CVP verified calculation capability

Special functions Calculation Capability

Real erf, erfc, Gamma, log-gamma, integer-order J_n, Y_n, I_n, and K_n, Lambert W_0 and W_{-1}, Euler beta B(x,y), incomplete beta B_x(a,b), unnormalized sinc, and normalized sinc. Discovery /error-function, /gamma-function, /bessel-function, /bessel-y, /bessel-i, /bessel-k, /lambert-w, /lambert-w-minus-1, /beta-function, /incomplete-beta, /sinc-function, /normalized-sinc. Not factorial, not CAS.

Math Capability ID math.special version 1.10.2 CVP VERIFIED STALE Core

Model

Error function

Error function: erf(x) = (2/√π) ∫₀ˣ e^{−t²} dt
Gamma: Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt
Log-gamma: lgamma(x) = ln Γ(x), x > 0
Bessel J: J_n(x) = (x/2)^n Σ_k (−1)^k (x/2)^{2k} / (k! (n+k)!)
Bessel Y: Y_n(x) Neumann, x > 0
Bessel I: I_n(x) = (x/2)^n Σ_k (x/2)^{2k} / (k! (n+k)!)
Bessel K: K_n(x) Macdonald, x > 0
Lambert W: W(x) e^{W(x)} = x
Lambert W₋₁: W_{-1}(x) e^{W_{-1}(x)} = x, −1/e ≤ x < 0
Beta: B(x, y) = Γ(x)Γ(y) / Γ(x+y)
Incomplete beta: B_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt
sinc: sinc(x) = sin(x)/x, sinc(0)=1
Normalized sinc: sinc(x) = sin(πx)/(πx), sinc(0)=1

Modes: erf · erfc · gamma · lgamma · bessel j · bessel y · bessel i · bessel k · lambert w · lambert wm1 · beta · incomplete beta · sinc · normalized sinc

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`x`	—	—	input / output
`y`	—	—	input / output
`a`	—	—	input / output
`b`	—	—	input / output
`n`	—	—	input / output
`value`	—	—	output
`regularized`	—	—	output
`erf`	—	—	output
`erfc`	—	—	output
`gamma`	—	—	output
`lgamma`	—	—	output
`branch`	—	—	output
`algorithm`	—	—	output

Trusted Math contract

Inspectable numeric contract for this Capability@Version. Not a rewrite of the engine.

Field	Declaration
`numeric_domain`	real
`exactness`	float64
`algorithm`	erf/erfc: Taylor / upper-gamma continued fraction; J/Y: series for \|x\|<4 then GL16 integral quadrature; K: log-scaled GL16 integral quadrature; I: positive series; published identities for Γ, W₀/W_{-1}, B, incomplete B, sinc
`valid_domain`	per-function real domain (poles at non-positive integers for Γ; Y,K singular at 0; W_{-1} on [−1/e, 0))
`singularity_policy`	INVALID_INPUT on poles / unpublished branches
`branch_policy`	real principal: W₀ vs W_{-1} are explicit modes, not a complex Lambert disc
`tolerance`	runtime_tolerance: Bessel integral refinement: \|fine−coarse\| ≤ 2e−13·\|fine\| + 2e−15 on each integral (K before unscaling); verification_tolerance: High-precision fixtures: relative 1e−10; J/Y near zeros add absolute 2e−14; I/K and J at \|x\|<1 add only 4 smallest binary64 subnormals. Other modes retain their Evidence policy.
`convergence`	Bessel integrals: up to 4 panel doublings then NO_CONVERGENCE; erfc continued fraction: up to 512 steps then NO_CONVERGENCE. Integral tail truncation is separate from refinement.
`conditioning`	Relative accuracy weakens near zeros of J/Y; compare absolute errors there. K is evaluated with log-scaled positive integrands to avoid subtractive tail cancellation.
`rounding`	ieee754_round_ties_to_even
`overflow`	ieee754_nonfinite_or_coded_error
`reproducibility`	deterministic algorithm + Capability@Version
`independent_oracle`	cvp:math.special O2 / Evidence published cases
`golden_vectors`	Evidence golden + boundary + property (and published_reference_case when listed)

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

Relative accuracy weakens near zeros of Bessel J/Y. Small positive K tails are retained within Float64 range.
Real argument
Gamma poles at non-positive integers
lgamma is real for x > 0
Bessel J integer n=0…8, |x|≤80
Bessel Y integer n=0…8, x>0
Bessel I integer n=0…8, |x|≤80
Bessel K integer n=0…8, x>0
Lambert W_0 for x ≥ −1/e
Lambert W_{-1} for −1/e ≤ x < 0
Complete Euler B(x,y) for x>0, y>0
Incomplete B_x(a,b) for a>0, b>0, 0≤x≤1
Unnormalized sinc(x)=sin(x)/x, sinc(0)=1
Normalized sinc(x)=sin(πx)/(πx), sinc(0)=1
The Result Card includes an engine-linked Live graph for erf, erfc, Γ, ln Γ, J_n / Y_n / I_n / K_n, W₀ / W₋₁, B(x, y) vs x (y held), incomplete B_x on [0, 1], sinc, and normalized sinc. Same special-functions engine as REST/MCP. Integer n! stays on /factorial. Not CAS.

Example

One numerical check. The calculator page holds worked steps and FAQ.

mode=erf x=1 → 0.84270079295

Open this example in the calculator

Trust

CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published modes. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Verified scope: all published modes
Assurance: Core
Declared partition coverage: PASS · 15/15 declared partitions (erf, erfc, gamma, lgamma, bessel-j, bessel-y, bessel-i, bessel-k, lambert-w, lambert-wm1, beta, incomplete-beta, sinc, normalized-sinc, invalid-domain)
Calculation version: 1.10.2
Verification: 58/58 calculation tests · 13/13 legacy golden · 8/8 legacy boundary · 37/37 legacy property
Production surface contract: 6/6
Versions: Calculation 1.10.2 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.bessel-integral-stability-domain2
Verification revision: 2026-10-07.bessel-integral-stability-domain2 · 1/1 property · digest c21a1fee5f33
Reference: O1 model · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP
ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
CVP suite: 482/482 golden · 8/8 CVP boundary · 10/10 invalid · 1/1 property · 3/3 metamorphic · 9/9 cross-interface · 1/1 cross-calculator · 6/6 CVP contract
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.10.2 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

STALE · Build schema 1.10.2 matches page · production binding STALE · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 6/6.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/special-functions' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"mode":"erf","x":1},"calculation_version":"1.10.2"}'

{
  "inputs": {
    "mode": "erf",
    "x": 1
  },
  "calculation_version": "1.10.2"
}

POST /api/v1/calc/special-functions. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: special-functions
Engine: special-functions
REST: POST /api/v1/calc/special-functions
MCP: execute_calculation with capability_id: "math.special"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.10.2)
Versioned output schema: Immutable pin (1.10.2)

References

Human-readable sources plus CVP reference evidence classes.

Sources

ISO 80000-2:2019 Quantities and units — Mathematics
Abramowitz and Stegun, Handbook of Mathematical Functions — erf, Gamma, Bessel, incomplete beta

Reference evidence classes: O1 model · O2 expected_values · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"math.special"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"math.special","inputs":{"mode":"erf","x":1},"calculation_version":"1.10.2"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.10.2) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.