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

↓

\[\frac{e \cdot \sin v}{(\left(e \cdot \cos v\right) \cdot \left(\cos v \cdot \left(-e\right)\right) + 1)_*} \cdot \left(1 - e \cdot \cos v\right)\]

Derivation

Initial program 0.1
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
Using strategy rm
Applied flip-+0.1
\[\leadsto \frac{e \cdot \sin v}{\color{blue}{\frac{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)}{1 - e \cdot \cos v}}}\]
Applied associate-/r/0.1
\[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 \cdot 1 - \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \left(1 - e \cdot \cos v\right)}\]
Applied simplify0.1
\[\leadsto \color{blue}{\frac{e \cdot \sin v}{(\left(e \cdot \cos v\right) \cdot \left(\cos v \cdot \left(-e\right)\right) + 1)_*}} \cdot \left(1 - e \cdot \cos v\right)\]

Runtime

herbie shell --seed 2018208 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))