Error

Derivation

1. Split input into 2 regimes

2. if (/ (sin (* (* (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re)))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))))) (/ (pow (exp y.im) (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re)))

   1. Initial program 58.7

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

   2. Applied simplify62.9

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

   4. Applied add-cube-cbrt62.9

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

   5. Applied pow-unpow62.9

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

   6. Taylor expanded around -inf 62.5

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

   7. Applied simplify62.0

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

   if (/ (sin (* (* (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re)))) (cbrt (fma (log (hypot x.im x.re)) y.im (* (atan2 x.im x.re) y.re))))) (/ (pow (exp y.im) (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re)))

   1. Initial program 32.8

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

   2. Applied simplify8.9

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

   4. Applied add-cube-cbrt8.9

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

   6. Applied add-cube-cbrt8.9

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

Runtime

Time bar (total: 46.9s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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)))))