\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -0.0087738037109375:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re} \cdot \left(1.0 \cdot \left(im \cdot im\right)\right)} \cdot 0.5\\ \end{array}\]

double f(double re, double im) {
        double r1134618 = 0.5;
        double r1134619 = 2.0;
        double r1134620 = re;
        double r1134621 = r1134620 * r1134620;
        double r1134622 = im;
        double r1134623 = r1134622 * r1134622;
        double r1134624 = r1134621 + r1134623;
        double r1134625 = sqrt(r1134624);
        double r1134626 = r1134625 - r1134620;
        double r1134627 = r1134619 * r1134626;
        double r1134628 = sqrt(r1134627);
        double r1134629 = r1134618 * r1134628;
        return r1134629;
}

↓

double f(double re, double im) {
        double r1134630 = re;
        double r1134631 = -0.0087738037109375;
        bool r1134632 = r1134630 <= r1134631;
        double r1134633 = 0.5;
        double r1134634 = 2.0;
        double r1134635 = r1134630 * r1134630;
        double r1134636 = im;
        double r1134637 = r1134636 * r1134636;
        double r1134638 = r1134635 + r1134637;
        double r1134639 = sqrt(r1134638);
        double r1134640 = r1134639 - r1134630;
        double r1134641 = r1134634 * r1134640;
        double r1134642 = sqrt(r1134641);
        double r1134643 = r1134633 * r1134642;
        double r1134644 = r1134637 + r1134635;
        double r1134645 = sqrt(r1134644);
        double r1134646 = r1134645 + r1134630;
        double r1134647 = r1134634 / r1134646;
        double r1134648 = 1.0;
        double r1134649 = r1134648 * r1134637;
        double r1134650 = r1134647 * r1134649;
        double r1134651 = sqrt(r1134650);
        double r1134652 = r1134651 * r1134633;
        double r1134653 = r1134632 ? r1134643 : r1134652;
        return r1134653;
}

0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}

↓

\begin{array}{l}
\mathbf{if}\;re \le -0.0087738037109375:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re} \cdot \left(1.0 \cdot \left(im \cdot im\right)\right)} \cdot 0.5\\

\end{array}

Derivation

Split input into 2 regimes
if re < -0.0087738037109375
1. Initial program 0.7
  \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
if -0.0087738037109375 < re
1. Initial program 3.2
  \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
2. Using strategy rm
3. Applied p16-flip--3.0
  \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \color{blue}{\left(\frac{\left(\left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)\right) - \left(re \cdot re\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)}\right)}\right)\]
4. Using strategy rm
5. Applied difference-of-squares3.2
  \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)\right)}\right)\]
6. Using strategy rm
7. Applied p16-flip--3.0
  \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right) \cdot \color{blue}{\left(\frac{\left(\left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)\right) - \left(re \cdot re\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)}\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)\right)}\right)\]
8. Applied associate-*r/3.6
  \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(\frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right) \cdot \left(\left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)\right) - \left(re \cdot re\right)\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)\right)}\right)\]
9. Applied associate-/l/3.6
  \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \color{blue}{\left(\frac{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right) \cdot \left(\left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)\right) - \left(re \cdot re\right)\right)\right)}{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)}\right)}\right)\]
10. Simplified1.7
  \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(\left(\frac{re}{\left(\sqrt{\left(\frac{\left(im \cdot im\right)}{\left(re \cdot re\right)}\right)}\right)}\right) \cdot \left(im \cdot im\right)\right)}}{\left(\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\right)}\right)\]
11. Simplified0.8
  \[\leadsto \color{blue}{\left(\sqrt{\left(\left(\frac{\left(2.0\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(im \cdot im\right)}{\left(re \cdot re\right)}\right)}\right)}{re}\right)}\right) \cdot \left(\left(1.0\right) \cdot \left(im \cdot im\right)\right)\right)}\right) \cdot \left(0.5\right)}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;re \le -0.0087738037109375:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re} \cdot \left(1.0 \cdot \left(im \cdot im\right)\right)} \cdot 0.5\\ \end{array}\]

Error

Derivation

`if re < -0.0087738037109375`

`if -0.0087738037109375 < re`

Reproduce