Average Error: 33.0 → 3.5
Time: 31.5s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\log \left(e^{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right) \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}\]
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)
↓
\log \left(e^{\cos \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) \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) \cdot \sqrt[3]{\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2917132 = x_re;
        double r2917133 = r2917132 * r2917132;
        double r2917134 = x_im;
        double r2917135 = r2917134 * r2917134;
        double r2917136 = r2917133 + r2917135;
        double r2917137 = sqrt(r2917136);
        double r2917138 = log(r2917137);
        double r2917139 = y_re;
        double r2917140 = r2917138 * r2917139;
        double r2917141 = atan2(r2917134, r2917132);
        double r2917142 = y_im;
        double r2917143 = r2917141 * r2917142;
        double r2917144 = r2917140 - r2917143;
        double r2917145 = exp(r2917144);
        double r2917146 = r2917138 * r2917142;
        double r2917147 = r2917141 * r2917139;
        double r2917148 = r2917146 + r2917147;
        double r2917149 = cos(r2917148);
        double r2917150 = r2917145 * r2917149;
        return r2917150;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2917151 = x_re;
        double r2917152 = x_im;
        double r2917153 = hypot(r2917151, r2917152);
        double r2917154 = log(r2917153);
        double r2917155 = y_im;
        double r2917156 = atan2(r2917152, r2917151);
        double r2917157 = y_re;
        double r2917158 = r2917156 * r2917157;
        double r2917159 = fma(r2917154, r2917155, r2917158);
        double r2917160 = cos(r2917159);
        double r2917161 = exp(r2917160);
        double r2917162 = log(r2917161);
        double r2917163 = r2917157 * r2917154;
        double r2917164 = r2917156 * r2917155;
        double r2917165 = cbrt(r2917164);
        double r2917166 = cbrt(r2917165);
        double r2917167 = r2917166 * r2917166;
        double r2917168 = r2917167 * r2917166;
        double r2917169 = r2917165 * r2917168;
        double r2917170 = r2917169 * r2917168;
        double r2917171 = r2917163 - r2917170;
        double r2917172 = exp(r2917171);
        double r2917173 = r2917162 * r2917172;
        return r2917173;
}

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 33.0

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

  2. Simplified3.5

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

  4. Applied add-cube-cbrt3.5

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

  6. Applied add-cube-cbrt3.5

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

  8. Applied add-cube-cbrt3.5

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

  10. Applied add-log-exp3.5

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

  11. Final simplification3.5

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (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)))))