taylorInvSqrt: Taylor Series Inverse Square Root Approximation

Fast Mathematical Approximation for Graphics Programming

The Taylor inverse square root function provides a rapid approximation of $\frac{1}{\sqrt{r}}$ using polynomial expansion. This technique sacrifices precision for computational speed, making it valuable in real-time graphics applications where normalization operations dominate performance.

The mathematical foundation rests on the Taylor series expansion around a carefully chosen point, yielding the linear approximation:

$$\text{taylorInvSqrt}(r) = 1.79284291400159 - 0.85373472095314 \cdot r$$

This approximation maintains reasonable accuracy within the range $r \in [0.5, 2.0]$, where the linear model closely follows the true inverse square root curve. Beyond this interval, deviation increases exponentially, creating distinct visual artifacts that reveal the approximation's mathematical boundaries.

Frequency Domain Analysis

The approximation's linear nature creates unique harmonic distortions when applied to oscillating functions. This mathematical property generates distinctive interference patterns that expose the polynomial's spectral characteristics:

ライブエディター

const fragment = () => {
      const t = iTime.mul(2)
      const freq = uv.x.mul(8).add(1)
      const amplitude = uv.y.mul(2).add(0.5)
      const signal = freq.mul(t).sin().mul(amplitude)
      const normalized = signal.mul(signal).add(1)
      const taylorNorm = taylorInvSqrt(normalized)
      const phase = taylorNorm.mul(freq).mul(t).sin()
      const spectral = phase.abs().pow(0.3)
      const color = vec3(
              spectral,
              spectral.mul(0.7),
              spectral.mul(0.4)
      )
      return vec4(color, 1)
}