Average Error: 33.3 → 3.4
Time: 8.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\]
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(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\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)\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14614 = x_re;
        double r14615 = r14614 * r14614;
        double r14616 = x_im;
        double r14617 = r14616 * r14616;
        double r14618 = r14615 + r14617;
        double r14619 = sqrt(r14618);
        double r14620 = log(r14619);
        double r14621 = y_re;
        double r14622 = r14620 * r14621;
        double r14623 = atan2(r14616, r14614);
        double r14624 = y_im;
        double r14625 = r14623 * r14624;
        double r14626 = r14622 - r14625;
        double r14627 = exp(r14626);
        double r14628 = r14620 * r14624;
        double r14629 = r14623 * r14621;
        double r14630 = r14628 + r14629;
        double r14631 = sin(r14630);
        double r14632 = r14627 * r14631;
        return r14632;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r14633 = x_re;
        double r14634 = x_im;
        double r14635 = hypot(r14633, r14634);
        double r14636 = log(r14635);
        double r14637 = exp(r14636);
        double r14638 = log(r14637);
        double r14639 = y_re;
        double r14640 = r14638 * r14639;
        double r14641 = atan2(r14634, r14633);
        double r14642 = y_im;
        double r14643 = r14641 * r14642;
        double r14644 = r14640 - r14643;
        double r14645 = exp(r14644);
        double r14646 = 1.0;
        double r14647 = r14646 * r14635;
        double r14648 = log(r14647);
        double r14649 = r14648 * r14642;
        double r14650 = r14641 * r14639;
        double r14651 = r14649 + r14650;
        double r14652 = sin(r14651);
        double r14653 = expm1(r14652);
        double r14654 = log1p(r14653);
        double r14655 = r14645 * r14654;
        return r14655;
}

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 33.3

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

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

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

  4. Applied sqrt-prod33.3

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

  5. Simplified33.3

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

  6. Simplified19.6

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

  8. Applied add-exp-log19.6

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

  9. Simplified3.4

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

  11. Applied log1p-expm1-u3.4

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

  12. Final simplification3.4

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

Reproduce

herbie shell --seed 2020049 +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)))))