Average Error: 32.8 → 4.0
Time: 8.0s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]
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)
↓
e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r19096 = x_re;
        double r19097 = r19096 * r19096;
        double r19098 = x_im;
        double r19099 = r19098 * r19098;
        double r19100 = r19097 + r19099;
        double r19101 = sqrt(r19100);
        double r19102 = log(r19101);
        double r19103 = y_re;
        double r19104 = r19102 * r19103;
        double r19105 = atan2(r19098, r19096);
        double r19106 = y_im;
        double r19107 = r19105 * r19106;
        double r19108 = r19104 - r19107;
        double r19109 = exp(r19108);
        double r19110 = r19102 * r19106;
        double r19111 = r19105 * r19103;
        double r19112 = r19110 + r19111;
        double r19113 = cos(r19112);
        double r19114 = r19109 * r19113;
        return r19114;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r19115 = y_re;
        double r19116 = cbrt(r19115);
        double r19117 = r19116 * r19116;
        double r19118 = 1.0;
        double r19119 = x_re;
        double r19120 = x_im;
        double r19121 = hypot(r19119, r19120);
        double r19122 = log(r19121);
        double r19123 = r19118 * r19122;
        double r19124 = r19117 * r19123;
        double r19125 = r19124 * r19116;
        double r19126 = atan2(r19120, r19119);
        double r19127 = y_im;
        double r19128 = r19126 * r19127;
        double r19129 = r19125 - r19128;
        double r19130 = exp(r19129);
        return r19130;
}

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 19.4

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

  4. Applied add-cube-cbrt19.4

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

  5. Applied associate-*r*19.4

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

  6. Simplified4.0

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

  7. Final simplification4.0

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

Reproduce

herbie shell --seed 2020056 +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)))))