\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

↓

\[\frac{\sin ky}{\sqrt{\left(\sqrt[3]{{\left(\sin kx\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

Error

The X axis uses an exponential scale — Bits error versus `kx`

Derivation

Initial program 3.8
\[\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Taylor expanded around -inf 3.8
\[\leadsto \frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + \color{blue}{{\left(\sin ky\right)}^{2}}}} \cdot \sin th\]
Using strategy rm
Applied add-cube-cbrt4.0
\[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\left(\sqrt[3]{{\left(\sin kx\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]
Final simplification4.0
\[\leadsto \frac{\sin ky}{\sqrt{\left(\sqrt[3]{{\left(\sin kx\right)}^{2}} \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}}\right) \cdot \sqrt[3]{{\left(\sin kx\right)}^{2}} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	4.0	4.0	2.5	1.5	0%

herbie shell --seed 2018285 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))

In
Out

Error

Try it out

Derivation

Runtime