Average Error: 33.0 → 9.6
Time: 8.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -6.451247643666555047151411443244956267451 \cdot 10^{153}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -3.011667906396995080273534682604245406099 \cdot 10^{-159}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -1.197440871000797137827287021764707966827 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]
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 -6.451247643666555047151411443244956267451 \cdot 10^{153}:\\
\;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\

\mathbf{elif}\;x.re \le -3.011667906396995080273534682604245406099 \cdot 10^{-159}:\\
\;\;\;\;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 1\\

\mathbf{elif}\;x.re \le -1.197440871000797137827287021764707966827 \cdot 10^{-310}:\\
\;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r16937 = x_re;
        double r16938 = r16937 * r16937;
        double r16939 = x_im;
        double r16940 = r16939 * r16939;
        double r16941 = r16938 + r16940;
        double r16942 = sqrt(r16941);
        double r16943 = log(r16942);
        double r16944 = y_re;
        double r16945 = r16943 * r16944;
        double r16946 = atan2(r16939, r16937);
        double r16947 = y_im;
        double r16948 = r16946 * r16947;
        double r16949 = r16945 - r16948;
        double r16950 = exp(r16949);
        double r16951 = r16943 * r16947;
        double r16952 = r16946 * r16944;
        double r16953 = r16951 + r16952;
        double r16954 = cos(r16953);
        double r16955 = r16950 * r16954;
        return r16955;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r16956 = x_re;
        double r16957 = -6.451247643666555e+153;
        bool r16958 = r16956 <= r16957;
        double r16959 = -1.0;
        double r16960 = r16959 * r16956;
        double r16961 = log(r16960);
        double r16962 = y_re;
        double r16963 = r16961 * r16962;
        double r16964 = x_im;
        double r16965 = atan2(r16964, r16956);
        double r16966 = y_im;
        double r16967 = r16965 * r16966;
        double r16968 = r16963 - r16967;
        double r16969 = exp(r16968);
        double r16970 = 1.0;
        double r16971 = r16969 * r16970;
        double r16972 = -3.011667906396995e-159;
        bool r16973 = r16956 <= r16972;
        double r16974 = r16956 * r16956;
        double r16975 = r16964 * r16964;
        double r16976 = r16974 + r16975;
        double r16977 = sqrt(r16976);
        double r16978 = log(r16977);
        double r16979 = r16978 * r16962;
        double r16980 = r16979 - r16967;
        double r16981 = exp(r16980);
        double r16982 = r16981 * r16970;
        double r16983 = -1.1974408710008e-310;
        bool r16984 = r16956 <= r16983;
        double r16985 = log(r16956);
        double r16986 = r16985 * r16962;
        double r16987 = r16986 - r16967;
        double r16988 = exp(r16987);
        double r16989 = r16988 * r16970;
        double r16990 = r16984 ? r16971 : r16989;
        double r16991 = r16973 ? r16982 : r16990;
        double r16992 = r16958 ? r16971 : r16991;
        return r16992;
}

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 < -6.451247643666555e+153 or -3.011667906396995e-159 < x.re < -1.1974408710008e-310

    1. Initial program 48.1

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

    2. Taylor expanded around 0 27.0

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

    3. Taylor expanded around -inf 5.3

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

    if -6.451247643666555e+153 < x.re < -3.011667906396995e-159

    1. Initial program 16.6

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

    2. Taylor expanded around 0 10.2

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

    if -1.1974408710008e-310 < x.re

    1. Initial program 33.9

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

    2. Taylor expanded around 0 21.3

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

    3. Taylor expanded around inf 11.4

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -6.451247643666555047151411443244956267451 \cdot 10^{153}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -3.011667906396995080273534682604245406099 \cdot 10^{-159}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{elif}\;x.re \le -1.197440871000797137827287021764707966827 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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)))))