Average Error: 37.3 → 26.9
Time: 20.8s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\
\;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\
\;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\
\;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r7101815 = 0.5;
        double r7101816 = 2.0;
        double r7101817 = re;
        double r7101818 = r7101817 * r7101817;
        double r7101819 = im;
        double r7101820 = r7101819 * r7101819;
        double r7101821 = r7101818 + r7101820;
        double r7101822 = sqrt(r7101821);
        double r7101823 = r7101822 + r7101817;
        double r7101824 = r7101816 * r7101823;
        double r7101825 = sqrt(r7101824);
        double r7101826 = r7101815 * r7101825;
        return r7101826;
}
↓
double f(double re, double im) {
        double r7101827 = re;
        double r7101828 = -1.0974932438808633e+26;
        bool r7101829 = r7101827 <= r7101828;
        double r7101830 = im;
        double r7101831 = r7101830 * r7101830;
        double r7101832 = 2.0;
        double r7101833 = r7101831 * r7101832;
        double r7101834 = sqrt(r7101833);
        double r7101835 = r7101827 * r7101827;
        double r7101836 = r7101831 + r7101835;
        double r7101837 = sqrt(r7101836);
        double r7101838 = r7101837 - r7101827;
        double r7101839 = sqrt(r7101838);
        double r7101840 = r7101834 / r7101839;
        double r7101841 = 0.5;
        double r7101842 = r7101840 * r7101841;
        double r7101843 = -4.4945327826415316e-20;
        bool r7101844 = r7101827 <= r7101843;
        double r7101845 = r7101827 + r7101830;
        double r7101846 = r7101845 * r7101832;
        double r7101847 = sqrt(r7101846);
        double r7101848 = r7101847 * r7101841;
        double r7101849 = -7.961223836723572e-96;
        bool r7101850 = r7101827 <= r7101849;
        double r7101851 = r7101831 / r7101838;
        double r7101852 = r7101832 * r7101851;
        double r7101853 = sqrt(r7101852);
        double r7101854 = r7101841 * r7101853;
        double r7101855 = -2.538815066158378e-267;
        bool r7101856 = r7101827 <= r7101855;
        double r7101857 = 1.8791426213625292e+66;
        bool r7101858 = r7101827 <= r7101857;
        double r7101859 = r7101827 + r7101837;
        double r7101860 = r7101859 * r7101832;
        double r7101861 = sqrt(r7101860);
        double r7101862 = r7101861 * r7101841;
        double r7101863 = r7101827 + r7101827;
        double r7101864 = r7101863 * r7101832;
        double r7101865 = sqrt(r7101864);
        double r7101866 = r7101865 * r7101841;
        double r7101867 = r7101858 ? r7101862 : r7101866;
        double r7101868 = r7101856 ? r7101848 : r7101867;
        double r7101869 = r7101850 ? r7101854 : r7101868;
        double r7101870 = r7101844 ? r7101848 : r7101869;
        double r7101871 = r7101829 ? r7101842 : r7101870;
        return r7101871;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target32.5
Herbie26.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.0974932438808633e+26

    1. Initial program 56.7

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

    3. Applied add-exp-log59.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]
    4. Using strategy rm

    5. Applied flip-+59.1

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

    6. Applied associate-*r/59.1

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

    7. Applied sqrt-div59.1

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

    8. Simplified39.5

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

    9. Simplified38.8

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

    if -1.0974932438808633e+26 < re < -4.4945327826415316e-20 or -7.961223836723572e-96 < re < -2.538815066158378e-267

    1. Initial program 33.8

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

    2. Taylor expanded around 0 38.9

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

    if -4.4945327826415316e-20 < re < -7.961223836723572e-96

    1. Initial program 38.9

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

    3. Applied flip-+38.9

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

    4. Simplified28.1

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

    if -2.538815066158378e-267 < re < 1.8791426213625292e+66

    1. Initial program 21.6

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

    if 1.8791426213625292e+66 < re

    1. Initial program 45.6

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

    2. Taylor expanded around inf 11.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))