Average Error: 0.1 → 0.1
Time: 13.6s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{1 + {\left(\cos v \cdot e\right)}^{3}} \cdot \left(\left(1 - \cos v \cdot e\right) + \left(\cos v \cdot e\right) \cdot \left(\cos v \cdot e\right)\right)\]

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. Initial simplification0.1

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

  4. Applied flip3-+0.1

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

  5. Applied associate-/r/0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Runtime

Time bar (total: 13.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.10.10.00.10%
herbie shell --seed 2018296 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))