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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

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

6. Applied add-cube-cbrt1.3

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

7. Applied prod-diff1.3

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

8. Simplified0.2

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

9. Simplified0.2

   \[\leadsto \left(\frac{1}{\sin B} - \frac{\cos B}{\frac{\sin B}{x}}\right) + \color{blue}{0}\]
10. Using strategy rm

11. Applied clear-num0.2

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

12. Final simplification0.2

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018351 +o rules:numerics
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  (+ (- (* x (/ 1 (tan B)))) (/ 1 (sin B))))