---
title: "algorithm-integer-sqrt-rust-approximant — Mova code model"
description: "Mova code model of algorithms/src/integer_sqrt/mod.rs.groupant/rust_approximant.rs: components with their labels, obernants and the model text."
url: "https://groupants.elamur.ai/free/en/mova/algorithms/src/integer_sqrt/mod.rs.groupant/rust_approximant.rs.groupant/"
---
Elamur: [Mova](https://md.mova.elamur.ai/free/en/) · [Dictionary](https://md.dictionary.elamur.ai/free/en/) · [Research](https://md.research.elamur.ai/free/en/) · **Groupants**

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# algorithm-integer-sqrt-rust-approximant

The groupant beside algorithms/src/integer_sqrt/mod.rs.groupant/rust_approximant.rs in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

## Facts

- **kind:** groupant
- **status:** partial source derived instance
- **source:** [`algorithms/src/integer_sqrt/mod.rs.groupant/rust_approximant.rs`](https://github.com/ElamurAI/mova-free/blob/main/algorithms/src/integer_sqrt/mod.rs.groupant/rust_approximant.rs)
- **source SHA-256:** `801e5fd8c8a93bb2…`
- **part of:** [algorithm-integer-sqrt](https://md.groupants.elamur.ai/free/en/mova/algorithms/src/integer_sqrt/mod.rs.groupant/)
- **repository folder:** [`algorithms/src/integer_sqrt/mod.rs.groupant/rust_approximant.rs.groupant`](https://github.com/ElamurAI/mova-free/tree/main/algorithms/src/integer_sqrt/mod.rs.groupant/rust_approximant.rs.groupant)
- **repository commit:** `855da7bd4a7b`

> **Implementation material.** A Mova code model describes this code at one source revision: partial models, finite checks and their stated scope. It is not a reviewed general exposition, and its labels are shown as written, never raised.

## Components

| component | label | where |
|---|---|---|
| **approximant** | analytical | [Forward](#forward) |
| **ast** | partial | `syntax.json` |
| **cost** | symbolic unmeasured | [Cost](#cost) |
| **descriptant** | present | [descriptant](#descriptant) |
| **invariant** | source derived | [Invariant](#invariant) |
| **tests** | bounded executed | [evidence](#evidence) |

## Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

| id | type | label | where |
|---|---|---|---|
| `behavior_complement` | behavior_complement | partial source derived | [behavior_complement](#behavior-complement) |
| `reverse_reconstruction` | reverse_reconstruction | partial source derived | [reverse_reconstruction](#reverse-reconstruction) |

## Model text


Migration: consolidated into model.md, 2026-11-04, operator decision; fresh hash-bound references to the former files were rebound to the sections below.

### descriptant

#### Executable integer_sqrt approximant

2026-10-29: real Rust companion with four budgets. The mathematical combiner connects source, domain, metric, runtime certificate and independent oracle. Source hashes alone establish no accuracy or speed claim.

2026-10-29, rustc 1.99.0: two tests pass. Four budgets on 0..100000 plus 100000 fixed-seed values each, square neighbors and extremes; independent u64::isqrt oracle; exact integer ratio comparison. u64::MAX witnesses nonexact execution.

Earlier exact observations remain in the parent. Expanded AST, formal proof and production evaluation remain absent.

### model

## Executable integer_sqrt model

### Forward

Let t=floor(log2(n)/2), k=7,6,5,4 for the four profiles, s=max(0,t-k), q=floor(sqrt(n >> 2s)), a=q*2^s. Integer checks certify q. Reduced n <=65535, so certification products fit u64. For R=floor(sqrt(n)), a<=R<(q+1)*2^s. If s=0, a=R. Otherwise q>=2^k and (R-a)/R<1/q<=2^-k, hence <=0.78125%,1.5625%,3.125%,6.25%. Zero returns zero. No minimum error is required.

### Invariant

All u64 inputs; zero is exact. Metric: abs(result-floor(sqrt(n)))/floor(sqrt(n)); zero input has zero error. Caps are upper bounds. Production source is unchanged; explicit standalone harness only.

### behavior_complement

For fixed b, exclude all (n,a) with a != F_b(n), where F is the deterministic certified-root expression above. In particular a>floor(sqrt(n)) and error exceeding b are impossible.

### reverse_reconstruction

For a,b, compatible inputs are the union over s in 0..31 with a divisible by 2^s, q=a/2^s, of [q^2*2^(2s),(q+1)^2*2^(2s)-1] intersect u64 and the shift-choice region s=max(0,floor(log2(n)/2)-k). Include n=0 when a=0. Set-valued preimage, not unique recovery.

### Cost

Constant space; one floating square root seed on integer <=65535 then integer certification. Prospective expectation: faster than unchanged binary search. Integration measured about 33x versus that reference and 3.36-3.38x versus std on the fixed batch; see docs/mathematical-modeling.md#executable-rust-approximants.

### Epoch notes

- Comments repointed to consolidated docs, 2026-11-04.

### evidence

2026-10-29, rustc 1.99.0: two tests pass. Four budgets on 0..100000 plus 100000 fixed-seed values each, square neighbors and extremes; independent u64::isqrt oracle; exact integer ratio comparison. u64::MAX witnesses nonexact execution.

Compile standalone with rustc --edition 2024 --test rust_approximant.rs --extern mova_approx=PATH_TO_RLIB, then execute. Runtime contract: tools/approx. Release timing belongs to the integrated harness; no speed factor claimed here.

The same tests passed on two executions. External compiled negative-control mutants were rejected: replacing the sqrt result with zero fails its differential bound test; unconditionally accepting sampled rank candidates fails both quickselect tests. Mutants were outside the repository and never changed production sources.

#### Current cap epoch, 2026-10-30

Supported profiles are 1%, 2%, 5%, 10%; prior 15%/25% profiles are retired.
The two optimized standalone Rust tests passed twice with the updated shared
contract. Independent isqrt and integer loss checks cover every profile;
u64::MAX remains a nonexact witness for all four. A compiled zero-result mutant
failed the differential test. The analytical bounds are 0.78125%, 1.5625%,
3.125%, 6.25%; finite tests are supporting evidence, not their derivation.
Classification: scalar_relative / quantized_numeric / analytical.
Earlier timing observations belong to the retired profile configuration and
are not fresh measurements of the current profiles.