Unveiling the Fast Inverse Square Root: How a 1999 Game Engine Cut Computation Time by 4×

John Carmack, a legendary programmer in the game industry, needed an extremely efficient way to compute the inverse square root for lighting and projection calculations while developing Quake III Arena in 1999, where millions of such operations were performed each second.

He devised the Fast Inverse Square Root algorithm, which starts with the hexadecimal constant 0x5f3759df and applies a single Newton‑Raphson iteration to obtain a close approximation of 1/√x. This method runs about four times faster than conventional approaches, making it ideal for high‑performance game graphics.

float Q_rsqrt( float number )
{
    long i;
    float x2, y;
    const float threehalfs = 1.5F;

    x2 = number * 0.5F;
    y  = number;
    i  = *( long * ) &y;               // evil floating point bit‑level hacking
    i  = 0x5f3759df - ( i >> 1 );       // what the fuck?
    y  = *( float * ) &i;
    y  = y * ( threehalfs - ( x2 * y * y ) ); // 1st iteration
    // y = y * ( threehalfs - ( x2 * y * y ) ); // 2nd iteration, optional

    return y;
}

The algorithm’s primary value lies in its ability to drastically cut the number of floating‑point operations required for real‑time rendering, which was a breakthrough for game graphics performance at the time.

The mysterious magic constant 0x5f3759df remains unexplained; later researchers have proposed alternative constants that can be even more efficient, yet the original constant’s effectiveness continues to be celebrated in the graphics community.