It’s not just Taylor series

This article examines the approximation exp(-x²) ≈ (1 + cos(sin(x) + x))/2, which is accurate over [-4, 4] while the Taylor series approximation 1 - x² + x⁴/2 fails beyond [-0.5, 0.5]. The author argues that Taylor series alone cannot explain the cosine approximation's extended range, as the error in the Taylor approximation at x=4 is 113 versus 0.002579 for the cosine version. The article also tests the approximation in the complex plane, showing the cosine approximation performs worse than Taylor for large imaginary arguments, debunking the notion that Taylor series is the sole reason for its accuracy.