Average Error: 38.9 → 27.9
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.7270300450622855 \cdot 10^{129}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\ \mathbf{elif}\;re \le -3.2947074803161398 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 2.93104225565989977 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;re \le 170.894038689016583:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.7270300450622855 \cdot 10^{129}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\

\mathbf{elif}\;re \le -3.2947074803161398 \cdot 10^{-260}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le 2.93104225565989977 \cdot 10^{-93}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\

\mathbf{elif}\;re \le 170.894038689016583:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\

\end{array}
double f(double re, double im) {
        double r12812 = 0.5;
        double r12813 = 2.0;
        double r12814 = re;
        double r12815 = r12814 * r12814;
        double r12816 = im;
        double r12817 = r12816 * r12816;
        double r12818 = r12815 + r12817;
        double r12819 = sqrt(r12818);
        double r12820 = r12819 - r12814;
        double r12821 = r12813 * r12820;
        double r12822 = sqrt(r12821);
        double r12823 = r12812 * r12822;
        return r12823;
}
↓
double f(double re, double im) {
        double r12824 = re;
        double r12825 = -8.727030045062286e+129;
        bool r12826 = r12824 <= r12825;
        double r12827 = 0.5;
        double r12828 = 2.0;
        double r12829 = -1.0;
        double r12830 = r12829 * r12824;
        double r12831 = r12830 - r12824;
        double r12832 = r12828 * r12831;
        double r12833 = sqrt(r12832);
        double r12834 = r12827 * r12833;
        double r12835 = -1.180833083521909e-161;
        bool r12836 = r12824 <= r12835;
        double r12837 = r12824 * r12824;
        double r12838 = im;
        double r12839 = r12838 * r12838;
        double r12840 = r12837 + r12839;
        double r12841 = sqrt(r12840);
        double r12842 = sqrt(r12841);
        double r12843 = cbrt(r12841);
        double r12844 = r12843 * r12843;
        double r12845 = r12844 * r12843;
        double r12846 = sqrt(r12845);
        double r12847 = r12842 * r12846;
        double r12848 = r12847 - r12824;
        double r12849 = r12828 * r12848;
        double r12850 = sqrt(r12849);
        double r12851 = r12827 * r12850;
        double r12852 = -3.29470748031614e-260;
        bool r12853 = r12824 <= r12852;
        double r12854 = r12838 - r12824;
        double r12855 = r12828 * r12854;
        double r12856 = sqrt(r12855);
        double r12857 = r12827 * r12856;
        double r12858 = 2.9310422556598998e-93;
        bool r12859 = r12824 <= r12858;
        double r12860 = 0.0;
        double r12861 = r12839 + r12860;
        double r12862 = r12824 + r12841;
        double r12863 = r12861 / r12862;
        double r12864 = r12828 * r12863;
        double r12865 = sqrt(r12864);
        double r12866 = r12827 * r12865;
        double r12867 = 170.89403868901658;
        bool r12868 = r12824 <= r12867;
        double r12869 = r12868 ? r12857 : r12866;
        double r12870 = r12859 ? r12866 : r12869;
        double r12871 = r12853 ? r12857 : r12870;
        double r12872 = r12836 ? r12851 : r12871;
        double r12873 = r12826 ? r12834 : r12872;
        return r12873;
}

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 4 regimes

  2. if re < -8.727030045062286e+129

    1. Initial program 57.0

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

    2. Taylor expanded around -inf 8.9

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

    if -8.727030045062286e+129 < re < -1.180833083521909e-161

    1. Initial program 16.7

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

    3. Applied add-sqr-sqrt16.7

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

    4. Applied sqrt-prod16.8

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

    6. Applied add-cube-cbrt17.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}}} - re\right)}\]

    if -1.180833083521909e-161 < re < -3.29470748031614e-260 or 2.9310422556598998e-93 < re < 170.89403868901658

    1. Initial program 36.3

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

    2. Taylor expanded around 0 39.4

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

    if -3.29470748031614e-260 < re < 2.9310422556598998e-93 or 170.89403868901658 < re

    1. Initial program 45.7

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

    3. Applied add-sqr-sqrt45.7

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

    4. Applied sqrt-prod46.4

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

    6. Applied flip--46.3

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

    7. Simplified36.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re}}\]

    8. Simplified36.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{\color{blue}{re + \sqrt{re \cdot re + im \cdot im}}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.7270300450622855 \cdot 10^{129}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\ \mathbf{elif}\;re \le -3.2947074803161398 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 2.93104225565989977 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;re \le 170.894038689016583:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)))))