Average Error: 33.5 → 9.1
Time: 32.2s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -7.316830147106013741421501992963017986241 \cdot 10^{-306}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]
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 -7.316830147106013741421501992963017986241 \cdot 10^{-306}:\\
\;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - 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 r1337954 = x_re;
        double r1337955 = r1337954 * r1337954;
        double r1337956 = x_im;
        double r1337957 = r1337956 * r1337956;
        double r1337958 = r1337955 + r1337957;
        double r1337959 = sqrt(r1337958);
        double r1337960 = log(r1337959);
        double r1337961 = y_re;
        double r1337962 = r1337960 * r1337961;
        double r1337963 = atan2(r1337956, r1337954);
        double r1337964 = y_im;
        double r1337965 = r1337963 * r1337964;
        double r1337966 = r1337962 - r1337965;
        double r1337967 = exp(r1337966);
        double r1337968 = r1337960 * r1337964;
        double r1337969 = r1337963 * r1337961;
        double r1337970 = r1337968 + r1337969;
        double r1337971 = cos(r1337970);
        double r1337972 = r1337967 * r1337971;
        return r1337972;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1337973 = x_re;
        double r1337974 = -7.316830147106014e-306;
        bool r1337975 = r1337973 <= r1337974;
        double r1337976 = -r1337973;
        double r1337977 = log(r1337976);
        double r1337978 = y_re;
        double r1337979 = r1337977 * r1337978;
        double r1337980 = y_im;
        double r1337981 = x_im;
        double r1337982 = atan2(r1337981, r1337973);
        double r1337983 = r1337980 * r1337982;
        double r1337984 = r1337979 - r1337983;
        double r1337985 = exp(r1337984);
        double r1337986 = log(r1337973);
        double r1337987 = r1337986 * r1337978;
        double r1337988 = r1337987 - r1337983;
        double r1337989 = exp(r1337988);
        double r1337990 = r1337975 ? r1337985 : r1337989;
        return r1337990;
}

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 x.re < -7.316830147106014e-306

    1. Initial program 31.5

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

    2. Taylor expanded around 0 17.1

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

    3. Taylor expanded around -inf 5.8

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

    4. Simplified5.8

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

    if -7.316830147106014e-306 < x.re

    1. Initial program 35.3

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

    2. Taylor expanded around 0 21.9

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

    3. Taylor expanded around inf 12.2

      \[\leadsto e^{\log \color{blue}{x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -7.316830147106013741421501992963017986241 \cdot 10^{-306}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))))