\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.3868507513807465 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2} \cdot \mathsf{fma}\left(\sqrt{1}, \mathsf{hypot}\left(re, im\right), re\right)}\right)\\ \end{array}\]

0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}

↓

\begin{array}{l}
\mathbf{if}\;re \le -5.3868507513807465 \cdot 10^{-10}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2} \cdot \mathsf{fma}\left(\sqrt{1}, \mathsf{hypot}\left(re, im\right), re\right)}\right)\\

\end{array}

double f(double re, double im) {
        double r195803 = 0.5;
        double r195804 = 2.0;
        double r195805 = re;
        double r195806 = r195805 * r195805;
        double r195807 = im;
        double r195808 = r195807 * r195807;
        double r195809 = r195806 + r195808;
        double r195810 = sqrt(r195809);
        double r195811 = r195810 + r195805;
        double r195812 = r195804 * r195811;
        double r195813 = sqrt(r195812);
        double r195814 = r195803 * r195813;
        return r195814;
}

↓

double f(double re, double im) {
        double r195815 = re;
        double r195816 = -5.386850751380747e-10;
        bool r195817 = r195815 <= r195816;
        double r195818 = 0.5;
        double r195819 = 2.0;
        double r195820 = 0.0;
        double r195821 = im;
        double r195822 = 2.0;
        double r195823 = pow(r195821, r195822);
        double r195824 = r195820 + r195823;
        double r195825 = hypot(r195815, r195821);
        double r195826 = r195825 - r195815;
        double r195827 = r195824 / r195826;
        double r195828 = r195819 * r195827;
        double r195829 = sqrt(r195828);
        double r195830 = r195818 * r195829;
        double r195831 = sqrt(r195819);
        double r195832 = sqrt(r195831);
        double r195833 = 1.0;
        double r195834 = sqrt(r195833);
        double r195835 = fma(r195834, r195825, r195815);
        double r195836 = r195831 * r195835;
        double r195837 = sqrt(r195836);
        double r195838 = r195832 * r195837;
        double r195839 = r195818 * r195838;
        double r195840 = r195817 ? r195830 : r195839;
        return r195840;
}

Original	38.4
Target	33.2
Herbie	11.7

Derivation

Split input into 2 regimes
if re < -5.386850751380747e-10
1. Initial program 56.4
  \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
2. Using strategy rm
3. Applied flip-+56.4
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]
4. Simplified39.4
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
5. Simplified30.7
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\color{blue}{\mathsf{hypot}\left(re, im\right) - re}}}\]
if -5.386850751380747e-10 < re
1. Initial program 32.1
  \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
2. Using strategy rm
3. Applied *-un-lft-identity32.1
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}} + re\right)}\]
4. Applied sqrt-prod32.1
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}} + re\right)}\]
5. Simplified4.9
  \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)}\]
6. Using strategy rm
7. Applied sqrt-prod5.2
  \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re}\right)}\]
8. Using strategy rm
9. Applied add-sqr-sqrt5.2
  \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \cdot \sqrt{\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re}\right)\]
10. Applied sqrt-prod5.4
  \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}\right)} \cdot \sqrt{\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re}\right)\]
11. Applied associate-*l*5.3
  \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re}\right)\right)}\]
12. Using strategy rm
13. Applied sqrt-unprod5.1
  \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \color{blue}{\sqrt{\sqrt{2} \cdot \left(\sqrt{1} \cdot \mathsf{hypot}\left(re, im\right) + re\right)}}\right)\]
14. Simplified5.1
  \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(\sqrt{1}, \mathsf{hypot}\left(re, im\right), re\right)}}\right)\]
Recombined 2 regimes into one program.
Final simplification11.7
\[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.3868507513807465 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2} \cdot \mathsf{fma}\left(\sqrt{1}, \mathsf{hypot}\left(re, im\right), re\right)}\right)\\ \end{array}\]

Error

Target

Derivation

`if re < -5.386850751380747e-10`

`if -5.386850751380747e-10 < re`

Reproduce