# `latmath.scalar` — Scalar Utilities

Mathematical constants, error measures, tolerance-based comparison, and approximate equality for scalar values. Pure stdlib, zero dependencies.

---

## Constants

```python
from latpy.latmath.scalar import pi, e, tau, inf, nan
```

| Symbol | Value | Description |
|--------|-------|-------------|
| `pi`   | `3.141592653589793` | Archimedes' constant (π) |
| `e`    | `2.718281828459045` | Euler's number |
| `tau`  | `6.283185307179586` | τ = 2π |
| `inf`  | `float('inf')` | IEEE 754 positive infinity |
| `nan`  | `float('nan')` | IEEE 754 Not-a-Number |

All constants are Python `float` values, safe for use in any expression.

---

## Error Measures

```python
from latpy.latmath.scalar import relative_error, absolute_error
```

| Signature | Description |
|-----------|-------------|
| `relative_error(a, b) -> float` | \|a − b\| / \|b\|; returns 0.0 if both are 0, inf if b = 0 and a ≠ 0 |
| `absolute_error(a, b) -> float` | \|a − b\| |

**Edge cases:**
- `relative_error(0.0, 0.0)` → `0.0` (not undefined)
- `relative_error(5.0, 0.0)` → `inf`
- `relative_error(0.0, 3.0)` → `0.0`

---

## Approximate Equality

```python
from latpy.latmath.scalar import isclose, allclose
```

| Signature | Description |
|-----------|-------------|
| `isclose(a, b, rel_tol=1e-9, abs_tol=0.0) -> bool` | Delegates to `math.isclose` |
| `allclose(a, b, rel_tol=1e-9, abs_tol=0.0) -> bool` | Pairwise `isclose` over iterables |

**Edge cases:**
- `isclose(nan, nan)` → `False` (IEEE 754 semantics)
- `isclose(inf, inf)` → `True`
- `allclose` stops early on first mismatch

---

## Robust Comparison

```python
from latpy.latmath.scalar import (
    approx_eq, approx_ne,
    approx_lt, approx_le,
    approx_gt, approx_ge,
)
```

| Signature | Description |
|-----------|-------------|
| `approx_eq(a, b, rel_tol, abs_tol) -> bool` | True if a ≈ b within tolerance |
| `approx_ne(a, b, rel_tol, abs_tol) -> bool` | True if a ≉ b |
| `approx_lt(a, b, rel_tol, abs_tol) -> bool` | True if a < b and a ≉ b |
| `approx_le(a, b, rel_tol, abs_tol) -> bool` | True if a < b or a ≈ b |
| `approx_gt(a, b, rel_tol, abs_tol) -> bool` | True if a > b and a ≉ b |
| `approx_ge(a, b, rel_tol, abs_tol) -> bool` | True if a > b or a ≈ b |

**Rationale:** Standard `<` / `>` break down when values differ by floating-point rounding. `approx_lt(1.0, 1.0 + 1e-16)` is `False` because the two are considered equal within tolerance — avoiding spurious branch misbehavior in numerical algorithms.

**Edge cases:**
- `approx_eq(1.0, 1.0 + 1e-10)` → `True` (default rel_tol=1e-9)
- `approx_lt(1.0, 1.0 + 1e-10)` → `False` (within tolerance, so not strictly less)
- `approx_ge(1.0 + 1e-10, 1.0)` → `True` (within tolerance, so "greater or equal")