Average Error: 0.8 → 0.5
Time: 3.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \le -0.682188176858241069:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)\right)\\ \end{array}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \le -0.682188176858241069:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)\right)\\

\end{array}
double f(double re, double im) {
        double r27468 = im;
        double r27469 = re;
        double r27470 = atan2(r27468, r27469);
        double r27471 = 10.0;
        double r27472 = log(r27471);
        double r27473 = r27470 / r27472;
        return r27473;
}
↓
double f(double re, double im) {
        double r27474 = im;
        double r27475 = re;
        double r27476 = atan2(r27474, r27475);
        double r27477 = 10.0;
        double r27478 = log(r27477);
        double r27479 = r27476 / r27478;
        double r27480 = -0.6821881768582411;
        bool r27481 = r27479 <= r27480;
        double r27482 = 3.0;
        double r27483 = pow(r27479, r27482);
        double r27484 = cbrt(r27483);
        double r27485 = log1p(r27479);
        double r27486 = expm1(r27485);
        double r27487 = r27481 ? r27484 : r27486;
        return r27487;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (atan2 im re) (log 10.0)) < -0.6821881768582411

    1. Initial program 1.0

      \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube1.6

      \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    4. Applied add-cbrt-cube1.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \tan^{-1}_* \frac{im}{re}}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

    5. Applied cbrt-undiv0.7

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\tan^{-1}_* \frac{im}{re} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \tan^{-1}_* \frac{im}{re}}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

    6. Simplified0.0

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}}\]

    if -0.6821881768582411 < (/ (atan2 im re) (log 10.0))

    1. Initial program 0.8

      \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
    2. Using strategy rm

    3. Applied expm1-log1p-u0.8

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan^{-1}_* \frac{im}{re}}{\log 10} \le -0.682188176858241069:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))