powFast: Fast Power Approximation through Rational Interpolation

Mathematical Foundation and Rational Function Theory

The powFast function implements a fast power approximation using rational interpolation theory. Instead of computing the expensive exponential-logarithm power operation $a^b$, it employs a rational function approximation:

$$\text{powFast}(a, b) = \frac{a}{(1-b) \cdot a + b}$$

This rational approximation exploits the mathematical property that for certain domains, a simple ratio can approximate power relationships with remarkable visual fidelity while maintaining computational efficiency.

Fast Power Approximation Usage

The powFast function provides a computationally cheaper alternative to the standard pow() function using a rational approximation. While less mathematically precise, it's useful for real-time graphics where performance matters more than perfect accuracy.

ライブエディター

const fragment = () => {
      const reality = uv.sub(0.5).mul(4)
      const time_warp = iTime.mul(0.6)

      const melt_factor = powFast(reality.x.abs().add(0.1), reality.y.add(time_warp).sin().mul(0.4).add(0.6))
      const liquid_space = reality.y.add(melt_factor.mul(3).sub(1.5))

      const flowing = powFast(liquid_space.abs().add(0.05), reality.x.add(time_warp.mul(0.7)).cos().mul(0.3).add(0.7))
      const dripping = flowing.mul(liquid_space.sign()).add(reality.x.mul(0.4))

      const surreal_bands = dripping.mul(5).fract()
      const psychedelic = powFast(surreal_bands.add(0.1), melt_factor.mul(0.6).add(0.2))

      const melted_hues = vec3(
              psychedelic.add(melt_factor.mul(0.8)),
              psychedelic.mul(0.7).add(flowing.mul(0.6)),
              psychedelic.mul(0.5).add(dripping.abs().mul(0.7))
      ).saturate()

      return vec4(melted_hues, 1)

}