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Derivation

1. Initial program 12.5

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

2. Initial simplification11.5

   \[\leadsto \frac{\sin th \cdot \sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]
3. Using strategy rm

4. Applied *-un-lft-identity11.5

   \[\leadsto \frac{\sin th \cdot \sin ky}{\color{blue}{1 \cdot \sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]

5. Applied times-frac8.8

   \[\leadsto \color{blue}{\frac{\sin th}{1} \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}\]

6. Simplified8.8

   \[\leadsto \color{blue}{\sin th} \cdot \frac{\sin ky}{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}\]
7. Using strategy rm

8. Applied add-sqr-sqrt9.1

   \[\leadsto \sin th \cdot \frac{\sin ky}{\color{blue}{\sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*} \cdot \sqrt{\sqrt{\left(\sin kx\right)^2 + \left(\sin ky\right)^2}^*}}}\]
9. Using strategy rm

10. Applied add-cube-cbrt9.2

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

11. Final simplification9.2

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

Runtime

Time bar (total: 37.9s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))