Average Error: 32.9 → 5.6
Time: 6.2s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -6.3846056748134043 \cdot 10^{81} \lor \neg \left(y.re \le 7.1591786557106491 \cdot 10^{162} \lor \neg \left(y.re \le 2.46354771067311571 \cdot 10^{232}\right)\right):\\ \;\;\;\;e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - 0}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)}\\ \end{array}\]
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}\;y.re \le -6.3846056748134043 \cdot 10^{81} \lor \neg \left(y.re \le 7.1591786557106491 \cdot 10^{162} \lor \neg \left(y.re \le 2.46354771067311571 \cdot 10^{232}\right)\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} - 0}\\

\mathbf{else}:\\
\;\;\;\;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} - \mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r19037 = x_re;
        double r19038 = r19037 * r19037;
        double r19039 = x_im;
        double r19040 = r19039 * r19039;
        double r19041 = r19038 + r19040;
        double r19042 = sqrt(r19041);
        double r19043 = log(r19042);
        double r19044 = y_re;
        double r19045 = r19043 * r19044;
        double r19046 = atan2(r19039, r19037);
        double r19047 = y_im;
        double r19048 = r19046 * r19047;
        double r19049 = r19045 - r19048;
        double r19050 = exp(r19049);
        double r19051 = r19043 * r19047;
        double r19052 = r19046 * r19044;
        double r19053 = r19051 + r19052;
        double r19054 = cos(r19053);
        double r19055 = r19050 * r19054;
        return r19055;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r19056 = y_re;
        double r19057 = -6.384605674813404e+81;
        bool r19058 = r19056 <= r19057;
        double r19059 = 7.159178655710649e+162;
        bool r19060 = r19056 <= r19059;
        double r19061 = 2.4635477106731157e+232;
        bool r19062 = r19056 <= r19061;
        double r19063 = !r19062;
        bool r19064 = r19060 || r19063;
        double r19065 = !r19064;
        bool r19066 = r19058 || r19065;
        double r19067 = cbrt(r19056);
        double r19068 = r19067 * r19067;
        double r19069 = 1.0;
        double r19070 = x_re;
        double r19071 = x_im;
        double r19072 = hypot(r19070, r19071);
        double r19073 = log(r19072);
        double r19074 = r19069 * r19073;
        double r19075 = r19068 * r19074;
        double r19076 = r19075 * r19067;
        double r19077 = 0.0;
        double r19078 = r19076 - r19077;
        double r19079 = exp(r19078);
        double r19080 = atan2(r19071, r19070);
        double r19081 = y_im;
        double r19082 = r19080 * r19081;
        double r19083 = log1p(r19082);
        double r19084 = expm1(r19083);
        double r19085 = r19076 - r19084;
        double r19086 = exp(r19085);
        double r19087 = r19066 ? r19079 : r19086;
        return r19087;
}

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 y.re < -6.384605674813404e+81 or 7.159178655710649e+162 < y.re < 2.4635477106731157e+232

    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 1.0

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

    4. Applied add-cube-cbrt1.0

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

    5. Applied associate-*r*1.0

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

    6. Simplified0

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

    8. Applied expm1-log1p-u9.5

      \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)}} \cdot 1\]

    9. Taylor expanded around inf 2.9

      \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \color{blue}{0}} \cdot 1\]

    if -6.384605674813404e+81 < y.re < 7.159178655710649e+162 or 2.4635477106731157e+232 < y.re

    1. Initial program 32.2

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

    2. Taylor expanded around 0 25.0

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

    4. Applied add-cube-cbrt25.0

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

    5. Applied associate-*r*25.0

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

    6. Simplified5.5

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

    8. Applied expm1-log1p-u6.5

      \[\leadsto e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)}} \cdot 1\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -6.3846056748134043 \cdot 10^{81} \lor \neg \left(y.re \le 7.1591786557106491 \cdot 10^{162} \lor \neg \left(y.re \le 2.46354771067311571 \cdot 10^{232}\right)\right):\\ \;\;\;\;e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - 0}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \left(1 \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right) \cdot \sqrt[3]{y.re} - \mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))))