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Derivation

1. Initial program 0.2

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\]

2. Initial simplification0.2

   \[\leadsto \frac{1}{\sin B} - \frac{x}{\tan B}\]

3. Taylor expanded around inf 0.2

   \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}}\]
4. Using strategy rm

5. Applied add-cube-cbrt0.6

   \[\leadsto \frac{1}{\sin B} - \frac{x \cdot \cos B}{\color{blue}{\left(\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}\right) \cdot \sqrt[3]{\sin B}}}\]

6. Applied times-frac0.7

   \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{\cos B}{\sqrt[3]{\sin B}}}\]

7. Applied div-inv0.7

   \[\leadsto \color{blue}{1 \cdot \frac{1}{\sin B}} - \frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}} \cdot \frac{\cos B}{\sqrt[3]{\sin B}}\]

8. Applied prod-diff0.7

   \[\leadsto \color{blue}{(1 \cdot \left(\frac{1}{\sin B}\right) + \left(-\frac{\cos B}{\sqrt[3]{\sin B}} \cdot \frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}\right))_* + (\left(-\frac{\cos B}{\sqrt[3]{\sin B}}\right) \cdot \left(\frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}\right) + \left(\frac{\cos B}{\sqrt[3]{\sin B}} \cdot \frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}\right))_*}\]

9. Simplified0.2

   \[\leadsto \color{blue}{\left(\frac{1}{\sin B} - \frac{\cos B}{\sin B} \cdot x\right)} + (\left(-\frac{\cos B}{\sqrt[3]{\sin B}}\right) \cdot \left(\frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}\right) + \left(\frac{\cos B}{\sqrt[3]{\sin B}} \cdot \frac{x}{\sqrt[3]{\sin B} \cdot \sqrt[3]{\sin B}}\right))_*\]

10. Simplified0.2

    \[\leadsto \left(\frac{1}{\sin B} - \frac{\cos B}{\sin B} \cdot x\right) + \color{blue}{0}\]

11. Final simplification0.2

    \[\leadsto \frac{1}{\sin B} - \frac{\cos B}{\sin B} \cdot x\]

Runtime

Time bar (total: 41.9s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))