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Derivation

1. Initial program 0.1

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

2. Initial simplification0.1

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

4. Applied flip3-+0.1

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

5. Applied associate-/r/0.1

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

6. Simplified0.1

   \[\leadsto \left(\frac{\sin v}{{\left(\cos v \cdot e\right)}^{3} + {1}^{3}} \cdot \color{blue}{\left(1 + \left(e \cdot \cos v - 1\right) \cdot \left(e \cdot \cos v\right)\right)}\right) \cdot e\]
7. Using strategy rm

8. Applied associate-*l/0.1

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

9. Applied associate-*l/0.1

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

10. Final simplification0.1

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

Runtime

Time bar (total: 36.8s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))