Average Error: 33.2 → 4.0
Time: 39.5s
Precision: 64
Internal Precision: 2112
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)} \cdot \left(\sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)} \cdot \sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}\right)}{e^{\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right) - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 33.2

    \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

  2. Initial simplification9.5

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{{\left(\sqrt{x.re^2 + x.im^2}^*\right)}^{y.re}}}\]
  3. Using strategy rm

  4. Applied pow-to-exp9.5

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{{\left(e^{y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}}{\color{blue}{e^{\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}}\]

  5. Applied pow-exp8.8

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\frac{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}{e^{\log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}\]

  6. Applied div-exp3.5

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt3.5

    \[\leadsto \frac{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}{e^{\color{blue}{\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt4.0

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)} \cdot \sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}\right) \cdot \sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}}}{e^{\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt4.0

    \[\leadsto \frac{\left(\sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)} \cdot \sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}\right) \cdot \sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}}{e^{\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)} - \log \left(\sqrt{x.re^2 + x.im^2}^*\right) \cdot y.re}}\]

  13. Final simplification4.0

    \[\leadsto \frac{\sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)} \cdot \left(\sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)} \cdot \sqrt[3]{\sin \left((y.im \cdot \left(\log \left(\sqrt{x.re^2 + x.im^2}^*\right)\right) + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right))_*\right)}\right)}{e^{\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right) - y.re \cdot \log \left(\sqrt{x.re^2 + x.im^2}^*\right)}}\]

Runtime

Time bar (total: 39.5s)Debug logProfile

herbie shell --seed 2018256 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))