Average Error: 0.1 → 0.2
Time: 44.4s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[(e^{\log_* (1 + \frac{\sin v}{\sqrt[3]{(e \cdot \left(\cos v\right) + 1)_*}})} - 1)^* \cdot \frac{\frac{e}{\sqrt[3]{(e \cdot \left(\cos v\right) + 1)_*}}}{\sqrt[3]{(e \cdot \left(\cos v\right) + 1)_*}}\]

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. 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. Simplified0.2

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

  6. Simplified0.2

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

  8. Applied expm1-log1p-u0.2

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

  9. Final simplification0.2

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

Runtime

Time bar (total: 44.4s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))