# Floating-Point Representation, Overflow, and Underflow

`D00-F12-A01` · Financial Mathematics, Statistics, and Data Foundations → Statistical Computing and Reproducibility · archetype `record-transform` · difficulty 1/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

```ts
import { floatingPointRepresentationOverflowAndUnderflow } from "fintech-algorithms/foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow";
```

## Signature

```ts
floatingPointRepresentationOverflowAndUnderflow(input)
```

Inspects one double-precision number: whether it is finite, how large one representable step at its magnitude is, and whether squaring it would stay inside a stated magnitude ceiling.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `D00Input` | yes | Reads `floatingValue`, the number under inspection, and `maxSafeMagnitude`, the largest magnitude the caller is willing to reach. A non-empty `values` list of finite numbers must also be present; it is validated but not used here. |

## Returns

`D00Output`

`isFinite` reports whether `floatingValue` is a finite number, `nextUp` adds one machine-epsilon step scaled to its magnitude, `ulp` is the gap between the two, and `overflowGuard` is true when the magnitude is at most the square root of `maxSafeMagnitude`.

## Errors

- When `values` is absent, empty, or holds a non-finite number — throws RangeError

## Complexity

Time `O(n)`, space `O(n)`.

## Worked example

This is the worked example published in the article, replayed by the test suite on every run. The output cannot drift.

### Input

`input`:

```json
{
  "values": [1, 2, 3, 4, 5],
  "floatingValue": 0.1,
  "maxSafeMagnitude": 1.7976931348623157e+308,
  "window": 3,
  "rawValues": [1, null, 2, "bad", 3],
  "invalidPolicy": "drop-and-report",
  "seed": 42,
  "sampleCount": 5,
  "leftVector": [
    {
      "index": "A",
      "value": 1
    },
    {
      "index": "B",
      "value": 2
    }
  ],
  "rightVector": [
    {
      "index": "B",
      "value": 20
    },
    {
      "index": "C",
      "value": 30
    }
  ],
  "train": [10, 12, 14, 16],
  "test": [18, 20],
  "actual": [1, 2.0000001, 3],
  "expected": [1, 2, 3]
}
```

Showing 14 of 18 fields.

### Call

```ts
floatingPointRepresentationOverflowAndUnderflow(input)
```

### Returns

object with 2 fields: isFinite, ulp

```json
{
  "isFinite": true,
  "ulp": 1.3877787807814457e-17
}
```

## Verification and provenance

Tier: **verified** (via input-expected).

The worked example below is the figure published in this algorithm's article, replayed and asserted by the test suite on every build. The arithmetic cannot drift without the build failing.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.0.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/foundations/statistical-computing-and-reproducibility/floating-point-representation-overflow-and-underflow/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/foundations/llms.txt