algorithm_analysis.json

{
  "explanation": "The Fast Inverse Square Root algorithm computes 1/√x using bit manipulation and Newton's method.",
  "original_use": "Used in Quake III Arena for lighting and reflection calculations.",
  "performance": "Approximately 30x faster than standard floating-point division and square root.",
  "steps": [
    "1. Treat the floating-point number as an integer",
    "2. Right shift the integer by 1 bit",
    "3. Subtract from the magic number (0x5f3759df)",
    "4. Treat the result as a floating-point number",
    "5. Apply one iteration of Newton's method: y = y * (1.5 - (x2 * y * y))"
  ],
  "mathematical_basis": "The algorithm exploits the linear relationship between the logarithm of a number and its floating-point representation.",
  "magic_number_derivation": "The magic number is derived from the IEEE 754 floating-point format and provides a good initial approximation."
}