Average Error: 0.1 → 0.2
Time: 26.4s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\frac{e \cdot \sin v}{\sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*} \cdot \sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*}}}{\sqrt[3]{(\left(\cos v\right) \cdot e + 1)_*}}\]

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 add-cube-cbrt0.2

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

  5. Applied associate-/r*0.2

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

  6. Final simplification0.2

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

Runtime

Time bar (total: 26.4s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))