Bug in AD implementation of product rule

[dsyme]

Here's a bug the AI found in the AD implementation for product rule: dsyme#46 (comment)

I agree with the assessment. The linked PR has test cases.

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🔧 Critical Bugs Fixed in Automatic Differentiation

I've identified and fixed two critical mathematical bugs in the Dual number implementation that were causing test failures.

🐛 Bugs Fixed

1. Product Rule Bug in operator*=:

// BEFORE (incorrect):
value *= other.value;
derivative = derivative * other.value + value * other.derivative;  // ❌ uses modified value

// AFTER (correct):
auto old_value = value;
value *= other.value;  
derivative = derivative * other.value + old_value * other.derivative;  // ✅ uses original value

2. Quotient Rule Bug in operator/=:

// BEFORE (incorrect):
value /= other.value;
derivative = (derivative * other.value - value * other.derivative) / (other.value * other.value);  // ❌ uses modified value

// AFTER (correct):  
auto old_value = value;
value /= other.value;
derivative = (derivative * other.value - old_value * other.derivative) / (other.value * other.value);  // ✅ uses original value

📊 Root Cause

The compound assignment operators were modifying the value first, then using the modified value in the derivative calculation. This violates the fundamental calculus rules:

• Product Rule: d(fg) = f'g + fg' (needs original f, not f*g)
• Quotient Rule: d(f/g) = (f'g - fg')/g² (needs original f, not f/g)

🚀 Expected Impact

• ✅ Automatic differentiation tests should now pass
• ✅ Correct derivative computation for all mathematical expressions
• ✅ No breaking API changes - purely internal bug fixes
• ⚠️ Half-precision tests may need separate investigation

These bugs were causing systematic errors in derivative calculations throughout the autodiff system. The fixes ensure mathematical correctness for all dual number arithmetic operations.

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