Average Error: 33.5 → 4.4
Time: 7.3s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[{\left(e^{\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}^{\left(\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)}\]
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)
↓
{\left(e^{\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \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}} \cdot \sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}^{\left(\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \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 r14294 = x_re;
        double r14295 = r14294 * r14294;
        double r14296 = x_im;
        double r14297 = r14296 * r14296;
        double r14298 = r14295 + r14297;
        double r14299 = sqrt(r14298);
        double r14300 = log(r14299);
        double r14301 = y_re;
        double r14302 = r14300 * r14301;
        double r14303 = atan2(r14296, r14294);
        double r14304 = y_im;
        double r14305 = r14303 * r14304;
        double r14306 = r14302 - r14305;
        double r14307 = exp(r14306);
        double r14308 = r14300 * r14304;
        double r14309 = r14303 * r14301;
        double r14310 = r14308 + r14309;
        double r14311 = cos(r14310);
        double r14312 = r14307 * r14311;
        return r14312;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14313 = x_re;
        double r14314 = x_im;
        double r14315 = hypot(r14313, r14314);
        double r14316 = 0.5;
        double r14317 = pow(r14315, r14316);
        double r14318 = r14317 * r14317;
        double r14319 = log(r14318);
        double r14320 = y_re;
        double r14321 = r14319 * r14320;
        double r14322 = atan2(r14314, r14313);
        double r14323 = y_im;
        double r14324 = r14322 * r14323;
        double r14325 = cbrt(r14324);
        double r14326 = r14325 * r14325;
        double r14327 = r14326 * r14325;
        double r14328 = r14321 - r14327;
        double r14329 = cbrt(r14328);
        double r14330 = r14321 - r14324;
        double r14331 = cbrt(r14330);
        double r14332 = r14329 * r14331;
        double r14333 = exp(r14332);
        double r14334 = pow(r14333, r14331);
        return r14334;
}

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. Initial program 33.5

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

  2. Taylor expanded around 0 19.8

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

  4. Applied add-sqr-sqrt19.8

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

  5. Applied sqrt-prod19.8

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

  6. Simplified19.8

    \[\leadsto e^{\log \left(\color{blue}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{\sqrt{x.re \cdot x.re + x.im \cdot x.im}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]

  7. Simplified4.4

    \[\leadsto e^{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\]
  8. Using strategy rm

  9. Applied add-cube-cbrt4.4

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

  10. Applied exp-prod4.4

    \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}^{\left(\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)}} \cdot 1\]
  11. Using strategy rm

  12. Applied add-cube-cbrt4.4

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

  13. Final simplification4.4

    \[\leadsto {\left(e^{\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)}^{\left(\sqrt[3]{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\frac{1}{2}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(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)))))