Average Error: 32.4 → 3.7
Time: 22.4s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]
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)
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24023 = x_re;
        double r24024 = r24023 * r24023;
        double r24025 = x_im;
        double r24026 = r24025 * r24025;
        double r24027 = r24024 + r24026;
        double r24028 = sqrt(r24027);
        double r24029 = log(r24028);
        double r24030 = y_re;
        double r24031 = r24029 * r24030;
        double r24032 = atan2(r24025, r24023);
        double r24033 = y_im;
        double r24034 = r24032 * r24033;
        double r24035 = r24031 - r24034;
        double r24036 = exp(r24035);
        double r24037 = r24029 * r24033;
        double r24038 = r24032 * r24030;
        double r24039 = r24037 + r24038;
        double r24040 = sin(r24039);
        double r24041 = r24036 * r24040;
        return r24041;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24042 = x_re;
        double r24043 = x_im;
        double r24044 = hypot(r24042, r24043);
        double r24045 = log(r24044);
        double r24046 = y_im;
        double r24047 = y_re;
        double r24048 = cbrt(r24047);
        double r24049 = r24048 * r24048;
        double r24050 = atan2(r24043, r24042);
        double r24051 = r24049 * r24050;
        double r24052 = r24051 * r24048;
        double r24053 = fma(r24045, r24046, r24052);
        double r24054 = sin(r24053);
        double r24055 = expm1(r24054);
        double r24056 = log1p(r24055);
        double r24057 = r24045 * r24047;
        double r24058 = r24050 * r24046;
        double r24059 = r24057 - r24058;
        double r24060 = exp(r24059);
        double r24061 = r24056 * r24060;
        return r24061;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 32.4

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

  2. Simplified9.0

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

  4. Applied add-exp-log9.0

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

  5. Applied pow-exp9.0

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

  6. Applied div-exp3.5

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

  8. Applied log1p-expm1-u3.5

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)} \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt3.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{y.re}\right)}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]

  11. Applied associate-*r*3.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right)\right) \cdot \sqrt[3]{y.re}}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]

  12. Simplified3.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{\left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt[3]{y.re}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]

  13. Final simplification3.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \left(\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.re}\right)\right)\right)\right) \cdot e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]

Reproduce

herbie shell --seed 2019212 +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)))))