Average Error: 32.8 → 8.8
Time: 7.6s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -6.9710965962323107 \cdot 10^{-309}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{x.re}^{y.re}}\right)}^{\left(\frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}\right) \cdot 1\\ \end{array}\]
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)
↓
\begin{array}{l}
\mathbf{if}\;x.re \le -6.9710965962323107 \cdot 10^{-309}:\\
\;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\

\mathbf{else}:\\
\;\;\;\;\log \left({\left(e^{{x.re}^{y.re}}\right)}^{\left(\frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}\right) \cdot 1\\

\end{array}
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) {
	double VAR;
	if ((x_46_re <= -6.97109659623231e-309)) {
		VAR = ((double) (((double) exp(((double) (((double) (-1.0 * ((double) (y_46_re * ((double) log(((double) (-1.0 / x_46_re)))))))) - ((double) (((double) atan2(x_46_im, x_46_re)) * y_46_im)))))) * 1.0));
	} else {
		VAR = ((double) (((double) log(((double) pow(((double) exp(((double) pow(x_46_re, y_46_re)))), ((double) (1.0 / ((double) exp(((double) (((double) atan2(x_46_im, x_46_re)) * y_46_im)))))))))) * 1.0));
	}
	return VAR;
}

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. Split input into 2 regimes

  2. if x.re < -6.97109659623231e-309

    1. Initial program 30.4

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

    2. Taylor expanded around 0 16.4

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

    3. Taylor expanded around -inf 6.1

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

    if -6.97109659623231e-309 < x.re

    1. Initial program 34.9

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

    2. Taylor expanded around 0 21.5

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

    4. Applied sub-neg21.5

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

    5. Applied exp-sum24.6

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

    6. Simplified24.6

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

    7. Taylor expanded around inf 15.5

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

    8. Simplified15.5

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

    10. Applied add-log-exp15.6

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

    11. Simplified11.2

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -6.9710965962323107 \cdot 10^{-309}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{{x.re}^{y.re}}\right)}^{\left(\frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(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)))))