Operator Precedence and Integer Division Pitfalls in C#

Problem Statement

When implementing a grid pathfinding algorithm to calculate routes in a $20 \times 20$ grid using the central binomial coefficient formula, two nearly identical approaches yield different results:

// This works correctly
result = result * (x + i) / i;

// This produces incorrect results
result *= (x + i) / i;

Despite appearing equivalent, these expressions behave differently due to subtle interactions between implicit order of operations and integer division truncation in C#.

Solution Explanation

The discrepancy stems from operator precedence rules combined with integer division truncation. In C#, the compound assignment operator *= has lower precedence than arithmetic operators (*, /).

Expression Breakdown

result *= (x + i) / i;

Is evaluated as:

result = result * ((x + i) / i);  // Integer division happens first

Whereas:

result = result * (x + i) / i;

Is evaluated as:

result = (result * (x + i)) / i;  // Multiplication happens first

Integer Division Truncation

The core difference lies in when the division occurs:

• (x + i) / i truncates decimal precision if (x + i) isn't divisible by i
• (result * (x + i)) / i produces exact results because the product is always divisible by i in binomial coefficient calculations

Concrete Example

Consider values where x = 20, i = 2:

result *= (20 + 2) / 2;   // → result * (22/2) → result * 11
result = result * (20 + 2) / 2;  // → (result * 22) / 2 → result * 11

Now consider x = 3, i = 2 at different result values:

Step	Correct Expression ((result*5)/2)	Incorrect Expression (result*(5/2))
Start	result = 3	result = 3
Calculation	(3*5)/2 = 15/2 = 7	3*(5/2) = 3*2 = 6
Difference	✅ Correct	❌ Off by 1

Why Order Matters Mathematically

In this specific binomial coefficient calculation:

result = result * (x + i) / i;

The expression (result * (x + i)) is always divisible by i. This algorithm effectively computes binomial coefficients $\binom{2x}{x}$, which are integers by definition.

Recommended Implementation

Always prioritize multiplication before division in integer arithmetic to minimize truncation errors. Modify your binomial coefficient formula as follows:

public long CalculateBinomialCoefficient(int gridSize)
{
    long result = 1;

    for (int i = 1; i <= gridSize; i++)
    {
        // Preferred: Multiplication before division
        result = result * (gridSize + i) / i;
    }

    return result;
}

Overflow Risk

While using multiplication-first improves accuracy, it increases the risk of integer overflow. For large grid sizes (> about 35), use System.Numerics.BigInteger instead:

BigInteger result = 1;
for (int i = 1; i <= size; i++)
{
    result = result * (size + i) / i;
}
return result;

Key Takeaways

1. Operator precedence matters: a *= b/c ≠ a = a*b/c
2. Integer division truncates decimal portions
3. Always prefer multiplication before division with integers
4. Verify intermediate values don't exceed data type limits
5. Use binomial properties to validate implementations

Understanding these precedence rules helps avoid subtle bugs beyond this formula—any computation mixing division, multiplication, and compound assignment could manifest similar issues.