Average Error: 39.2 → 27.1
Time: 16.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;re \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 2.840429153857787148913368668369147336488 \cdot 10^{-258}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\

\mathbf{elif}\;re \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\

\mathbf{elif}\;re \le 2.840429153857787148913368668369147336488 \cdot 10^{-258}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r24864 = 0.5;
        double r24865 = 2.0;
        double r24866 = re;
        double r24867 = r24866 * r24866;
        double r24868 = im;
        double r24869 = r24868 * r24868;
        double r24870 = r24867 + r24869;
        double r24871 = sqrt(r24870);
        double r24872 = r24871 - r24866;
        double r24873 = r24865 * r24872;
        double r24874 = sqrt(r24873);
        double r24875 = r24864 * r24874;
        return r24875;
}
↓
double f(double re, double im) {
        double r24876 = re;
        double r24877 = -5.292452664608309e+126;
        bool r24878 = r24876 <= r24877;
        double r24879 = 0.5;
        double r24880 = 2.0;
        double r24881 = -r24876;
        double r24882 = r24881 - r24876;
        double r24883 = r24880 * r24882;
        double r24884 = sqrt(r24883);
        double r24885 = r24879 * r24884;
        double r24886 = -3.6872979313797757e-267;
        bool r24887 = r24876 <= r24886;
        double r24888 = r24876 * r24876;
        double r24889 = im;
        double r24890 = r24889 * r24889;
        double r24891 = r24888 + r24890;
        double r24892 = cbrt(r24891);
        double r24893 = fabs(r24892);
        double r24894 = sqrt(r24892);
        double r24895 = r24893 * r24894;
        double r24896 = r24895 - r24876;
        double r24897 = r24880 * r24896;
        double r24898 = sqrt(r24897);
        double r24899 = r24879 * r24898;
        double r24900 = 2.840429153857787e-258;
        bool r24901 = r24876 <= r24900;
        double r24902 = r24889 - r24876;
        double r24903 = r24880 * r24902;
        double r24904 = sqrt(r24903);
        double r24905 = r24879 * r24904;
        double r24906 = sqrt(r24891);
        double r24907 = r24906 + r24876;
        double r24908 = r24890 / r24907;
        double r24909 = r24880 * r24908;
        double r24910 = sqrt(r24909);
        double r24911 = r24879 * r24910;
        double r24912 = r24901 ? r24905 : r24911;
        double r24913 = r24887 ? r24899 : r24912;
        double r24914 = r24878 ? r24885 : r24913;
        return r24914;
}

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 < -5.292452664608309e+126

    1. Initial program 57.1

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

    2. Taylor expanded around -inf 8.8

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

    3. Simplified8.8

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

    if -5.292452664608309e+126 < re < -3.6872979313797757e-267

    1. Initial program 19.9

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

    3. Applied add-cube-cbrt20.1

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

    4. Applied sqrt-prod20.1

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

    5. Simplified20.1

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

    if -3.6872979313797757e-267 < re < 2.840429153857787e-258

    1. Initial program 30.2

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

    2. Taylor expanded around 0 32.9

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

    if 2.840429153857787e-258 < re

    1. Initial program 48.3

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

    3. Applied flip--48.2

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

    4. Simplified36.7

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(-re\right) - re\right)}\\ \mathbf{elif}\;re \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 2.840429153857787148913368668369147336488 \cdot 10^{-258}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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)))))