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Derivation

1. Initial program 0.1

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

2. Initial simplification0.1

   \[\leadsto \frac{e \cdot \sin v}{\cos v \cdot e + 1}\]
3. Using strategy rm

4. Applied add-cube-cbrt0.2

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\left(\sqrt[3]{\cos v \cdot e + 1} \cdot \sqrt[3]{\cos v \cdot e + 1}\right) \cdot \sqrt[3]{\cos v \cdot e + 1}}}\]

5. Applied times-frac0.2

   \[\leadsto \color{blue}{\frac{e}{\sqrt[3]{\cos v \cdot e + 1} \cdot \sqrt[3]{\cos v \cdot e + 1}} \cdot \frac{\sin v}{\sqrt[3]{\cos v \cdot e + 1}}}\]
6. Using strategy rm

7. Applied flip-+0.2

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

8. Applied cbrt-div0.2

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

9. Applied flip-+0.2

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

10. Applied cbrt-div0.2

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

11. Applied frac-times0.2

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

12. Applied associate-/r/0.2

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

13. Applied associate-*l*0.2

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

14. Simplified0.2

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

15. Final simplification0.2

    \[\leadsto \left(\left(\sqrt[3]{e \cdot \cos v - 1} \cdot \sqrt[3]{e \cdot \cos v - 1}\right) \cdot \frac{\sin v}{\sqrt[3]{1 + e \cdot \cos v}}\right) \cdot \frac{e}{\sqrt[3]{-1 + \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \sqrt[3]{-1 + \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}}\]

Runtime

Time bar (total: 55.6s)Debug logProfile

herbie shell --seed 2018336 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))