Average Error: 0.1 → 0.2
Time: 25.4s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \left(\sin v \cdot \frac{\frac{1}{\sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*} \cdot \sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*}}}{\sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*}}\right)\]

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 div-inv0.1

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

  5. Applied associate-*l*0.1

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

  7. Applied add-cube-cbrt0.2

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

  8. Applied associate-/r*0.2

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

  9. Final simplification0.2

    \[\leadsto e \cdot \left(\sin v \cdot \frac{\frac{1}{\sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*} \cdot \sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*}}}{\sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*}}\right)\]

Reproduce

herbie shell --seed 2019050 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))