Average Error: 33.1 → 6.0
Time: 18.0s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -2.3684273757276652 \cdot 10^{74} \lor \neg \left(y.re \le 20.400440132882025\right):\\ \;\;\;\;e^{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}\\ \end{array}\]
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}\;y.re \le -2.3684273757276652 \cdot 10^{74} \lor \neg \left(y.re \le 20.400440132882025\right):\\
\;\;\;\;e^{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26444 = x_re;
        double r26445 = r26444 * r26444;
        double r26446 = x_im;
        double r26447 = r26446 * r26446;
        double r26448 = r26445 + r26447;
        double r26449 = sqrt(r26448);
        double r26450 = log(r26449);
        double r26451 = y_re;
        double r26452 = r26450 * r26451;
        double r26453 = atan2(r26446, r26444);
        double r26454 = y_im;
        double r26455 = r26453 * r26454;
        double r26456 = r26452 - r26455;
        double r26457 = exp(r26456);
        double r26458 = r26450 * r26454;
        double r26459 = r26453 * r26451;
        double r26460 = r26458 + r26459;
        double r26461 = sin(r26460);
        double r26462 = r26457 * r26461;
        return r26462;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26463 = y_re;
        double r26464 = -2.368427375727665e+74;
        bool r26465 = r26463 <= r26464;
        double r26466 = 20.400440132882025;
        bool r26467 = r26463 <= r26466;
        double r26468 = !r26467;
        bool r26469 = r26465 || r26468;
        double r26470 = x_re;
        double r26471 = x_im;
        double r26472 = hypot(r26470, r26471);
        double r26473 = pow(r26472, r26463);
        double r26474 = log(r26472);
        double r26475 = y_im;
        double r26476 = atan2(r26471, r26470);
        double r26477 = r26476 * r26463;
        double r26478 = fma(r26474, r26475, r26477);
        double r26479 = sin(r26478);
        double r26480 = r26473 * r26479;
        double r26481 = log(r26480);
        double r26482 = r26476 * r26475;
        double r26483 = r26481 - r26482;
        double r26484 = exp(r26483);
        double r26485 = exp(r26482);
        double r26486 = r26485 / r26479;
        double r26487 = r26473 / r26486;
        double r26488 = r26469 ? r26484 : r26487;
        return r26488;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 2 regimes

  2. if y.re < -2.368427375727665e+74 or 20.400440132882025 < y.re

    1. Initial program 29.8

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

    2. Simplified15.3

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\]
    3. Using strategy rm

    4. Applied add-exp-log39.5

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

    5. Applied add-exp-log39.5

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

    6. Applied pow-exp39.5

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

    7. Applied prod-exp39.5

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

    8. Applied div-exp32.0

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

    9. Simplified7.1

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

    if -2.368427375727665e+74 < y.re < 20.400440132882025

    1. Initial program 34.7

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

    2. Simplified5.4

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\]
    3. Using strategy rm

    4. Applied associate-/l*5.4

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.3684273757276652 \cdot 10^{74} \lor \neg \left(y.re \le 20.400440132882025\right):\\ \;\;\;\;e^{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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)))))