Average Error: 0.1 → 0.1
Time: 29.2s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{(\left(\cos v\right) \cdot e + 1)_*}\]

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 add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied pow10.4

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

  8. Applied pow10.4

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

  9. Applied pow-prod-down0.4

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

  10. Applied pow10.4

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

  11. Applied pow-prod-down0.4

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))