Average Error: 0.1 → 0.1
Time: 43.6s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\sqrt{\left(1 - e \cdot \cos v\right) + \left(e \cdot \cos v\right) \cdot \left(e \cdot \cos v\right)} \cdot \frac{e}{\sqrt{{\left(e \cdot \cos v\right)}^{3} + 1}}\right) \cdot \frac{\sin v}{\sqrt{e \cdot \cos v + 1}}\]

Error

Bits error versus e

Bits error versus v

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Results

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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-sqr-sqrt0.1

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\sqrt{\cos v \cdot e + 1} \cdot \sqrt{\cos v \cdot e + 1}}}\]

  5. Applied times-frac0.1

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

  7. Applied flip3-+0.1

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

  8. Applied sqrt-div0.1

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

  9. Applied associate-/r/0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Runtime

Time bar (total: 43.6s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))