Average Error: 0.1 → 0.3
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\frac{(\left(\cos v\right) \cdot e + 1)_*}{\sin v}}\]

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 *-un-lft-identity0.1

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

  5. Applied associate-/r*0.1

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

  6. Applied associate-*l/0.1

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

  7. Simplified0.1

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

  9. Applied *-commutative0.1

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

  10. Applied associate-/l*0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))