Choose an operation by its inputs
All 73 examples below are executed by the same backend analysis registry used in chat. The result shows the supplied example, not a promise about arbitrary inputs. Type “find tools for probability” in chat to discover a family, or “tool help NAME” for one example.
Scaled attention
tool: scaled_attention {"queries": [[1, 0]], "keys": [[1, 0], [0, 1]], "values": [[10], [20]]}Computed result and method
{
"weights": [
[
0.6697615493266569,
0.3302384506733431
]
],
"output": [
[
13.302384506733432
]
],
"method": "softmax(QKᵀ/√d)V; supplied Q/K/V, not a trained transformer."
}Boltzmann distribution
tool: boltzmann_distribution {"energies": [0, 1, 2], "temperature": 1}Computed result and method
{
"probabilities": [
0.6652409557748218,
0.24472847105479764,
0.09003057317038046
],
"assumptions": "Dimensionless energy, k_B = 1; this is a supplied energy model, not evidence about LLM physics."
}Gaussian pdf
tool: gaussian_pdf {"x": 0, "mean": 0, "sd": 1}Computed result and method
{
"density": 0.3989422804014327,
"note": "A probability density can exceed 1; interval probability requires integration."
}Double integral polynomial
tool: double_integral_polynomial {"terms": [{"coefficient": 1, "xPower": 1, "yPower": 1}], "xBounds": [0, 2], "yBounds": [0, 3]}Computed result and method
{
"value": 9,
"method": "Analytical polynomial integral over a rectangle, evaluated in floating point."
}Quadratic descent
tool: quadratic_descent {"a": 1, "b": 0, "start": 10, "rate": 0.1, "steps": 50}Computed result and method
{
"x": 0.00014272476927059603,
"loss": 2.0370359763344873e-08,
"exactMinimizer": 0,
"trace": [
{
"step": 0,
"x": 10,
"loss": 100
},
{
"step": 1,
"x": 8,
"loss": 64
},
{
"step": 2,
"x": 6.4,
"loss": 40.96000000000001
},
{
"step": 3,
"x": 5.12,
"loss": 26.2144
},
{
"step": 4,
"x": 4.096,
"loss": 16.777216
},
{
"step": 5,
"x": 3.2768,
"loss": 10.73741824
},
{
"step": 6,
"x": 2.62144,
"loss": 6.871947673600001
},
{
"step": 7,
"x": 2.0971520000000003,
"loss": 4.398046511104002
},
{
"step": 8,
"x": 1.6777216000000004,
"loss": 2.8147497671065613
},
{
"step": 9,
"x": 1.3421772800000003,
"loss": 1.801439850948199
},
{
"step": 10,
"x": 1.0737418240000003,
"loss": 1.1529215046068475
},
{
"step": 50,
"x": 0.00014272476927059603,
"loss": 2.0370359763344873e-08
}
],
"method": "Gradient descent on supplied scalar quadratic ax²+bx+c."
}Hopfield recall
tool: hopfield_recall {"patterns": [[1, 1, -1, -1]], "state": [1, 1, 1, -1]}Computed result and method
{
"state": [
1,
1,
-1,
-1
],
"stable": true,
"sweeps": 2,
"energies": [
0,
-1.5,
-1.5
],
"matchedPattern": 0,
"method": "Hebbian symmetric weights, zero diagonal, asynchronous deterministic updates. Spurious attractors are possible."
}Percentage change
tool: percentage_change {"from": 80, "to": 100}Computed result and method
{
"difference": 20,
"percent": 25,
"method": "Change divided by absolute starting value."
}Polynomial derivative
tool: polynomial_derivative {"coefficients": [3, 2, 7]}Computed result and method
{
"coefficients": [
6,
2
],
"coefficientOrder": "highest power first"
}Polynomial integral
tool: polynomial_integral {"coefficients": [3, 0, 0], "low": 0, "high": 2}Computed result and method
{
"value": 8,
"antiderivative": [
1,
0,
0,
0
],
"method": "Analytical polynomial integration evaluated in floating point."
}Interpolate
tool: interpolate {"x": [0, 2, 4], "y": [0, 4, 0], "at": 3}Computed result and method
{
"value": 2,
"method": "Piecewise linear interpolation; no extrapolation."
}Z scores
tool: z_scores {"values": [1, 3]}Computed result and method
{
"values": [
-1,
1
],
"mean": 2,
"standardDeviation": 1,
"method": "Population standard deviation."
}Softmax
tool: softmax {"values": [1, 2, 3]}Computed result and method
{
"probabilities": [
0.09003057317038046,
0.24472847105479764,
0.6652409557748218
]
}Entropy
tool: entropy {"probabilities": [0.5, 0.5]}Computed result and method
{
"bits": 1
}Binomial probability
tool: binomial_probability {"n": 4, "k": 2, "p": 0.5}Computed result and method
{
"probability": 0.3749999999999999,
"assumptions": "Exactly k successes in n independent trials with constant probability p."
}Poisson probability
tool: poisson_probability {"k": 0, "lambda": 2}Computed result and method
{
"probability": 0.1353352832366127,
"assumptions": "Poisson count model at the supplied mean; model fit is not verified."
}Distance
tool: distance {"x": [0, 0], "y": [3, 4]}Computed result and method
{
"euclidean": 5,
"manhattan": 7
}Gcd lcm
tool: gcd_lcm {"a": 24, "b": 18}Computed result and method
{
"gcd": "6",
"lcm": "72",
"exact": true
}Prime factors
tool: prime_factors {"n": 360}Computed result and method
{
"factors": [
{
"prime": 2,
"power": 3
},
{
"prime": 3,
"power": 2
},
{
"prime": 5,
"power": 1
}
]
}Combinations
tool: combinations {"n": 10, "k": 3}Computed result and method
{
"value": "120",
"exact": true
}Matrix multiply
tool: matrix_multiply {"a": [[1, 2]], "b": [[3], [4]]}Computed result and method
{
"matrix": [
[
11
]
]
}Matrix solve
tool: matrix_solve {"a": [[2, 0], [0, 3]], "b": [4, 9]}Computed result and method
{
"solution": [
2,
3
],
"maxResidual": 0,
"method": "Gaussian elimination with partial pivoting; floating point."
}Polynomial
tool: polynomial {"coefficients": [1, 0, -2], "x": 3}Computed result and method
{
"value": 7,
"coefficientOrder": "highest power first",
"method": "Horner evaluation"
}Correlation
tool: correlation {"x": [1, 2, 3], "y": [2, 4, 6]}Computed result and method
{
"value": 0.9999999999999999,
"method": "Pearson correlation; association does not establish causation."
}Quantiles
tool: quantiles {"values": [1, 2, 3, 4], "probabilities": [0.25, 0.5, 0.75]}Computed result and method
{
"quantiles": [
{
"p": 0.25,
"value": 1.75
},
{
"p": 0.5,
"value": 2.5
},
{
"p": 0.75,
"value": 3.25
}
],
"method": "Linear interpolation at (n-1)p"
}Weighted mean
tool: weighted_mean {"values": [10, 20], "weights": [1, 3]}Computed result and method
{
"value": 17.5
}Compound growth
tool: compound_growth {"principal": 100, "rate": 0.1, "periods": 2}Computed result and method
{
"value": 121.00000000000001,
"assumptions": "Rate is a decimal per period; no deposits, fees or taxes."
}Loan payment
tool: loan_payment {"principal": 1200, "rate": 0, "periods": 12}Computed result and method
{
"value": 100,
"total": 1200,
"interest": 0,
"assumptions": "Fixed rate per payment period; equal end-of-period payments; no fees."
}Descriptive counts
tool: descriptive_counts {"values": ["apple", "pear", "apple"]}Computed result and method
{
"counts": [
{
"value": "apple",
"count": 2
},
{
"value": "pear",
"count": 1
}
]
}Dot product
tool: dot_product {"x": [1, 2], "y": [3, 4]}Computed result and method
{
"value": 11
}Vector norm
tool: vector_norm {"values": [3, 4]}Computed result and method
{
"value": 5,
"method": "Euclidean norm"
}Matrix determinant
tool: matrix_determinant {"a": [[1, 2], [3, 4]]}Computed result and method
{
"value": -2,
"method": "Elimination with partial pivoting; floating point."
}Polynomial root
tool: polynomial_root {"coefficients": [1, 0, -2], "low": 0, "high": 2}Computed result and method
{
"value": 1.414213562372879,
"residual": 6.108447081487611e-13,
"iterations": 42,
"bracket": [
1.4142135623724243,
1.4142135623733338
],
"method": "Bisection of a continuous polynomial; one bracketed root, not all roots."
}Integrate samples
tool: integrate_samples {"x": [0, 1, 2], "y": [0, 2, 4]}Computed result and method
{
"value": 4,
"method": "Composite trapezoidal approximation; not symbolic integration."
}Spearman
tool: spearman {"x": [1, 2, 3], "y": [3, 2, 1]}Computed result and method
{
"value": -0.9999999999999999,
"method": "Spearman rank correlation with average ranks for ties."
}Covariance
tool: covariance {"x": [1, 2, 3], "y": [2, 4, 6]}Computed result and method
{
"sample": 2,
"population": 1.3333333333333333
}Robust statistics
tool: robust_statistics {"values": [1, 2, 3, 100]}Computed result and method
{
"median": 2.5,
"mad": 1,
"count": 4,
"method": "Median and unscaled median absolute deviation."
}Text profile
tool: text_profile "A clear example. Another sentence."Computed result and method
{
"characters": 34,
"words": 5,
"uniqueWords": 5,
"sentences": 2,
"averageWordsPerSentence": 2.5
}Keywords
tool: keywords "orchard orchard river"Computed result and method
{
"keywords": [
{
"word": "orchard",
"count": 2
},
{
"word": "river",
"count": 1
}
]
}Text similarity
tool: text_similarity {"left": "orchard river", "right": "orchard road"}Computed result and method
{
"value": 0.3333333333333333,
"sharedWords": [
"orchard"
],
"method": "Jaccard similarity of normalized content-word sets; not semantic equivalence or truth."
}Extract summary
tool: extract_summary {"text": "Trees need water. Rain provides water. Roots absorb water.", "count": 2}Computed result and method
{
"text": "Trees need water. Rain provides water.",
"sentences": [
"Trees need water.",
"Rain provides water."
],
"method": "Extracted source sentences; no new factual claims."
}Sentences
tool: sentences "One sentence. Another sentence!"Computed result and method
{
"sentences": [
"One sentence.",
"Another sentence!"
]
}Unique lines
tool: unique_lines "apple\napple\npear"Computed result and method
{
"text": "apple\npear",
"lines": [
"apple",
"pear"
],
"removed": 1
}Sort numbers
tool: sort_numbers [3, 1, 2]Computed result and method
{
"values": [
1,
2,
3
]
}Moving average
tool: moving_average {"values": [1, 2, 3, 4], "window": 2}Computed result and method
{
"values": [
1.5,
2.5,
3.5
],
"window": 2,
"method": "Trailing arithmetic mean; no padded observations."
}Linear fit
tool: linear_fit {"x": [1, 2, 3], "y": [3, 5, 7]}Computed result and method
{
"slope": 2,
"intercept": 1,
"rSquared": 1,
"count": 3,
"method": "Ordinary least squares; association does not establish causation or reliable extrapolation."
}Minmax scale
tool: minmax_scale [1, 2, 3]Computed result and method
{
"values": [
0,
0.5,
1
],
"min": 1,
"max": 3
}Shortest path
tool: shortest_path {"from": "A", "to": "C", "edges": [{"from": "A", "to": "B", "cost": 2}, {"from": "B", "to": "C", "cost": 3}]}Computed result and method
{
"found": true,
"cost": 5,
"path": [
"A",
"B",
"C"
],
"directed": true
}Word definition
tool: word_definition "cow"Computed result and method
{
"term": "cow",
"senses": [
{
"id": "n:02406106",
"pos": "n",
"gloss": "female of domestic cattle: \"`moo-cow' is a child's term\"",
"words": [
"cow",
"moo-cow"
]
},
{
"id": "n:01890428",
"pos": "n",
"gloss": "mature female of mammals of which the male is called `bull'",
"words": [
"cow"
]
},
{
"id": "n:09992117",
"pos": "n",
"gloss": "a large unpleasant woman",
"words": [
"cow"
]
},
{
"id": "v:01783720",
"pos": "v",
"gloss": "subdue, restrain, or overcome by affecting with a feeling of awe; frighten (as with threats)",
"words": [
"overawe",
"cow"
]
}
],
"source": "Offline WordNet 3.1"
}Category proof
tool: category_proof {"child": "cow", "parent": "mammal"}Computed result and method
{
"supported": true,
"path": [
{
"id": "n:02406106",
"word": "cow",
"gloss": "female of domestic cattle: \"`moo-cow' is a child's term\""
},
{
"id": "n:02405077",
"word": "cattle",
"gloss": "domesticated bovine animals as a group regardless of sex or age; \"so many head of cattle\"; \"wait till the cows come home\"; \"seven thin and ill-favored kine\"- Bible; \"a team of oxen\""
},
{
"id": "n:02404662",
"word": "bovine",
"gloss": "any of various members of the genus Bos"
},
{
"id": "n:02403683",
"word": "bovid",
"gloss": "hollow-horned ruminants"
},
{
"id": "n:02401652",
"word": "ruminant",
"gloss": "any of various cud-chewing hoofed mammals having a stomach divided into four (occasionally three) compartments"
},
{
"id": "n:02397129",
"word": "even-toed ungulate",
"gloss": "placental mammal having hooves with an even number of functional toes on each foot"
},
{
"id": "n:02373458",
"word": "ungulate",
"gloss": "any of a number of mammals with hooves that are superficially similar but not necessarily closely related taxonomically"
},
{
"id": "n:01889397",
"word": "placental",
"gloss": "mammals having a placenta; all mammals except monotremes and marsupials"
},
{
"id": "n:01864419",
"word": "mammal",
"gloss": "any warm-blooded vertebrate having the skin more or less covered with hair; young are born alive except for the small subclass of monotremes and nourished with milk"
}
],
"sense": {
"term": "cow",
"pos": "n",
"rank": 1,
"tagged": 2,
"id": "n:02406106",
"domain": 5,
"gloss": "female of domestic cattle: \"`moo-cow' is a child's term\"",
"words": [
"cow",
"moo-cow"
]
}
}Cosine similarity
tool: cosine_similarity {"left": [1, 0], "right": [0, 1]}Computed result and method
{
"value": 0,
"method": "Cosine of supplied vectors; similarity is not factual equivalence."
}Normal cdf
tool: normal_cdf {"x": 0}Computed result and method
{
"probability": 0.5,
"method": "Normal CDF, composite Simpson approximation; standardized tails beyond ±10 are clipped."
}Normal quantile
tool: normal_quantile {"probability": 0.975}Computed result and method
{
"value": 1.9599639845420187,
"method": "Bisection of numerical normal CDF; approximate quantile."
}Wilson interval
tool: wilson_interval {"successes": 50, "trials": 100}Computed result and method
{
"estimate": 0.5,
"lower": 0.4038315303659028,
"upper": 0.5961684696340972,
"confidence": 0.95,
"method": "Wilson score interval under independent Bernoulli sampling; not a probability that the fixed parameter lies in this interval."
}Hypergeometric probability
tool: hypergeometric_probability {"population": 10, "successes": 4, "draws": 3, "observed": 2}Computed result and method
{
"probability": 0.29999999999999977,
"method": "Sampling without replacement from a supplied finite population."
}Markov steps
tool: markov_steps {"initial": [1, 0], "transition": [[0.8, 0.2], [0.4, 0.6]], "steps": 2}Computed result and method
{
"probabilities": [
0.7200000000000002,
0.28
],
"steps": 2,
"method": "Row-stochastic finite Markov propagation; transition probabilities are supplied, not inferred."
}Pagerank
tool: pagerank {"nodes": ["A", "B", "C"], "edges": [["A", "B"], ["B", "C"], ["C", "A"]]}Computed result and method
{
"ranking": [
{
"node": "A",
"score": 0.3333333333333333
},
{
"node": "B",
"score": 0.3333333333333333
},
{
"node": "C",
"score": 0.3333333333333333
}
],
"iterations": 1,
"converged": true,
"residual": 0,
"method": "Uniform teleportation and dangling-node redistribution on supplied graph; not Google ranking data."
}Edit distance
tool: edit_distance {"left": "kitten", "right": "sitting"}Computed result and method
{
"distance": 3,
"method": "Case-sensitive Levenshtein code-point distance, not semantic distance."
}Topological sort
tool: topological_sort {"nodes": ["build", "test", "deploy"], "edges": [["build", "test"], ["test", "deploy"]]}Computed result and method
{
"acyclic": true,
"order": [
"build",
"test",
"deploy"
],
"blocked": [],
"method": "Kahn topological sorting; blocked nodes may lie in a cycle or depend on one."
}Cross entropy
tool: cross_entropy {"p": [0.5, 0.5], "q": [0.5, 0.5]}Computed result and method
{
"value": 0.6931471805599453,
"infinite": false,
"unit": "nats"
}Kl divergence
tool: kl_divergence {"p": [0.25, 0.75], "q": [0.25, 0.75]}Computed result and method
{
"value": 0,
"infinite": false,
"unit": "nats"
}Vector projection
tool: vector_projection {"left": [3, 4], "right": [1, 0]}Computed result and method
{
"values": [
3,
0
],
"method": "Orthogonal projection of left onto right in Euclidean space."
}Permutations
tool: permutations {"n": 10, "k": 3}Computed result and method
{
"value": "720",
"exact": true,
"method": "Ordered selections without replacement: n!/(n-k)!."
}Json format
tool: json_format {"text": "{\"ready\":true,\"items\":[1,2]}"}Computed result and method
{
"text": "{\n \"ready\": true,\n \"items\": [\n 1,\n 2\n ]\n}"
}Json pointer
tool: json_pointer {"value": {"items": [{"name": "kernel"}]}, "pointer": "/items/0/name"}Computed result and method
{
"found": true,
"value": "kernel"
}Json diff
tool: json_diff {"before": {"version": 1}, "after": {"version": 2, "ready": true}}Computed result and method
{
"equal": false,
"changes": [
{
"op": "add",
"path": "/ready",
"value": true
},
{
"op": "replace",
"path": "/version",
"before": 1,
"value": 2
}
],
"note": "Structural comparison; descriptive entries are not an executable JSON Patch sequence."
}Url inspect
tool: url_inspect {"text": "https://example.com/tools/?q=hello&q=world#start"}Computed result and method
{
"origin": "https://example.com",
"hostname": "example.com",
"port": "",
"path": "/tools/",
"query": [
[
"q",
"hello"
],
[
"q",
"world"
]
],
"fragment": "#start",
"networkAccess": false
}Base64 encode
tool: base64_encode {"text": "Hello, HuLaHo!"}Computed result and method
{
"text": "SGVsbG8sIEh1TGFIbyE=",
"encoding": "UTF-8"
}Base64 decode
tool: base64_decode {"text": "SGVsbG8="}Computed result and method
{
"text": "Hello",
"encoding": "UTF-8"
}Sha256
tool: sha256 {"text": "abc"}Computed result and method
{
"hex": "ba7816bf8f01cfea414140de5dae2223b00361a396177a9cb410ff61f20015ad",
"encoding": "UTF-8",
"note": "Checksum, not encryption or password storage."
}Text patch
tool: text_patch {"text": "Hello world", "edits": [{"find": "world", "replace": "HuLaHo", "expected": 1}]}Computed result and method
{
"text": "Hello HuLaHo",
"replaced": 1,
"filesModified": false
}Integer base
tool: integer_base {"text": "ff", "from": 16, "to": 2}Computed result and method
{
"text": "11111111",
"from": 16,
"to": 2,
"exact": true
}Html escape
tool: html_escape {"text": "<a title=\"hello\">&</a>"}Computed result and method
{
"text": "<a title="hello">&</a>",
"context": "HTML text or quoted attribute; not a sanitizer for CSS, JavaScript or URLs."
}Line diff
tool: line_diff {"before": "alpha\nbeta", "after": "alpha\ngamma"}Computed result and method
{
"lines": [
{
"op": "equal",
"text": "alpha"
},
{
"op": "remove",
"text": "beta"
},
{
"op": "add",
"text": "gamma"
}
],
"equal": false,
"method": "Line LCS; exact whitespace, no file writes."
}