Average Error: 38.6 → 23.5
Time: 5.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -6.9498658840754837 \cdot 10^{41}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -1.49541069512477158 \cdot 10^{-157}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 8.69141323217306704 \cdot 10^{-176}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.0921861446497823 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{elif}\;im \le 1.9398757964769447 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -6.9498658840754837 \cdot 10^{41}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\

\mathbf{elif}\;im \le -1.49541069512477158 \cdot 10^{-157}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;im \le 8.69141323217306704 \cdot 10^{-176}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;im \le 1.0921861446497823 \cdot 10^{-69}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\

\mathbf{elif}\;im \le 1.9398757964769447 \cdot 10^{-46}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\end{array}
double f(double re, double im) {
        double r20855 = 0.5;
        double r20856 = 2.0;
        double r20857 = re;
        double r20858 = r20857 * r20857;
        double r20859 = im;
        double r20860 = r20859 * r20859;
        double r20861 = r20858 + r20860;
        double r20862 = sqrt(r20861);
        double r20863 = r20862 - r20857;
        double r20864 = r20856 * r20863;
        double r20865 = sqrt(r20864);
        double r20866 = r20855 * r20865;
        return r20866;
}
↓
double f(double re, double im) {
        double r20867 = im;
        double r20868 = -6.949865884075484e+41;
        bool r20869 = r20867 <= r20868;
        double r20870 = 0.5;
        double r20871 = 2.0;
        double r20872 = re;
        double r20873 = r20872 + r20867;
        double r20874 = -r20873;
        double r20875 = r20871 * r20874;
        double r20876 = sqrt(r20875);
        double r20877 = r20870 * r20876;
        double r20878 = -1.4954106951247716e-157;
        bool r20879 = r20867 <= r20878;
        double r20880 = 2.0;
        double r20881 = pow(r20867, r20880);
        double r20882 = r20871 * r20881;
        double r20883 = sqrt(r20882);
        double r20884 = r20872 * r20872;
        double r20885 = r20867 * r20867;
        double r20886 = r20884 + r20885;
        double r20887 = sqrt(r20886);
        double r20888 = r20887 + r20872;
        double r20889 = sqrt(r20888);
        double r20890 = r20883 / r20889;
        double r20891 = r20870 * r20890;
        double r20892 = 8.691413232173067e-176;
        bool r20893 = r20867 <= r20892;
        double r20894 = -1.0;
        double r20895 = r20894 * r20872;
        double r20896 = r20895 - r20872;
        double r20897 = r20871 * r20896;
        double r20898 = sqrt(r20897);
        double r20899 = r20870 * r20898;
        double r20900 = 1.0921861446497823e-69;
        bool r20901 = r20867 <= r20900;
        double r20902 = 1.0;
        double r20903 = pow(r20867, r20902);
        double r20904 = r20888 / r20867;
        double r20905 = r20903 / r20904;
        double r20906 = r20871 * r20905;
        double r20907 = sqrt(r20906);
        double r20908 = r20870 * r20907;
        double r20909 = 1.9398757964769447e-46;
        bool r20910 = r20867 <= r20909;
        double r20911 = r20867 - r20872;
        double r20912 = r20871 * r20911;
        double r20913 = sqrt(r20912);
        double r20914 = r20870 * r20913;
        double r20915 = r20910 ? r20899 : r20914;
        double r20916 = r20901 ? r20908 : r20915;
        double r20917 = r20893 ? r20899 : r20916;
        double r20918 = r20879 ? r20891 : r20917;
        double r20919 = r20869 ? r20877 : r20918;
        return r20919;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if im < -6.949865884075484e+41

    1. Initial program 44.7

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--45.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Simplified44.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Taylor expanded around -inf 12.8

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

    if -6.949865884075484e+41 < im < -1.4954106951247716e-157

    1. Initial program 25.7

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--36.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Simplified27.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
    5. Using strategy rm

    6. Applied associate-*r/27.2

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot {im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    7. Applied sqrt-div26.2

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    if -1.4954106951247716e-157 < im < 8.691413232173067e-176 or 1.0921861446497823e-69 < im < 1.9398757964769447e-46

    1. Initial program 43.4

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

    2. Taylor expanded around -inf 36.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} - re\right)}\]

    if 8.691413232173067e-176 < im < 1.0921861446497823e-69

    1. Initial program 29.9

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--44.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Simplified35.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
    5. Using strategy rm

    6. Applied sqr-pow35.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{\left(\frac{2}{2}\right)} \cdot {im}^{\left(\frac{2}{2}\right)}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    7. Applied associate-/l*34.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{\left(\frac{2}{2}\right)}}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{{im}^{\left(\frac{2}{2}\right)}}}}}\]

    8. Simplified34.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}}\]

    if 1.9398757964769447e-46 < im

    1. Initial program 39.9

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

    2. Taylor expanded around 0 16.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -6.9498658840754837 \cdot 10^{41}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -1.49541069512477158 \cdot 10^{-157}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 8.69141323217306704 \cdot 10^{-176}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.0921861446497823 \cdot 10^{-69}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{elif}\;im \le 1.9398757964769447 \cdot 10^{-46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))