Average Error: 33.0 → 11.3
Time: 24.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -2.194259725539570386691584397856890606327 \cdot 10^{-310}:\\ \;\;\;\;\frac{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\frac{-1}{x.re}\right)}^{y.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]
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 -2.194259725539570386691584397856890606327 \cdot 10^{-310}:\\
\;\;\;\;\frac{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\frac{-1}{x.re}\right)}^{y.re}}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24110 = x_re;
        double r24111 = r24110 * r24110;
        double r24112 = x_im;
        double r24113 = r24112 * r24112;
        double r24114 = r24111 + r24113;
        double r24115 = sqrt(r24114);
        double r24116 = log(r24115);
        double r24117 = y_re;
        double r24118 = r24116 * r24117;
        double r24119 = atan2(r24112, r24110);
        double r24120 = y_im;
        double r24121 = r24119 * r24120;
        double r24122 = r24118 - r24121;
        double r24123 = exp(r24122);
        double r24124 = r24116 * r24120;
        double r24125 = r24119 * r24117;
        double r24126 = r24124 + r24125;
        double r24127 = cos(r24126);
        double r24128 = r24123 * r24127;
        return r24128;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24129 = x_re;
        double r24130 = -2.19425972553957e-310;
        bool r24131 = r24129 <= r24130;
        double r24132 = x_im;
        double r24133 = atan2(r24132, r24129);
        double r24134 = y_im;
        double r24135 = r24133 * r24134;
        double r24136 = -r24135;
        double r24137 = exp(r24136);
        double r24138 = -1.0;
        double r24139 = r24138 / r24129;
        double r24140 = y_re;
        double r24141 = pow(r24139, r24140);
        double r24142 = r24137 / r24141;
        double r24143 = log(r24129);
        double r24144 = r24143 * r24140;
        double r24145 = r24144 - r24135;
        double r24146 = exp(r24145);
        double r24147 = r24131 ? r24142 : r24146;
        return r24147;
}

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 < -2.19425972553957e-310

    1. Initial program 31.7

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

    2. Taylor expanded around 0 17.5

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

    3. Taylor expanded around -inf 5.7

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

    4. Simplified11.2

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

    if -2.19425972553957e-310 < x.re

    1. Initial program 34.2

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

    2. Taylor expanded around 0 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 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -2.194259725539570386691584397856890606327 \cdot 10^{-310}:\\ \;\;\;\;\frac{e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\frac{-1}{x.re}\right)}^{y.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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)))))