Average Error: 37.6 → 23.6
Time: 23.0s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.9406267605871937 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.1044997925898934 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.7825065207715073 \cdot 10^{+78}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{re + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.9406267605871937 \cdot 10^{+138}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 1.1044997925898934 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\

\mathbf{elif}\;re \le 1.7825065207715073 \cdot 10^{+78}:\\
\;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\

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

\end{array}
double f(double re, double im) {
        double r788815 = 0.5;
        double r788816 = 2.0;
        double r788817 = re;
        double r788818 = r788817 * r788817;
        double r788819 = im;
        double r788820 = r788819 * r788819;
        double r788821 = r788818 + r788820;
        double r788822 = sqrt(r788821);
        double r788823 = r788822 - r788817;
        double r788824 = r788816 * r788823;
        double r788825 = sqrt(r788824);
        double r788826 = r788815 * r788825;
        return r788826;
}
↓
double f(double re, double im) {
        double r788827 = re;
        double r788828 = -1.9406267605871937e+138;
        bool r788829 = r788827 <= r788828;
        double r788830 = -2.0;
        double r788831 = r788830 * r788827;
        double r788832 = 2.0;
        double r788833 = r788831 * r788832;
        double r788834 = sqrt(r788833);
        double r788835 = 0.5;
        double r788836 = r788834 * r788835;
        double r788837 = 1.1044997925898934e-17;
        bool r788838 = r788827 <= r788837;
        double r788839 = im;
        double r788840 = r788839 * r788839;
        double r788841 = r788827 * r788827;
        double r788842 = r788840 + r788841;
        double r788843 = sqrt(r788842);
        double r788844 = r788843 - r788827;
        double r788845 = r788832 * r788844;
        double r788846 = sqrt(r788845);
        double r788847 = r788846 * r788835;
        double r788848 = 1.7825065207715073e+78;
        bool r788849 = r788827 <= r788848;
        double r788850 = sqrt(r788832);
        double r788851 = r788839 * r788850;
        double r788852 = r788843 + r788827;
        double r788853 = sqrt(r788852);
        double r788854 = r788851 / r788853;
        double r788855 = r788835 * r788854;
        double r788856 = r788832 * r788840;
        double r788857 = sqrt(r788856);
        double r788858 = r788827 + r788827;
        double r788859 = sqrt(r788858);
        double r788860 = r788857 / r788859;
        double r788861 = r788835 * r788860;
        double r788862 = r788849 ? r788855 : r788861;
        double r788863 = r788838 ? r788847 : r788862;
        double r788864 = r788829 ? r788836 : r788863;
        return r788864;
}

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 < -1.9406267605871937e+138

    1. Initial program 57.0

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

    2. Taylor expanded around -inf 8.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

    if -1.9406267605871937e+138 < re < 1.1044997925898934e-17

    1. Initial program 25.4

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity25.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}\]

    if 1.1044997925898934e-17 < re < 1.7825065207715073e+78

    1. Initial program 46.7

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--46.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/46.7

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

    5. Applied sqrt-div46.7

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

    6. Simplified28.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    7. Taylor expanded around inf 38.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    if 1.7825065207715073e+78 < re

    1. Initial program 58.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--58.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/58.6

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

    5. Applied sqrt-div58.6

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

    6. Simplified42.1

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    7. Taylor expanded around inf 22.2

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\color{blue}{re} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.9406267605871937 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.1044997925898934 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.7825065207715073 \cdot 10^{+78}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{re + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))