Average Error: 0.1 → 0.2
Time: 28.0s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\frac{\sin v}{(\left(\cos v\right) \cdot e + 1)_*}}{\frac{1}{e}}\]

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. Simplified0.1

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

  4. Applied clear-num0.2

    \[\leadsto \color{blue}{\frac{1}{\frac{(\left(\cos v\right) \cdot e + 1)_*}{e}}} \cdot \sin v\]
  5. Using strategy rm

  6. Applied associate-*l/0.2

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

  7. Simplified0.2

    \[\leadsto \frac{\color{blue}{\sin v}}{\frac{(\left(\cos v\right) \cdot e + 1)_*}{e}}\]
  8. Using strategy rm

  9. Applied div-inv0.2

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

  10. Applied associate-/r*0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))