Average Error: 29.4 → 22.5
Time: 30.5s
Precision: 64
Internal Precision: 1408
\[\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)\]
↓
\[e^{\left(\sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \left(\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}}\]

Error

Bits error versus r

Bits error versus d

Derivation

  1. Initial program 29.4

    \[\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)\]
  2. Using strategy rm

  3. Applied add-log-exp22.5

    \[\leadsto \color{blue}{\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)}\]
  4. Using strategy rm

  5. Applied add-exp-log22.5

    \[\leadsto \color{blue}{e^{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt22.5

    \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt22.5

    \[\leadsto e^{\left(\sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\log \left(\log \left(e^{\color{blue}{\left(\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}}}\right)\right)}}\]

  10. Applied exp-prod22.5

    \[\leadsto e^{\left(\sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\log \left(\log \color{blue}{\left({\left(e^{\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}}\right)}^{\left(\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)}\right)}\right)}}\]

  11. Applied log-pow22.5

    \[\leadsto e^{\left(\sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\log \color{blue}{\left(\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \log \left(e^{\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}}\right)\right)}}}\]

  12. Applied simplify22.5

    \[\leadsto e^{\left(\sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)} \cdot \sqrt[3]{\log \left(\log \left(e^{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \color{blue}{\left(\sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} \left(\frac{r}{d}\right)\right)}\right)}\right)}}\]
  13. Removed slow pow expressions.

Runtime

Time bar (total: 30.5s)Debug log

herbie shell --seed '#(428700651 1106906792 470491210 3500543625 1808163884 651714402)' -o rules:trigonometry -o setup:simplify -o generate:taylor -o generate:simplify
(FPCore (r d)
  :name "cos atan frac"
  (cos (atan (/ r d))))