/

ライセンス

MIT

Available Tools

2 tools

evaluateEvaluate & cross-check a math expressionA

Read-only

Verify any numeric result that has to be trusted, via the Ultimath API (https://ultimath.ai). Use this whenever the correctness of a number matters — floating-point-sensitive expressions, catastrophic cancellation, values near singularities or branch cuts, ties, or any computation where a single method could be silently wrong. Prefer it over computing by hand or running code: it evaluates the expression on 8 independent engines (exact symbolic, multiprecision ball, interval arithmetic, three different arbitrary-precision floats, deferred symbolic rewriting, IEEE 754 double) in parallel and returns every result plus per-engine precision diagnostics, so disagreement exposes an unreliable value that a single calculation would hide.

The 8 engines are independent and each computes with its own native primitive — so divergence between them (at poles, singularities, branch cuts, or ties like round(2.5)) is the backends' real behavior, NOT a bug. Agreement across engines is a confidence signal; divergence flags a value that is convention-dependent or near a domain edge — verify it rather than trust a single column.

One engine (calcium) is exact rather than numeric: it decides a value instead of approximating it, so it reaches poles the others can only approach (tan(pi/2) is uinf), and when the printed digits are only a truncation it also returns the closed form in symbolic / symbolic_latex (sqrt(2)+1 carries Add(Sqrt(2), 1)). Prefer that closed form when quoting an exact value.

A comparison does not mean the same thing on every engine. On the enclosure engines (flint, mpfi) !=, < and > return true only once the property is PROVEN, while ==, <= and >= return true as soon as it is not disproven. So a true == there is not a proof of equality — calcium decides that one exactly — but a true != IS a proof of difference. Each pair is complementary: a == b and a != b are never both false.

A value that is real comes back as a plain real number: an engine drops a zero imaginary part only when it PROVES the value real. So a trailing +0i is NOT decoration — it means that engine did not establish realness, and a visible residue (e.g. 4.3e-17i on the double engine) is genuine numerical error. Do not strip either one when reporting a result.

Supports: arithmetic, trigonometry (sin, cos, tan), exponentials and logs (exp, ln, log), roots and powers (sqrt, x^y), factorial, complex numbers (3+2i), arbitrary bases (0xFF, 0b1010), constants (pi, e, the golden ratio PHI). Multiplication must be EXPLICIT: write 2pi, 2(3+4), (a+b)(c+d), 2sin(x) — adjacency is NOT a product (2pi is an error). Precedence, tightest first: postfix factorial ! (2^3! = 2^(3!) = 64), then powers ^, then * / %, then + -. Note: expressions are mathematical only — there are no type casts ((int)x) or constructors (complex(1,2)); write a complex number as 1+2i or (re, im). Call list_functions for the authoritative list of available functions.

Name	Required	Description
`format`	No	Output format (default 'fixed')
`precision`	No	Decimal digits of precision (default 50). The ceiling is the PLAN behind the key — the free beta caps at 50 and refuses more — and 999 above every plan, the length of the built-in constants.
`expression`	Yes	Math expression, e.g. '0.1 + 0.2', 'sin(pi/4)', 'sqrt(2)^2 - 2', '2pi', '2(3+4)', 'factorial(10)'

Output Schema

Name	Required	Description
`ms`	No
`trace`	No
`result`	No	The retained reading — read `accuracy` before quoting any of its digits
`engines`	Yes	Per-engine results, compare to detect floating-point error
`inexact`	Yes	True if any engine emitted a warning (precision loss or domain issue somewhere)
`accuracy`	No	What the reading is worth once the other engines have been read against it. `exact` and `proved` are different KINDS of fact and stand together here
`angle_unit`	No
`expression`	Yes

TDQS

A5/5.0

Behavior5/5

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

The description explains nontrivial behavior far beyond the readOnlyHint annotation: engine divergence is real behavior, not a bug; comparison semantics differ per engine (e.g., `!=` can be a proof of difference while `==` is not a proof of equality); and trailing `+0i` indicates unproven realness rather than decoration. This is precisely the kind of behavioral nuance an agent needs.

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

Conciseness5/5

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

The description is long but every section earns its place: purpose, when-to-use, engine behavior, comparison semantics, realness, syntax, and precedence are all distinct and clearly organized. It is front-loaded with the core purpose and usage guidance before diving into technical details.

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

Completeness5/5

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

Given the tool's complexity, the description is unusually complete. It covers return content (per-engine results and precision diagnostics), engine semantics, exact closed-form output, precision limits, supported operations, syntax constraints, and points to list_functions for the authoritative list. Nothing essential is missing for an agent to invoke it correctly.

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

Parameters5/5

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

Schema coverage is 100%, but the description adds substantial meaning beyond the schema: explicit-multiplication rules, precedence order, complex-number syntax, unsupported constructs like type casts, the precision ceiling tied to the plan, and examples. This makes the expression parameter far more usable than the schema alone.

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

Purpose5/5

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

The description opens with a specific verb and resource: 'Verify any numeric result that has to be trusted, via the Ultimath API.' It clearly states what the tool does, names its multi-engine evaluation behavior, and differentiates it from computing by hand, running code, and the sibling list_functions tool.

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

Usage Guidelines5/5

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

It gives explicit when-to-use guidance: 'Use this whenever the correctness of a number matters,' and enumerates concrete scenarios such as floating-point-sensitive expressions, cancellation, singularities, branch cuts, and ties. It also says 'Prefer it over computing by hand or running code,' directly routing the agent away from alternatives.

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

list_functionsList available functionsA

Read-only

List all mathematical functions available via the Ultimath API (https://ultimath.ai). Returns name, arity, category, and description for each function. Use this to discover what functions you can pass to the evaluate tool. Optionally filter by category (algebraic, trigonometric, hyperbolic, exponential, special, rounding, introspection). Every function is available on all 8 engines, but each computes it with its own native primitive — so results may diverge at poles, branch cuts, or ties, and that divergence is honest backend behavior, not a bug.

Name	Required	Description	Default
`category`	No	Filter by category, e.g. 'special', 'trigonometric'. Omit to list all.

TDQS

A4.7/5.0

Behavior5/5

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

Beyond the readOnlyHint annotation, the description reveals important behavioral nuances: every function exists on all 8 engines but is computed via native primitives, so results may diverge at poles, branch cuts, or ties, and that divergence is intended backend behavior. This is rich, honest context that an agent needs before trusting listed functions.

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

Conciseness5/5

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

The description is front-loaded with purpose and return fields, then usage guidance, then filter options, then behavioral caveats. Every sentence earns its place; there is no filler or repetition of schema/annotation content.

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

Completeness5/5

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

For a simple read-only list tool with one optional parameter and no output schema, the description provides everything an agent needs: what it lists, what each entry contains, how to filter, how it relates to evaluate, and a warning about cross-engine divergence. No material gap remains.

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

Parameters4/5

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

The schema already covers the single optional 'category' parameter at 100% coverage, so the baseline is 3. The description adds value by enumerating all valid category values (algebraic, trigonometric, hyperbolic, exponential, special, rounding, introspection), which is especially useful since the schema provides no enum constraint.

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

Purpose5/5

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

The description states a specific verb and resource: 'List all mathematical functions available via the Ultimath API,' and explicitly lists the returned fields (name, arity, category, description). It clearly distinguishes itself from the sibling 'evaluate' tool by framing itself as the discovery step for functions to pass to evaluate.

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

Usage Guidelines4/5

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

It explicitly tells the agent when to use this tool: 'Use this to discover what functions you can pass to the evaluate tool.' It names the sibling evaluate as the downstream consumer, but it does not explicitly state a when-not condition or directly contrast with evaluate in an 'instead' formulation.

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

Tool Schema Changelog

Recent tool additions, removals, and schema changes observed during successful MCP inspections.

1 tool updatev0.6.0Aug 27, 2026

Changedevaluate10 fields changed

changedInput schema / properties / precision / description

Previous value: -"Decimal digits of precision (default 50). The free beta caps at 50; higher values are rejected server-side."New value: +"Decimal digits of precision (default 50). The ceiling is the PLAN behind the key — the free beta caps at 50 and refuses more — and 999 above every plan, the length of the built-in constants."

changedInput schema / properties / precision / maximum
```
Previous value: -50New value: +999
```

addedOutput schema / properties / accuracy

Added value: +{
+  "additionalProperties": false,
+  "description": "What the reading is worth once the other engines have been read against it. `exact` and `proved` are different KINDS of fact and stand together here",
+  "properties": {
+    "agree": {
+      "description": "Engines returning this same reading",
+      "type": "number"
+    },
+    "engines": {
+      "description": "Engines asked — refusals included, since a refusal is part of the answer",
+      "type": "number"
+    },
+    "exact": {
+      "description": "An agreeing engine HOLDS this value — the reading is a truncation of it",
+      "type": "boolean"
+    },
+    "proved": {
+      "description": "Leading significant digits of `result` covered by a DIFFERENT engine's own error bound. Past this point the digits were printed at the precision asked for and no engine underwrites them — quoting them is quoting one implementation. 0 is a real answer; absent means no other engine carries a bound at all, which is not the same statement",
+      "type": "number"
+    },
+    "source": {
+      "description": "Engine the reading came from",
+      "type": "string"
+    }
+  },
+  "required": [
+    "source",
+    "agree",
+    "engines"
+  ],
+  "type": "object"
+}

addedOutput schema / properties / engines / items / properties / digits

Added value: +{
+  "description": "Leading significant digits of `result` this engine backs with its OWN error bound; 0 means the bound covers none of them. With `exact` and `digits` BOTH absent, the engine carries no error bound at all and can say nothing about any digit it printed — which is not 0, and is the single most useful thing to read before trusting a number",
+  "type": "number"
+}

addedOutput schema / properties / engines / items / properties / diverges

Added value: +{
+  "description": "This engine's reading CONTRADICTS the consensus — a finding, not a formatting difference. Absent on an engine that agrees or cannot be compared",
+  "type": "boolean"
+}

addedOutput schema / properties / engines / items / properties / exact

Added value: +{
+  "description": "True when this engine holds the value EXACTLY — the reading is a truncation of the number itself, so every further digit you ask for is a digit of it. Not a very large `digits`: a different statement, never set alongside it",
+  "type": "boolean"
+}

changedOutput schema / properties / engines / items / properties / name / description

Previous value: -"Engine id: flint, dec, cpp, mpfi, or ntl"New value: +"Engine id: calcium, flint, mpfi, or cpp"

addedOutput schema / properties / engines / items / properties / symbolic

Added value: +{
+  "description": "Closed form of the value, e.g. Add(Sqrt(2), 1) — present only when the engine proved one AND the printed digits are a truncation of it",
+  "type": "string"
+}

addedOutput schema / properties / engines / items / properties / symbolic_latex

Added value: +{
+  "description": "The closed form as LaTeX",
+  "type": "string"
+}

addedOutput schema / properties / result

Added value: +{
+  "description": "The retained reading — read `accuracy` before quoting any of its digits",
+  "type": "string"
+}

1 tool updatev0.2.0Jul 5, 2026
- Changedevaluate1 field changed
  - changedOutput schema / properties / engines / items / properties / name / description
```
Previous value: -"Engine id: flint, dec, cpp, or mpfi"New value: +"Engine id: flint, dec, cpp, mpfi, or ntl"
```
2 tool updatesv0.1.5Jul 1, 2026
- First observedevaluate
- First observedlist_functions

Deploy Server

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TDQS

A4.6/5.0

Scored 2026-08-27 16:55 across 2 tools

Disambiguation5/5

evaluate performs mathematical computation and returns multi-engine results, while list_functions only returns metadata about supported functions. There is no overlap or ambiguity between the two operations.

Naming Consistency4/5

Both names are imperative and clear, but evaluate is a bare verb while list_functions follows a verb_noun pattern. This is a minor stylistic inconsistency that does not create confusion.

Tool Count4/5

Two tools is below the typical 3–15 range, but the server's scope is narrow: one computational entry point plus one discovery helper. Each tool is necessary, so the count is slightly under but reasonable.

Completeness4/5

The core workflow is complete: discover functions with list_functions, then evaluate expressions with evaluate. Since this is a stateless computation service, lifecycle operations do not apply; the main minor gap is that supported constants are documented only in evaluate's description rather than being discoverable through list_functions.

Maintenance

ActivityStale

ResponsivenessNo recent community issues

Resources

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Ultimath MCP

各エンジンが保証するもの

正確エンジン

ツール

セットアップ

要件

リンク

内部構造

プライバシー

ライセンス

Available Tools

Output Schema

TDQS

TDQS

Tool Schema Changelog

TDQS

Maintenance

Resources

Related MCP Connectors

Related MCP Servers

math-logic-mcp

mcp-abacus

loadbench-mcp

calc-mcp-server