Reverse Payroll Solver (Newton + Brent)

A numerical engine that computes gross salary from a target net salary using a hybrid root-finding approach.

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Overview

This project solves:

net_salary(gross) = target_net

Rewritten as:

f(gross) = net_salary(gross) - target_net = 0

It combines:

• Newton–Raphson (fast convergence)
• Brent’s Method (guaranteed convergence)

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Features

• Hybrid solver (Newton + Brent)
• Adaptive bracketing
• Progressive tax model support
• Contribution caps
• High numerical precision
• Robust handling of non-linear, piecewise functions

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Payroll Model

Income Tax

Progressive brackets:

(0.0,    0.0)
(1000.0, 0.10)
(2000.0, 0.20)
(4000.0, 0.30)

Tax is computed cumulatively per bracket.

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Social Contribution

min(gross * SOCIAL_RATE, SOCIAL_CAP)

Constants:

SOCIAL_RATE = 0.08
SOCIAL_CAP  = 500.0

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Net Salary

net = gross - income_tax(gross) - social_contribution(gross)

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Solver Architecture

1. Adaptive Bracketing

Finds an interval [low, high] such that:

f(low) * f(high) < 0

Expands high exponentially if needed.

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2. Newton Pre-pass

Iteration:

g_next = g - f(g) / f'(g)

Derivative approximated via central difference:

f'(x) ≈ (f(x + h) - f(x - h)) / (2h)

Used as a fast initial convergence attempt.

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3. Brent Solver

Fallback solver combining:

• Bisection
• Secant method
• Inverse quadratic interpolation

Guarantees convergence when a root is bracketed.

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Solver Flow

Target Net
↓
Bracketing
↓
Newton (fast path)
↓ success
Return result

↓ failure
Brent solver
↓
Return result

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Convergence Criteria

• Newton tolerance: 1e-6
• Brent tolerance: 1e-6
• Maximum iterations: configurable (default 100)

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Output Types

SolveResult:

* Converged(f64)
* NoBracket
* MaxIterations

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Example

Target Net: 2500
Estimated Gross: 3055.56
Verified Net: 2499.999999999954

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Complexity

• Newton: O(log n) near solution
• Brent: O(log n) guaranteed convergence
• Hybrid: typically < 20 iterations total

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Use Cases

• Payroll systems
• Tax simulation
• Financial modeling
• Inverse pricing problems
• Embedded numerical computation

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Notes

• Works with non-linear and piecewise functions
• Handles discontinuities in derivatives
• Designed for numerical stability
• No analytical inversion required

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