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Derivation

1. Split input into 2 regimes

2. if (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) < 1.0

   1. Initial program 2.2

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

   2. Taylor expanded around inf 2.2

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

   if 1.0 < (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))

   1. Initial program 61.6

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

   2. Taylor expanded around inf 61.6

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

   3. Taylor expanded around 0 29.8

      \[\leadsto \frac{\sin ky}{\color{blue}{\left(\frac{1}{12} \cdot \left({kx}^{2} \cdot ky\right) + ky\right) - \frac{1}{6} \cdot {ky}^{3}}} \cdot \sin th\]
3. Recombined 2 regimes into one program.

4. Final simplification3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 1.0:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(\frac{1}{12} \cdot \left(ky \cdot {kx}^{2}\right) + ky\right) - {ky}^{3} \cdot \frac{1}{6}}\\ \end{array}\]

Runtime

Time bar (total: 20.1s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))