powComplex, imaginary part

Average Error: 33.0 → 17.6

Time: 30.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x.re \le -4.000236435680563415231517513561000379886 \cdot 10^{-43}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|x.im\right|\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \le 1.405848435471205693683660544267189981616 \cdot 10^{-293} \lor \neg \left(x.re \le 2.698797435107739892751313988383445168574 \cdot 10^{-254}\right) \land x.re \le 6.089823661559195862886324010710708065133 \cdot 10^{-145}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|x.im\right|\right) \cdot y.im + \left(\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}\right) \cdot \sqrt[3]{\sqrt[3]{y.re}}\right) \cdot \sqrt[3]{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|x.im\right|\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r23605 = x_re;
        double r23606 = r23605 * r23605;
        double r23607 = x_im;
        double r23608 = r23607 * r23607;
        double r23609 = r23606 + r23608;
        double r23610 = sqrt(r23609);
        double r23611 = log(r23610);
        double r23612 = y_re;
        double r23613 = r23611 * r23612;
        double r23614 = atan2(r23607, r23605);
        double r23615 = y_im;
        double r23616 = r23614 * r23615;
        double r23617 = r23613 - r23616;
        double r23618 = exp(r23617);
        double r23619 = r23611 * r23615;
        double r23620 = r23614 * r23612;
        double r23621 = r23619 + r23620;
        double r23622 = sin(r23621);
        double r23623 = r23618 * r23622;
        return r23623;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r23624 = x_re;
        double r23625 = -4.0002364356805634e-43;
        bool r23626 = r23624 <= r23625;
        double r23627 = -r23624;
        double r23628 = log(r23627);
        double r23629 = y_re;
        double r23630 = r23628 * r23629;
        double r23631 = x_im;
        double r23632 = atan2(r23631, r23624);
        double r23633 = y_im;
        double r23634 = r23632 * r23633;
        double r23635 = r23630 - r23634;
        double r23636 = exp(r23635);
        double r23637 = fabs(r23631);
        double r23638 = log(r23637);
        double r23639 = r23638 * r23633;
        double r23640 = r23632 * r23629;
        double r23641 = r23639 + r23640;
        double r23642 = sin(r23641);
        double r23643 = r23636 * r23642;
        double r23644 = 1.4058484354712057e-293;
        bool r23645 = r23624 <= r23644;
        double r23646 = 2.69879743510774e-254;
        bool r23647 = r23624 <= r23646;
        double r23648 = !r23647;
        double r23649 = 6.089823661559196e-145;
        bool r23650 = r23624 <= r23649;
        bool r23651 = r23648 && r23650;
        bool r23652 = r23645 || r23651;
        double r23653 = r23624 * r23624;
        double r23654 = r23631 * r23631;
        double r23655 = r23653 + r23654;
        double r23656 = sqrt(r23655);
        double r23657 = log(r23656);
        double r23658 = r23657 * r23629;
        double r23659 = r23658 - r23634;
        double r23660 = exp(r23659);
        double r23661 = cbrt(r23629);
        double r23662 = r23632 * r23661;
        double r23663 = r23661 * r23661;
        double r23664 = cbrt(r23663);
        double r23665 = r23662 * r23664;
        double r23666 = cbrt(r23661);
        double r23667 = r23665 * r23666;
        double r23668 = r23667 * r23661;
        double r23669 = r23639 + r23668;
        double r23670 = sin(r23669);
        double r23671 = r23660 * r23670;
        double r23672 = log(r23624);
        double r23673 = r23672 * r23629;
        double r23674 = r23673 - r23634;
        double r23675 = exp(r23674);
        double r23676 = r23675 * r23642;
        double r23677 = r23652 ? r23671 : r23676;
        double r23678 = r23626 ? r23643 : r23677;
        return r23678;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 3 regimes

2. if x.re < -4.0002364356805634e-43

   1. Initial program 38.4

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

   3. Applied add-sqr-sqrt 38.4

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

   4. Applied rem-sqrt-square 38.4

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

   5. Taylor expanded around 0 25.1

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

   6. Taylor expanded around -inf 10.9

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

   7. Simplified 10.9

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

   if -4.0002364356805634e-43 < x.re < 1.4058484354712057e-293 or 2.69879743510774e-254 < x.re < 6.089823661559196e-145

   1. Initial program 25.1

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

   3. Applied add-sqr-sqrt 25.1

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

   4. Applied rem-sqrt-square 25.1

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

   5. Taylor expanded around 0 15.3

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

   7. Applied add-cube-cbrt 15.5

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

   8. Applied associate-*r* 15.5

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

   9. Simplified 15.5

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

   11. Applied add-cube-cbrt 15.5

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

   12. Applied cbrt-prod 15.5

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

   13. Applied associate-*r* 15.5

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

   if 1.4058484354712057e-293 < x.re < 2.69879743510774e-254 or 6.089823661559196e-145 < x.re

   1. Initial program 35.5

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

   3. Applied add-sqr-sqrt 35.5

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

   4. Applied rem-sqrt-square 35.5

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

   5. Taylor expanded around 0 29.0

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

   6. Taylor expanded around inf 23.5

      \[\leadsto e^{\log \color{blue}{x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|x.im\right|\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -4.000236435680563415231517513561000379886 \cdot 10^{-43}:\\ \;\;\;\;e^{\log \left(-x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|x.im\right|\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \le 1.405848435471205693683660544267189981616 \cdot 10^{-293} \lor \neg \left(x.re \le 2.698797435107739892751313988383445168574 \cdot 10^{-254}\right) \land x.re \le 6.089823661559195862886324010710708065133 \cdot 10^{-145}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|x.im\right|\right) \cdot y.im + \left(\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}\right) \cdot \sqrt[3]{\sqrt[3]{y.re}}\right) \cdot \sqrt[3]{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|x.im\right|\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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)))))