Amortized Noise

A new noise generation algorithm called amortized noise generates smooth noise using dynamic programming techniques to amortize the cost of floating point multiplications over neighboring points. The amortized 2D noise algorithm uses a factor of 17/3 ≈ 5.67 fewer floating point multiplications than the 2D Perlin noise algorithm, resulting in a speedup by a factor of 3 in practice on current desktop computing hardware. The amortized 3D noise algorithm uses a factor of 40/7 ≈ 5.71 fewer floating point multiplications than the 3D Perlin noise algorithm, but the increasing overhead for the initialization of tables limits the speedup factor achieved in practice to around 2. In addition to being significantly faster, amortized noise is less prone to repeat at regular intervals than Perlin noise, and it is smoother since it uses quintic splines in place of cubic splines.