Average Error: 33.4 → 3.9
Time: 8.9s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right)\right) \cdot \sqrt[3]{y.re}\right)\]
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)
↓
e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \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)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r16142 = x_re;
        double r16143 = r16142 * r16142;
        double r16144 = x_im;
        double r16145 = r16144 * r16144;
        double r16146 = r16143 + r16145;
        double r16147 = sqrt(r16146);
        double r16148 = log(r16147);
        double r16149 = y_re;
        double r16150 = r16148 * r16149;
        double r16151 = atan2(r16144, r16142);
        double r16152 = y_im;
        double r16153 = r16151 * r16152;
        double r16154 = r16150 - r16153;
        double r16155 = exp(r16154);
        double r16156 = r16148 * r16152;
        double r16157 = r16151 * r16149;
        double r16158 = r16156 + r16157;
        double r16159 = sin(r16158);
        double r16160 = r16155 * r16159;
        return r16160;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r16161 = 1.0;
        double r16162 = x_re;
        double r16163 = x_im;
        double r16164 = hypot(r16162, r16163);
        double r16165 = r16161 * r16164;
        double r16166 = log(r16165);
        double r16167 = y_re;
        double r16168 = r16166 * r16167;
        double r16169 = atan2(r16163, r16162);
        double r16170 = y_im;
        double r16171 = r16169 * r16170;
        double r16172 = r16168 - r16171;
        double r16173 = exp(r16172);
        double r16174 = r16166 * r16170;
        double r16175 = cbrt(r16167);
        double r16176 = r16175 * r16175;
        double r16177 = r16169 * r16176;
        double r16178 = r16177 * r16175;
        double r16179 = r16174 + r16178;
        double r16180 = sin(r16179);
        double r16181 = r16173 * r16180;
        return r16181;
}

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 33.4

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

  3. Applied *-un-lft-identity33.4

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

  4. Applied sqrt-prod33.4

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

  5. Simplified33.4

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

  6. Simplified19.4

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

  8. Applied *-un-lft-identity19.4

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

  9. Applied sqrt-prod19.4

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

  10. Simplified19.4

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

  11. Simplified3.7

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

  13. Applied add-cube-cbrt3.9

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

  14. Applied associate-*r*3.9

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

  15. Final simplification3.9

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

Reproduce

herbie shell --seed 2019353 +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)))))