Error

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Derivation

1. Split input into 2 regimes

2. if (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) < 0.9999953171152953

   1. Initial program 14.1

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

   3. Applied clear-num14.1

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

   4. Taylor expanded around inf 14.1

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

   if 0.9999953171152953 < (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2))))

   1. Initial program 9.1

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

   3. Applied clear-num9.1

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

   4. Taylor expanded around inf 9.1

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

   5. Taylor expanded around 0 5.2

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

4. Final simplification12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}} \le 0.9999953171152953:\\ \;\;\;\;\frac{1}{\frac{\sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}}{\sin ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(1 - \frac{1}{6} \cdot {kx}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2) (pow (sin ky) 2)))) (sin th)))

Details

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