Average Error: 0.1 → 0.1
Time: 31.5s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + {\left(\cos v \cdot e\right)}^{3}} \cdot \left(\left(\cos v \cdot e + -1\right) \cdot \left(\cos v \cdot e\right) - -1\right)\]

Error

Bits error versus e

Bits error versus v

Derivation

  1. Initial program 0.1

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

  3. Applied flip3-+0.1

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

  4. Applied associate-/r/0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019090 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))