Average Error: 33.7 → 23.2
Time: 25.9s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -4.484553050797035804670682450978201199884 \cdot 10^{64}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{elif}\;x.re \le -1.197356869030033423932962593195270915285 \cdot 10^{-68}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\left(\sqrt[3]{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \sqrt[3]{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot \sqrt[3]{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)\\ \mathbf{elif}\;x.re \le -5.879793377467557445546507040277763544615 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]
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 \sin \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 -4.484553050797035804670682450978201199884 \cdot 10^{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 \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\

\mathbf{elif}\;x.re \le -1.197356869030033423932962593195270915285 \cdot 10^{-68}:\\
\;\;\;\;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 \left(\left(\sqrt[3]{\sin \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]{\sin \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]{\sin \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)\\

\mathbf{elif}\;x.re \le -5.879793377467557445546507040277763544615 \cdot 10^{-309}:\\
\;\;\;\;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 \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26071 = x_re;
        double r26072 = r26071 * r26071;
        double r26073 = x_im;
        double r26074 = r26073 * r26073;
        double r26075 = r26072 + r26074;
        double r26076 = sqrt(r26075);
        double r26077 = log(r26076);
        double r26078 = y_re;
        double r26079 = r26077 * r26078;
        double r26080 = atan2(r26073, r26071);
        double r26081 = y_im;
        double r26082 = r26080 * r26081;
        double r26083 = r26079 - r26082;
        double r26084 = exp(r26083);
        double r26085 = r26077 * r26081;
        double r26086 = r26080 * r26078;
        double r26087 = r26085 + r26086;
        double r26088 = sin(r26087);
        double r26089 = r26084 * r26088;
        return r26089;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26090 = x_re;
        double r26091 = -4.484553050797036e+64;
        bool r26092 = r26090 <= r26091;
        double r26093 = r26090 * r26090;
        double r26094 = x_im;
        double r26095 = r26094 * r26094;
        double r26096 = r26093 + r26095;
        double r26097 = sqrt(r26096);
        double r26098 = log(r26097);
        double r26099 = y_re;
        double r26100 = r26098 * r26099;
        double r26101 = atan2(r26094, r26090);
        double r26102 = y_im;
        double r26103 = r26101 * r26102;
        double r26104 = r26100 - r26103;
        double r26105 = exp(r26104);
        double r26106 = r26101 * r26099;
        double r26107 = -1.0;
        double r26108 = r26107 / r26090;
        double r26109 = log(r26108);
        double r26110 = r26102 * r26109;
        double r26111 = r26106 - r26110;
        double r26112 = sin(r26111);
        double r26113 = r26105 * r26112;
        double r26114 = -1.1973568690300334e-68;
        bool r26115 = r26090 <= r26114;
        double r26116 = r26098 * r26102;
        double r26117 = r26116 + r26106;
        double r26118 = sin(r26117);
        double r26119 = cbrt(r26118);
        double r26120 = r26119 * r26119;
        double r26121 = r26120 * r26119;
        double r26122 = r26105 * r26121;
        double r26123 = -5.879793377467557e-309;
        bool r26124 = r26090 <= r26123;
        double r26125 = log(r26090);
        double r26126 = r26125 * r26102;
        double r26127 = r26126 + r26106;
        double r26128 = sin(r26127);
        double r26129 = r26105 * r26128;
        double r26130 = r26124 ? r26113 : r26129;
        double r26131 = r26115 ? r26122 : r26130;
        double r26132 = r26092 ? r26113 : r26131;
        return r26132;
}

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 < -4.484553050797036e+64 or -1.1973568690300334e-68 < x.re < -5.879793377467557e-309

    1. Initial program 36.2

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

    2. Taylor expanded around -inf 22.5

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

    if -4.484553050797036e+64 < x.re < -1.1973568690300334e-68

    1. Initial program 17.5

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

    3. Applied add-cube-cbrt17.9

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

    if -5.879793377467557e-309 < x.re

    1. Initial program 34.9

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

    2. Taylor expanded around inf 24.6

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

  4. Final simplification23.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -4.484553050797035804670682450978201199884 \cdot 10^{64}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{elif}\;x.re \le -1.197356869030033423932962593195270915285 \cdot 10^{-68}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\left(\sqrt[3]{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot \sqrt[3]{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right) \cdot \sqrt[3]{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)\\ \mathbf{elif}\;x.re \le -5.879793377467557445546507040277763544615 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log x.re \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))