Average Error: 0.1 → 0.1
Time: 23.8s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + {\left(\cos v \cdot e\right)}^{3}} \cdot \left(\log \left(e^{\left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)}\right) + \left(1 - \cos v \cdot e\right)\right)\]

Error

Bits error versus e

Bits error versus v

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Your Program's Arguments

Results

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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

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

  7. Applied add-log-exp0.1

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

  8. Final simplification0.1

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

Runtime

Time bar (total: 23.8s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))