Perform aggregation operations on 1D arrays.

Examples:

SUMPRODUCT: operation="sumproduct", array1=[1,2,3], array2=[4,5,6] Result: 32 (1×4 + 2×5 + 3×6)

WEIGHTED AVERAGE: operation="weighted_average", array1=[10,20,30], weights=[1,2,3] Result: 23.33... ((10×1 + 20×2 + 30×3) / (1+2+3))

DOT PRODUCT: operation="dot_product", array1=[1,2], array2=[3,4] Result: 11 (1×3 + 2×4)

GRADE CALCULATION: operation="weighted_average", array1=[85,92,78], weights=[0.3,0.5,0.2] Result: 86.5

Perform element-wise operations on arrays using Polars.

Supports array-array and array-scalar operations.

Examples:

SCALAR MULTIPLICATION: operation="multiply", array1=[[1,2],[3,4]], array2=2 Result: [[2,4],[6,8]]

ARRAY ADDITION: operation="add", array1=[[1,2]], array2=[[3,4]] Result: [[4,6]]

POWER OPERATION: operation="power", array1=[[2,3]], array2=2 Result: [[4,9]]

ARRAY DIVISION: operation="divide", array1=[[10,20],[30,40]], array2=[[2,4],[5,8]] Result: [[5,5],[6,5]]

Calculate statistical measures on arrays using Polars.

Supports computation across entire array, rows, or columns.

Examples:

COLUMN-WISE MEANS: data=[[1,2,3],[4,5,6]], operations=["mean"], axis=0 Result: [2.5, 3.5, 4.5] (average of each column)

ROW-WISE MEANS: data=[[1,2,3],[4,5,6]], operations=["mean"], axis=1 Result: [2.0, 5.0] (average of each row)

OVERALL STATISTICS: data=[[1,2,3],[4,5,6]], operations=["mean","std"], axis=None Result: {mean: 3.5, std: 1.71}

MULTIPLE STATISTICS: data=[[1,2,3],[4,5,6]], operations=["min","max","mean"], axis=0 Result: {min: [1,2,3], max: [4,5,6], mean: [2.5,3.5,4.5]}

Transform arrays for ML preprocessing and data normalization.

Transformations: - normalize: L2 normalization (unit vector) - standardize: Z-score (mean=0, std=1) - minmax_scale: Scale to [0,1] range - log_transform: Natural log transform

Examples:

L2 NORMALIZATION: data=[[3,4]], transform="normalize" Result: [[0.6,0.8]] (3²+4²=25, √25=5, 3/5=0.6, 4/5=0.8)

STANDARDIZATION (Z-SCORE): data=[[1,2],[3,4]], transform="standardize" Result: Values with mean=0, std=1

MIN-MAX SCALING: data=[[1,2],[3,4]], transform="minmax_scale" Result: [[0,0.33],[0.67,1]] (scaled to [0,1])

LOG TRANSFORM: data=[[1,10,100]], transform="log_transform" Result: [[0,2.3,4.6]] (natural log)

Execute multiple math operations in a single request with automatic dependency chaining.

USE THIS TOOL when you need 2+ calculations where outputs feed into inputs (bond pricing, statistical workflows, multi-step formulas). Don't make sequential individual tool calls.

Benefits: 90-95% token reduction, single API call, highly flexible workflows

Quick Start

Available tools (20): • Basic: calculate, percentage, round, convert_units • Arrays: array_operations, array_statistics, array_aggregate, array_transform • Statistics: statistics, pivot_table, correlation • Financial: financial_calcs, compound_interest, perpetuity • Linear Algebra: matrix_operations, solve_linear_system, matrix_decomposition • Calculus: derivative, integral, limits_series

Result referencing:

Pass $op_id.result directly in any parameter:

• $op_id.result - Use output from prior operation

• $op_id.result[0] - Array indexing

• $op_id.metadata.field - Nested fields

Example: "payment": "$coupon.result" or "variables": {"x": "$op1.result"}

Example - Bond valuation:

{
  "operations": [
    {"id": "coupon", "tool": "calculate",
     "context": "Calculate annual coupon payment",
     "arguments": {"expression": "principal * 0.04", "variables": {"principal": 8306623.86}}},
    {"id": "fv", "tool": "financial_calcs",
     "context": "Future value of coupon payments",
     "arguments": {"calculation": "fv", "rate": 0.04, "periods": 10,
                   "payment": "$coupon.result", "present_value": 0}},
    {"id": "total", "tool": "calculate",
     "context": "Total bond maturity value",
     "arguments": {"expression": "fv + principal",
                   "variables": {"fv": "$fv.result", "principal": 8306623.86}}}
  ],
  "execution_mode": "auto",
  "output_mode": "minimal",
  "context": "Bond A 10-year valuation"
}

When to Use

✅ Multi-step calculations (financial models, statistics, transformations) ✅ Data pipelines where step N needs output from step N-1 ✅ Any workflow requiring 2+ operations from the tools above

❌ Single standalone calculation ❌ Need to inspect/validate intermediate results before proceeding

Execution Modes

• auto (recommended): DAG-based optimization, parallel where possible

• sequential: Strict order

• parallel: All concurrent (only if truly independent)

Output Modes

• full: Complete metadata (default)

• compact: Remove nulls/whitespace

• minimal: Basic operation objects with values

• value: Flat {id: value} map (~90% smaller) - use this for most cases

• final: Sequential chains only, returns terminal result (~95% smaller)

Structure

Each operation:

• tool: Tool name (required)

• arguments: Tool parameters (required)

• id: Unique identifier (auto-generated if omitted)

• context: Optional label for this operation

Batch-level context parameter labels entire workflow across all output modes.

Response includes: per-operation status, result/error, execution_time_ms, dependency wave, summary stats.

Evaluate mathematical expressions using SymPy.

Supports: - Arithmetic: +, -, *, /, ^ - Trigonometry: sin, cos, tan, asin, acos, atan - Logarithms: log, ln, exp - Constants: pi, e - Functions: sqrt, abs

Examples:

SIMPLE ARITHMETIC: expression="2 + 2" Result: 4

TRIGONOMETRY: expression="sin(pi/2)" Result: 1.0

WITH VARIABLES: expression="x^2 + 2*x + 1", variables={"x": 3} Result: 16

MULTIPLE VARIABLES: expression="x^2 + y^2", variables={"x": 3, "y": 4} Result: 25

Calculate compound interest with various compounding frequencies.

Formulas: Discrete: A = P(1 + r/n)^(nt) Continuous: A = Pe^(rt)

Examples:

ANNUAL COMPOUNDING: £1000 at 5% for 10 years principal=1000, rate=0.05, time=10, frequency="annual" Result: £1628.89

MONTHLY COMPOUNDING: £1000 at 5% for 10 years principal=1000, rate=0.05, time=10, frequency="monthly" Result: £1647.01

CONTINUOUS COMPOUNDING: £1000 at 5% for 10 years principal=1000, rate=0.05, time=10, frequency="continuous" Result: £1648.72

Convert between angle units: degrees ↔ radians.

Examples:

DEGREES TO RADIANS: value=180, from_unit="degrees", to_unit="radians" Result: 3.14159... (π)

RADIANS TO DEGREES: value=3.14159, from_unit="radians", to_unit="degrees" Result: 180

RIGHT ANGLE: value=90, from_unit="degrees", to_unit="radians" Result: 1.5708... (π/2)

Calculate correlation matrices between multiple variables using Polars.

Methods: - pearson: Linear correlation (-1 to +1, 0 = no linear relationship) - spearman: Rank-based correlation (monotonic, robust to outliers)

Examples:

PEARSON CORRELATION: data={"x":[1,2,3], "y":[2,4,6], "z":[1,1,1]}, method="pearson", output_format="matrix" Result: { "x": {"x":1.0, "y":1.0, "z":NaN}, "y": {"x":1.0, "y":1.0, "z":NaN}, "z": {"x":NaN, "y":NaN, "z":NaN} }

PAIRWISE FORMAT: data={"height":[170,175,168], "weight":[65,78,62]}, method="pearson", output_format="pairs" Result: [{"var1":"height", "var2":"weight", "correlation":0.89}]

SPEARMAN (RANK): data={"x":[1,2,100], "y":[2,4,200]}, method="spearman" Result: Perfect correlation (1.0) despite non-linear relationship

Compute symbolic and numerical derivatives with support for higher orders and partial derivatives.

Examples:

FIRST DERIVATIVE: expression="x^3 + 2x^2", variable="x", order=1 Result: derivative="3x^2 + 4*x"

SECOND DERIVATIVE (acceleration/concavity): expression="x^3", variable="x", order=2 Result: derivative="6*x"

EVALUATE AT POINT: expression="sin(x)", variable="x", order=1, point=0 Result: derivative="cos(x)", value_at_point=1.0

PRODUCT RULE: expression="sin(x)*cos(x)", variable="x", order=1 Result: derivative="cos(x)^2 - sin(x)^2"

PARTIAL DERIVATIVE: expression="x^2*y", variable="y", order=1 Result: derivative="x^2" (treating x as constant)

Time Value of Money (TVM) calculations: solve for PV, FV, PMT, rate, IRR, or NPV.

The TVM equation has 5 variables - know 4, solve for the 5th: PV = Present Value (lump sum now) FV = Future Value (lump sum at maturity) PMT = Payment (regular periodic cash flow) N = Number of periods I/Y = Interest rate per period

Sign convention: negative = cash out (you pay), positive = cash in (you receive)

Examples:

ZERO-COUPON BOND: PV of £1000 in 10 years at 5% calculation="pv", rate=0.05, periods=10, future_value=1000 Result: £613.91

COUPON BOND: PV of £30 annual coupons + £1000 face value at 5% yield calculation="pv", rate=0.05, periods=10, payment=30, future_value=1000 Result: £845.57

RETIREMENT SAVINGS: FV with £500/month for 30 years at 7% calculation="fv", rate=0.07/12, periods=360, payment=-500, present_value=0 Result: £566,764

MORTGAGE PAYMENT: Monthly payment on £200k loan, 30 years, 4% APR calculation="pmt", rate=0.04/12, periods=360, present_value=-200000, future_value=0 Result: £954.83

INTEREST RATE: What rate grows £613.81 to £1000 in 10 years? calculation="rate", periods=10, present_value=-613.81, future_value=1000 Result: 0.05 (5%)

GROWING ANNUITY: Salary stream with 3.5% raises, discounted at 12% calculation="pv", rate=0.12, periods=25, payment=-45000, growth_rate=0.035 Result: £402,586

Compute symbolic and numerical integrals (definite and indefinite).

Examples:

INDEFINITE INTEGRAL (antiderivative): expression="x^2", variable="x" Result: "x^3/3"

DEFINITE INTEGRAL (area): expression="x^2", variable="x", lower_bound=0, upper_bound=1 Result: 0.333

TRIGONOMETRIC: expression="sin(x)", variable="x", lower_bound=0, upper_bound=3.14159 Result: 2.0 (area under one period)

NUMERICAL METHOD (non-elementary): expression="exp(-x^2)", variable="x", lower_bound=0, upper_bound=1, method="numerical" Result: 0.746824 (Gaussian integral approximation)

SYMBOLIC ANTIDERIVATIVE: expression="1/x", variable="x" Result: "log(x)"

Compute limits and series expansions using SymPy.

Examples:

CLASSIC LIMIT: expression="sin(x)/x", variable="x", point=0, operation="limit" Result: limit=1

LIMIT AT INFINITY: expression="1/x", variable="x", point="oo", operation="limit" Result: limit=0

ONE-SIDED LIMIT: expression="1/x", variable="x", point=0, operation="limit", direction="+" Result: limit=+∞ (approaching from right)

REMOVABLE DISCONTINUITY: expression="(x^2-1)/(x-1)", variable="x", point=1, operation="limit" Result: limit=2

MACLAURIN SERIES (at 0): expression="exp(x)", variable="x", point=0, operation="series", order=4 Result: "1 + x + x^2/2 + x^3/6 + O(x^4)"

TAYLOR SERIES (at point): expression="sin(x)", variable="x", point=3.14159, operation="series", order=4 Result: expansion around π

Matrix decompositions: eigenvalues/vectors, SVD, QR, Cholesky, LU.

Examples:

EIGENVALUE DECOMPOSITION: matrix=[[4,2],[1,3]], decomposition="eigen" Result: {eigenvalues: [5, 2], eigenvectors: [[0.89,0.45],[0.71,-0.71]]}

SINGULAR VALUE DECOMPOSITION (SVD): matrix=[[1,2],[3,4],[5,6]], decomposition="svd" Result: {U: 3×3, singular_values: [9.5, 0.77], Vt: 2×2}

QR FACTORISATION: matrix=[[1,2],[3,4]], decomposition="qr" Result: {Q: orthogonal, R: upper triangular}

CHOLESKY (symmetric positive definite): matrix=[[4,2],[2,3]], decomposition="cholesky" Result: {L: [[2,0],[1,1.41]]} where A=LL^T

LU DECOMPOSITION: matrix=[[2,1],[4,3]], decomposition="lu" Result: {P: permutation, L: lower, U: upper} where A=PLU

Core matrix operations using NumPy BLAS.

Examples:

MATRIX MULTIPLICATION: operation="multiply", matrix1=[[1,2],[3,4]], matrix2=[[5,6],[7,8]] Result: [[19,22],[43,50]]

MATRIX INVERSE: operation="inverse", matrix1=[[1,2],[3,4]] Result: [[-2,1],[1.5,-0.5]]

TRANSPOSE: operation="transpose", matrix1=[[1,2],[3,4]] Result: [[1,3],[2,4]]

DETERMINANT: operation="determinant", matrix1=[[1,2],[3,4]] Result: -2.0

TRACE: operation="trace", matrix1=[[1,2],[3,4]] Result: 5.0 (1+4)

Perform percentage calculations: of, increase, decrease, or change.

Examples:

PERCENTAGE OF: 15% of 200 operation="of", value=200, percentage=15 Result: 30

INCREASE: 100 increased by 20% operation="increase", value=100, percentage=20 Result: 120

DECREASE: 100 decreased by 20% operation="decrease", value=100, percentage=20 Result: 80

PERCENTAGE CHANGE: from 80 to 100 operation="change", value=80, percentage=100 Result: 25 (25% increase)

Calculate present value of a perpetuity (infinite series of payments).

A perpetuity is an annuity that continues forever. Common in: - Preferred stock dividends - Endowment funds - Real estate with infinite rental income - UK Consol bonds (historically)

Formulas: Level Ordinary: PV = C / r Level Due: PV = C / r × (1 + r) Growing: PV = C / (r - g), where r > g

Examples:

LEVEL PERPETUITY: £1000 annual payment at 5% payment=1000, rate=0.05 Result: PV = £20,000

GROWING PERPETUITY: £1000 payment growing 3% annually at 8% discount payment=1000, rate=0.08, growth_rate=0.03 Result: PV = £20,000

PERPETUITY DUE: £1000 at period start at 5% payment=1000, rate=0.05, when='begin' Result: PV = £21,000

Create pivot tables from tabular data using Polars.

Like Excel pivot tables: reshape data with row/column dimensions and aggregated values.

Example:

SALES BY REGION AND PRODUCT: data=[ {"region":"North","product":"A","sales":100}, {"region":"North","product":"B","sales":150}, {"region":"South","product":"A","sales":80}, {"region":"South","product":"B","sales":120} ], index="region", columns="product", values="sales", aggfunc="sum" Result: product | A | B --------|------|------ North | 100 | 150 South | 80 | 120

COUNT AGGREGATION: Same data with aggfunc="count" Result: Count of entries per region-product combination

AVERAGE SCORES: data=[{"dept":"Sales","role":"Manager","score":85}, ...] index="dept", columns="role", values="score", aggfunc="mean" Result: Average scores by department and role

Advanced rounding operations with multiple methods.

Methods: - round: Round to nearest (3.145 → 3.15 at 2dp) - floor: Always round down (3.149 → 3.14) - ceil: Always round up (3.141 → 3.15) - trunc: Truncate towards zero (-3.7 → -3, 3.7 → 3)

Examples:

ROUND TO NEAREST: values=3.14159, method="round", decimals=2 Result: 3.14

FLOOR (DOWN): values=3.14159, method="floor", decimals=2 Result: 3.14

CEIL (UP): values=3.14159, method="ceil", decimals=2 Result: 3.15

MULTIPLE VALUES: values=[3.14159, 2.71828], method="round", decimals=2 Result: [3.14, 2.72]

Solve systems of linear equations (Ax = b) using SciPy's optimised solver.

Examples:

SQUARE SYSTEM (2 equations, 2 unknowns): coefficients=[[2,3],[1,1]], constants=[8,3], method="direct" Solves: 2x+3y=8, x+y=3 Result: [x=1, y=2]

OVERDETERMINED SYSTEM (3 equations, 2 unknowns): coefficients=[[1,2],[3,4],[5,6]], constants=[5,6,7], method="least_squares" Finds best-fit x minimizing ||Ax-b|| Result: [x≈-6, y≈5.5]

3x3 SYSTEM: coefficients=[[2,1,-1],[1,3,2],[-1,2,1]], constants=[8,13,5], method="direct" Result: [x=3, y=2, z=1]

Comprehensive statistical analysis using Polars.

Analysis types: - describe: Count, mean, std, min, max, median - quartiles: Q1, Q2, Q3, IQR - outliers: IQR-based detection (values beyond Q1-1.5×IQR or Q3+1.5×IQR)

Examples:

DESCRIPTIVE STATISTICS: data=[1,2,3,4,5,100], analyses=["describe"] Result: {count:6, mean:19.17, std:39.25, min:1, max:100, median:3.5}

QUARTILES: data=[1,2,3,4,5], analyses=["quartiles"] Result: {Q1:2, Q2:3, Q3:4, IQR:2}

OUTLIER DETECTION: data=[1,2,3,4,5,100], analyses=["outliers"] Result: {outlier_values:[100], outlier_count:1, lower_bound:-1, upper_bound:8.5}

FULL ANALYSIS: data=[1,2,3,4,5,100], analyses=["describe","quartiles","outliers"] Result: All three analyses combined