Average Error: 32.9 → 22.1
Time: 33.3s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -1.2685618014437514 \cdot 10^{-33}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le -4.748647839754185 \cdot 10^{-157}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \left(\log \left(\sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) + \log \left(\sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)\right)}\\ \mathbf{elif}\;x.re \le -1.2253344719312 \cdot 10^{-312}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\\ \end{array}\]
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 -1.2685618014437514 \cdot 10^{-33}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\

\mathbf{elif}\;x.re \le -4.748647839754185 \cdot 10^{-157}:\\
\;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \left(\log \left(\sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) + \log \left(\sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)\right)}\\

\mathbf{elif}\;x.re \le -1.2253344719312 \cdot 10^{-312}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1392479 = x_re;
        double r1392480 = r1392479 * r1392479;
        double r1392481 = x_im;
        double r1392482 = r1392481 * r1392481;
        double r1392483 = r1392480 + r1392482;
        double r1392484 = sqrt(r1392483);
        double r1392485 = log(r1392484);
        double r1392486 = y_re;
        double r1392487 = r1392485 * r1392486;
        double r1392488 = atan2(r1392481, r1392479);
        double r1392489 = y_im;
        double r1392490 = r1392488 * r1392489;
        double r1392491 = r1392487 - r1392490;
        double r1392492 = exp(r1392491);
        double r1392493 = r1392485 * r1392489;
        double r1392494 = r1392488 * r1392486;
        double r1392495 = r1392493 + r1392494;
        double r1392496 = sin(r1392495);
        double r1392497 = r1392492 * r1392496;
        return r1392497;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1392498 = x_re;
        double r1392499 = -1.2685618014437514e-33;
        bool r1392500 = r1392498 <= r1392499;
        double r1392501 = y_re;
        double r1392502 = r1392498 * r1392498;
        double r1392503 = x_im;
        double r1392504 = r1392503 * r1392503;
        double r1392505 = r1392502 + r1392504;
        double r1392506 = sqrt(r1392505);
        double r1392507 = log(r1392506);
        double r1392508 = r1392501 * r1392507;
        double r1392509 = atan2(r1392503, r1392498);
        double r1392510 = y_im;
        double r1392511 = r1392509 * r1392510;
        double r1392512 = r1392508 - r1392511;
        double r1392513 = exp(r1392512);
        double r1392514 = r1392509 * r1392501;
        double r1392515 = -r1392498;
        double r1392516 = log(r1392515);
        double r1392517 = r1392510 * r1392516;
        double r1392518 = r1392514 + r1392517;
        double r1392519 = sin(r1392518);
        double r1392520 = r1392513 * r1392519;
        double r1392521 = -4.748647839754185e-157;
        bool r1392522 = r1392498 <= r1392521;
        double r1392523 = r1392510 * r1392507;
        double r1392524 = r1392523 + r1392514;
        double r1392525 = sin(r1392524);
        double r1392526 = exp(r1392511);
        double r1392527 = cbrt(r1392526);
        double r1392528 = log(r1392527);
        double r1392529 = r1392527 * r1392527;
        double r1392530 = log(r1392529);
        double r1392531 = r1392528 + r1392530;
        double r1392532 = r1392508 - r1392531;
        double r1392533 = exp(r1392532);
        double r1392534 = r1392525 * r1392533;
        double r1392535 = -1.2253344719312e-312;
        bool r1392536 = r1392498 <= r1392535;
        double r1392537 = log(r1392498);
        double r1392538 = r1392537 * r1392510;
        double r1392539 = r1392514 + r1392538;
        double r1392540 = sin(r1392539);
        double r1392541 = r1392513 * r1392540;
        double r1392542 = r1392536 ? r1392520 : r1392541;
        double r1392543 = r1392522 ? r1392534 : r1392542;
        double r1392544 = r1392500 ? r1392520 : r1392543;
        return r1392544;
}

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. Split input into 3 regimes

  2. if x.re < -1.2685618014437514e-33 or -4.748647839754185e-157 < x.re < -1.2253344719312e-312

    1. Initial program 35.3

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

    2. Taylor expanded around -inf 21.1

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

    3. Simplified21.1

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

    if -1.2685618014437514e-33 < x.re < -4.748647839754185e-157

    1. Initial program 16.1

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

    3. Applied add-log-exp18.7

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

    5. Applied add-cube-cbrt18.7

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

    6. Applied log-prod18.7

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

    if -1.2253344719312e-312 < x.re

    1. Initial program 34.3

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

    2. Taylor expanded around inf 23.6

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -1.2685618014437514 \cdot 10^{-33}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le -4.748647839754185 \cdot 10^{-157}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \left(\log \left(\sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right) + \log \left(\sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sqrt[3]{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\right)\right)}\\ \mathbf{elif}\;x.re \le -1.2253344719312 \cdot 10^{-312}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (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)))))