Average Error: 33.2 → 4.2
Time: 7.0s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}\]
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)
↓
e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4} \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 \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14092 = x_re;
        double r14093 = r14092 * r14092;
        double r14094 = x_im;
        double r14095 = r14094 * r14094;
        double r14096 = r14093 + r14095;
        double r14097 = sqrt(r14096);
        double r14098 = log(r14097);
        double r14099 = y_re;
        double r14100 = r14098 * r14099;
        double r14101 = atan2(r14094, r14092);
        double r14102 = y_im;
        double r14103 = r14101 * r14102;
        double r14104 = r14100 - r14103;
        double r14105 = exp(r14104);
        double r14106 = r14098 * r14102;
        double r14107 = r14101 * r14099;
        double r14108 = r14106 + r14107;
        double r14109 = cos(r14108);
        double r14110 = r14105 * r14109;
        return r14110;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14111 = 1.0;
        double r14112 = x_re;
        double r14113 = x_im;
        double r14114 = hypot(r14112, r14113);
        double r14115 = r14111 * r14114;
        double r14116 = log(r14115);
        double r14117 = y_re;
        double r14118 = r14116 * r14117;
        double r14119 = atan2(r14113, r14112);
        double r14120 = cbrt(r14119);
        double r14121 = cbrt(r14120);
        double r14122 = 4.0;
        double r14123 = pow(r14121, r14122);
        double r14124 = r14121 * r14121;
        double r14125 = r14123 * r14124;
        double r14126 = y_im;
        double r14127 = r14120 * r14126;
        double r14128 = r14125 * r14127;
        double r14129 = r14118 - r14128;
        double r14130 = exp(r14129);
        return r14130;
}

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 33.2

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

  2. Taylor expanded around 0 19.9

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

  4. Applied *-un-lft-identity19.9

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

  5. Applied sqrt-prod19.9

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

  6. Simplified19.9

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

  7. Simplified4.2

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

  9. Applied add-cube-cbrt4.2

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

  10. Applied associate-*l*4.2

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

  12. Applied add-cube-cbrt4.2

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

  13. Applied add-cube-cbrt4.2

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

  14. Applied swap-sqr4.2

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

  15. Simplified4.2

    \[\leadsto e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)} \cdot 1\]

  16. Final simplification4.2

    \[\leadsto e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \left({\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \cdot \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)}\]

Reproduce

herbie shell --seed 2020036 +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)))))