Average Error: 33.8 → 9.6
Time: 7.4s
Precision: binary64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -4.7133602580677282 \cdot 10^{-75}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\\ \mathbf{elif}\;x.re \le -3.36363180591155713 \cdot 10^{-124}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -5.36067953850043814 \cdot 10^{-216}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\\ \mathbf{elif}\;x.re \le 2.7537773299991192 \cdot 10^{-46}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x.re}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1\\ \end{array}\]
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 -4.7133602580677282 \cdot 10^{-75}:\\
\;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\\

\mathbf{elif}\;x.re \le -3.36363180591155713 \cdot 10^{-124}:\\
\;\;\;\;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 1\\

\mathbf{elif}\;x.re \le -5.36067953850043814 \cdot 10^{-216}:\\
\;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\\

\mathbf{elif}\;x.re \le 2.7537773299991192 \cdot 10^{-46}:\\
\;\;\;\;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 1\\

\mathbf{else}:\\
\;\;\;\;\frac{{x.re}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \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 <= -4.713360258067728e-75)) {
		VAR = ((double) (((double) exp(((double) (((double) (((double) log(((double) (-1.0 * x_46_re)))) * y_46_re)) - ((double) (((double) (((double) pow(((double) cbrt(((double) cbrt(((double) atan2(x_46_im, x_46_re)))))), 5.0)) * ((double) cbrt(((double) cbrt(((double) atan2(x_46_im, x_46_re)))))))) * ((double) (((double) cbrt(((double) atan2(x_46_im, x_46_re)))) * y_46_im)))))))) * 1.0));
	} else {
		double VAR_1;
		if ((x_46_re <= -3.363631805911557e-124)) {
			VAR_1 = ((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)))))) * 1.0));
		} else {
			double VAR_2;
			if ((x_46_re <= -5.360679538500438e-216)) {
				VAR_2 = ((double) (((double) exp(((double) (((double) (((double) log(((double) (-1.0 * x_46_re)))) * y_46_re)) - ((double) (((double) (((double) pow(((double) cbrt(((double) cbrt(((double) atan2(x_46_im, x_46_re)))))), 5.0)) * ((double) cbrt(((double) cbrt(((double) atan2(x_46_im, x_46_re)))))))) * ((double) (((double) cbrt(((double) atan2(x_46_im, x_46_re)))) * y_46_im)))))))) * 1.0));
			} else {
				double VAR_3;
				if ((x_46_re <= 2.7537773299991192e-46)) {
					VAR_3 = ((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)))))) * 1.0));
				} else {
					VAR_3 = ((double) (((double) (((double) pow(x_46_re, y_46_re)) / ((double) exp(((double) (((double) atan2(x_46_im, x_46_re)) * y_46_im)))))) * 1.0));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if x.re < -4.7133602580677282e-75 or -3.36363180591155713e-124 < x.re < -5.36067953850043814e-216

    1. Initial program 34.7

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

    2. Taylor expanded around 0 16.7

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

    3. Taylor expanded around -inf 4.3

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

    5. Applied add-cube-cbrt4.3

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

    6. Applied associate-*l*4.3

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

    8. Applied add-cube-cbrt4.3

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

    9. Applied associate-*r*4.3

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

    10. Simplified4.3

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

    if -4.7133602580677282e-75 < x.re < -3.36363180591155713e-124 or -5.36067953850043814e-216 < x.re < 2.7537773299991192e-46

    1. Initial program 26.5

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

    2. Taylor expanded around 0 14.3

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

    if 2.7537773299991192e-46 < x.re

    1. Initial program 41.2

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

    2. Taylor expanded around 0 26.1

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

    3. Taylor expanded around inf 10.9

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

    4. Simplified13.8

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -4.7133602580677282 \cdot 10^{-75}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\\ \mathbf{elif}\;x.re \le -3.36363180591155713 \cdot 10^{-124}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -5.36067953850043814 \cdot 10^{-216}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{5} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\\ \mathbf{elif}\;x.re \le 2.7537773299991192 \cdot 10^{-46}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x.re}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(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)))))