Average Error: 32.4 → 9.5
Time: 32.8s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.8717104734621963 \cdot 10^{-281}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 1.601505551461043 \cdot 10^{-26}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
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 -1.8717104734621963 \cdot 10^{-281}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\

\mathbf{elif}\;x.re \le 1.601505551461043 \cdot 10^{-26}:\\
\;\;\;\;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{else}:\\
\;\;\;\;e^{\log x.re \cdot y.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 r1439961 = x_re;
        double r1439962 = r1439961 * r1439961;
        double r1439963 = x_im;
        double r1439964 = r1439963 * r1439963;
        double r1439965 = r1439962 + r1439964;
        double r1439966 = sqrt(r1439965);
        double r1439967 = log(r1439966);
        double r1439968 = y_re;
        double r1439969 = r1439967 * r1439968;
        double r1439970 = atan2(r1439963, r1439961);
        double r1439971 = y_im;
        double r1439972 = r1439970 * r1439971;
        double r1439973 = r1439969 - r1439972;
        double r1439974 = exp(r1439973);
        double r1439975 = r1439967 * r1439971;
        double r1439976 = r1439970 * r1439968;
        double r1439977 = r1439975 + r1439976;
        double r1439978 = cos(r1439977);
        double r1439979 = r1439974 * r1439978;
        return r1439979;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1439980 = x_re;
        double r1439981 = -1.8717104734621963e-281;
        bool r1439982 = r1439980 <= r1439981;
        double r1439983 = -r1439980;
        double r1439984 = log(r1439983);
        double r1439985 = y_re;
        double r1439986 = r1439984 * r1439985;
        double r1439987 = y_im;
        double r1439988 = x_im;
        double r1439989 = atan2(r1439988, r1439980);
        double r1439990 = r1439987 * r1439989;
        double r1439991 = r1439986 - r1439990;
        double r1439992 = exp(r1439991);
        double r1439993 = 1.601505551461043e-26;
        bool r1439994 = r1439980 <= r1439993;
        double r1439995 = r1439980 * r1439980;
        double r1439996 = r1439988 * r1439988;
        double r1439997 = r1439995 + r1439996;
        double r1439998 = sqrt(r1439997);
        double r1439999 = log(r1439998);
        double r1440000 = r1439985 * r1439999;
        double r1440001 = r1440000 - r1439990;
        double r1440002 = exp(r1440001);
        double r1440003 = log(r1439980);
        double r1440004 = r1440003 * r1439985;
        double r1440005 = r1440004 - r1439990;
        double r1440006 = exp(r1440005);
        double r1440007 = r1439994 ? r1440002 : r1440006;
        double r1440008 = r1439982 ? r1439992 : r1440007;
        return r1440008;
}

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 < -1.8717104734621963e-281

    1. Initial program 31.0

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

    2. Taylor expanded around 0 17.0

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

    3. Taylor expanded around -inf 5.4

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

    4. Simplified5.4

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

    if -1.8717104734621963e-281 < x.re < 1.601505551461043e-26

    1. Initial program 24.9

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

    2. Taylor expanded around 0 14.6

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

    if 1.601505551461043e-26 < x.re

    1. Initial program 41.1

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

    2. Taylor expanded around 0 25.9

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

    3. Taylor expanded around inf 11.7

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.8717104734621963 \cdot 10^{-281}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \le 1.601505551461043 \cdot 10^{-26}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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)))))