Average Error: 32.8 → 4.5
Time: 8.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\log \left({\left({\left(e^{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\right)}^{\left({\left(\sqrt[3]{e^{-y.im}} \cdot \sqrt[3]{e^{-y.im}}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}\right)}^{\left({\left(\sqrt[3]{e^{-y.im}}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}\right)\]
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 \cos \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)
↓
\log \left({\left({\left(e^{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\right)}^{\left({\left(\sqrt[3]{e^{-y.im}} \cdot \sqrt[3]{e^{-y.im}}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}\right)}^{\left({\left(\sqrt[3]{e^{-y.im}}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}\right)
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((double) (((double) exp(((double) (((double) (((double) log(((double) sqrt(((double) (((double) (x_46_re * x_46_re)) + ((double) (x_46_im * x_46_im)))))))) * y_46_re)) - ((double) (((double) atan2(x_46_im, x_46_re)) * y_46_im)))))) * ((double) cos(((double) (((double) (((double) log(((double) sqrt(((double) (((double) (x_46_re * x_46_re)) + ((double) (x_46_im * x_46_im)))))))) * y_46_im)) + ((double) (((double) atan2(x_46_im, x_46_re)) * y_46_re))))))));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((double) log(((double) pow(((double) pow(((double) exp(((double) pow(((double) hypot(x_46_re, x_46_im)), y_46_re)))), ((double) pow(((double) (((double) cbrt(((double) exp(((double) -(y_46_im)))))) * ((double) cbrt(((double) exp(((double) -(y_46_im)))))))), ((double) atan2(x_46_im, x_46_re)))))), ((double) pow(((double) cbrt(((double) exp(((double) -(y_46_im)))))), ((double) atan2(x_46_im, x_46_re))))))));
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.8

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

  2. Taylor expanded around 0 19.1

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

  4. Applied add-log-exp19.1

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

  5. Simplified4.3

    \[\leadsto \log \color{blue}{\left({\left(e^{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\right)}^{\left(e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)}\right)} \cdot 1\]
  6. Using strategy rm

  7. Applied *-commutative4.3

    \[\leadsto \log \left({\left(e^{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\right)}^{\left(e^{-\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}\right) \cdot 1\]

  8. Applied distribute-lft-neg-in4.3

    \[\leadsto \log \left({\left(e^{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\right)}^{\left(e^{\color{blue}{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}\right) \cdot 1\]

  9. Applied exp-prod4.5

    \[\leadsto \log \left({\left(e^{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\right)}^{\color{blue}{\left({\left(e^{-y.im}\right)}^{\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}}\right) \cdot 1\]
  10. Using strategy rm

  11. Applied add-cube-cbrt4.5

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

  12. Applied unpow-prod-down4.5

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

  13. Applied pow-unpow4.5

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

  14. Final simplification4.5

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))