Average Error: 32.8 → 9.7
Time: 29.7s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -4.9835237568442066 \cdot 10^{-17}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -3.7737597763103814 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \sqrt[3]{\cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot \sqrt[3]{\cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -1.91205330543206 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 9.798541471285326 \cdot 10^{-63}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 3.102132787630003 \cdot 10^{-33}:\\ \;\;\;\;e^{\log \left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
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 -4.9835237568442066 \cdot 10^{-17}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le -3.7737597763103814 \cdot 10^{-161}:\\
\;\;\;\;\left(\left(\sqrt[3]{\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)} \cdot \sqrt[3]{\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)}\right) \cdot \sqrt[3]{\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)}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le -1.91205330543206 \cdot 10^{-310}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le 9.798541471285326 \cdot 10^{-63}:\\
\;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le 3.102132787630003 \cdot 10^{-33}:\\
\;\;\;\;e^{\log \left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1374987 = x_re;
        double r1374988 = r1374987 * r1374987;
        double r1374989 = x_im;
        double r1374990 = r1374989 * r1374989;
        double r1374991 = r1374988 + r1374990;
        double r1374992 = sqrt(r1374991);
        double r1374993 = log(r1374992);
        double r1374994 = y_re;
        double r1374995 = r1374993 * r1374994;
        double r1374996 = atan2(r1374989, r1374987);
        double r1374997 = y_im;
        double r1374998 = r1374996 * r1374997;
        double r1374999 = r1374995 - r1374998;
        double r1375000 = exp(r1374999);
        double r1375001 = r1374993 * r1374997;
        double r1375002 = r1374996 * r1374994;
        double r1375003 = r1375001 + r1375002;
        double r1375004 = cos(r1375003);
        double r1375005 = r1375000 * r1375004;
        return r1375005;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1375006 = x_re;
        double r1375007 = -4.9835237568442066e-17;
        bool r1375008 = r1375006 <= r1375007;
        double r1375009 = -r1375006;
        double r1375010 = log(r1375009);
        double r1375011 = y_re;
        double r1375012 = r1375010 * r1375011;
        double r1375013 = y_im;
        double r1375014 = x_im;
        double r1375015 = atan2(r1375014, r1375006);
        double r1375016 = r1375013 * r1375015;
        double r1375017 = r1375012 - r1375016;
        double r1375018 = exp(r1375017);
        double r1375019 = -3.7737597763103814e-161;
        bool r1375020 = r1375006 <= r1375019;
        double r1375021 = r1375006 * r1375006;
        double r1375022 = r1375014 * r1375014;
        double r1375023 = r1375021 + r1375022;
        double r1375024 = sqrt(r1375023);
        double r1375025 = log(r1375024);
        double r1375026 = r1375025 * r1375013;
        double r1375027 = r1375015 * r1375011;
        double r1375028 = r1375026 + r1375027;
        double r1375029 = cos(r1375028);
        double r1375030 = cbrt(r1375029);
        double r1375031 = r1375030 * r1375030;
        double r1375032 = r1375031 * r1375030;
        double r1375033 = r1375011 * r1375025;
        double r1375034 = r1375033 - r1375016;
        double r1375035 = exp(r1375034);
        double r1375036 = r1375032 * r1375035;
        double r1375037 = -1.91205330543206e-310;
        bool r1375038 = r1375006 <= r1375037;
        double r1375039 = 9.798541471285326e-63;
        bool r1375040 = r1375006 <= r1375039;
        double r1375041 = log(r1375006);
        double r1375042 = r1375011 * r1375041;
        double r1375043 = r1375042 - r1375016;
        double r1375044 = exp(r1375043);
        double r1375045 = 3.102132787630003e-33;
        bool r1375046 = r1375006 <= r1375045;
        double r1375047 = exp(r1375024);
        double r1375048 = log(r1375047);
        double r1375049 = log(r1375048);
        double r1375050 = r1375049 * r1375011;
        double r1375051 = r1375050 - r1375016;
        double r1375052 = exp(r1375051);
        double r1375053 = r1375046 ? r1375052 : r1375044;
        double r1375054 = r1375040 ? r1375044 : r1375053;
        double r1375055 = r1375038 ? r1375018 : r1375054;
        double r1375056 = r1375020 ? r1375036 : r1375055;
        double r1375057 = r1375008 ? r1375018 : r1375056;
        return r1375057;
}

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 4 regimes

  2. if x.re < -4.9835237568442066e-17 or -3.7737597763103814e-161 < x.re < -1.91205330543206e-310

    1. Initial program 35.6

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

    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. 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. Simplified4.3

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

    if -4.9835237568442066e-17 < x.re < -3.7737597763103814e-161

    1. Initial program 15.1

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

    3. Applied add-cube-cbrt15.1

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

    if -1.91205330543206e-310 < x.re < 9.798541471285326e-63 or 3.102132787630003e-33 < x.re

    1. Initial program 35.2

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

    2. Taylor expanded around 0 22.2

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

    3. Taylor expanded around inf 11.5

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

    if 9.798541471285326e-63 < x.re < 3.102132787630003e-33

    1. Initial program 20.5

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

    2. Taylor expanded around 0 13.9

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

    4. Applied add-log-exp32.0

      \[\leadsto e^{\log \color{blue}{\left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -4.9835237568442066 \cdot 10^{-17}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -3.7737597763103814 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \sqrt[3]{\cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot \sqrt[3]{\cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le -1.91205330543206 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 9.798541471285326 \cdot 10^{-63}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 3.102132787630003 \cdot 10^{-33}:\\ \;\;\;\;e^{\log \left(\log \left(e^{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (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)))))