Average Error: 0.1 → 0.2
Time: 27.1s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\left(\left(\sqrt[3]{\frac{1}{(\left(\cos v\right) \cdot e + 1)_*}} \cdot \sqrt[3]{\frac{1}{(\left(\cos v\right) \cdot e + 1)_*}}\right) \cdot \sqrt[3]{\frac{1}{(\left(\cos v\right) \cdot e + 1)_*}}\right) \cdot \left(e \cdot \sin v\right)\]

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. Initial simplification0.1

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

  4. Applied div-inv0.1

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

  6. Applied add-cube-cbrt0.2

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

  7. Final simplification0.2

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

Runtime

Time bar (total: 27.1s)Debug logProfile

herbie shell --seed 2018285 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))