Description

This PR introduces a fast square root approximation using least squares linear regression,
optimized for the interval [1.0, 2.0]. It is designed for performance-critical environments
such as game engines, signal processors, and embedded systems where calling the standard
sqrt() is too expensive.

The core idea is to fit a linear equation y = αx + β that minimizes the squared error
against √x over the target range, then use bit-level tricks to extend it to all positive
floats via IEEE 754 exponent manipulation.

How It Works

1. Coefficient Calculation

Uses the normal equations of linear regression over N = 1000 sampled points in [1.0, 2.0]
to compute the optimal slope α and intercept β:

$$\alpha = \frac{N \sum x_i y_i - \sum x_i \sum y_i}{N \sum x_i^2 - (\sum x_i)^2}, \quad \beta = \frac{\sum y_i - \alpha \sum x_i}{N}$$

2. IEEE 754 Bit-Level Representation

The coefficients are inspected at the bit level, which is useful for low-level systems
that need hardcoded hex constants instead of runtime computation:

alpha = 0.411859  →  0x3ED2DF36
beta  = 0.601160  →  0x3F19E5A4

3. Fixed-Point Conversion (16.16 format)

For hardware without an FPU, the coefficients are scaled by 2^16 = 65536 and stored as integers:

alpha_fixed = 26991  →  0x696F
beta_fixed  = 39397  →  0x99E5

4. Range Extension via Bit Manipulation

To handle inputs outside [1.0, 2.0]:

1. Extract the IEEE 754 exponent to normalize x into [1.0, 2.0)
2. Apply the linear approximation on the mantissa
3. Rescale the result by 2^(exp/2)
4. Handle odd exponents by multiplying by the hardcoded constant √2 ≈ 1.41421356

Performance vs Accuracy

Sub-1% error across the primary range, with no calls to math.h.

Use Cases

• Game engines and real-time graphics — similar in spirit to the Quake III fast inverse sqrt
• Embedded systems and microcontrollers — runs on hardware without an FPU
• Signal processing pipelines — where throughput matters more than exact precision
• Education — demonstrates how linear regression applies to function approximation

References

• Linear Regression — Wikipedia
• Methods of Computing Square Roots — Wikipedia
• Fast Inverse Square Root — Quake III

Checklist

• Description of change added
• Filename matches file name guidelines
• Tests and examples added, tests pass
• Relevant documentation and comments updated
• PR title follows semantic commit guidelines
• Checked for duplicate suggestions

I acknowledge that all my contributions will be made under the project's license.