Average Error: 0.1 → 0.5
Time: 49.8s
Precision: 64
Internal Precision: 384
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\sqrt{\frac{e}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}}} \cdot \left(\sqrt{\frac{e}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}}} \cdot \frac{\sin v}{\sqrt[3]{1 + e \cdot \cos v}}\right)\]

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. Using strategy rm

  3. Applied add-cube-cbrt0.2

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

  4. Applied times-frac0.2

    \[\leadsto \color{blue}{\frac{e}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt[3]{1 + e \cdot \cos v}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.5

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

  7. Applied associate-*l*0.5

    \[\leadsto \color{blue}{\sqrt{\frac{e}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}}} \cdot \left(\sqrt{\frac{e}{\sqrt[3]{1 + e \cdot \cos v} \cdot \sqrt[3]{1 + e \cdot \cos v}}} \cdot \frac{\sin v}{\sqrt[3]{1 + e \cdot \cos v}}\right)}\]
  8. Removed slow pow expressions.

Runtime

Time bar (total: 49.8s)Debug log

herbie shell --seed '#(2961832646 520228599 1275628947 1047906571 1774476463 2890033825)' 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))