Average Error: 38.0 → 31.7
Time: 4.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -2.5585399092543814 \cdot 10^{-148}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \left(\sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) - re\right)}\\ \mathbf{elif}\;im \le -4.0771067353042325 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.1321272697325898 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 2.19133113958616996 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 2.9805355872812398 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -2.5585399092543814 \cdot 10^{-148}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \left(\sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) - re\right)}\\

\mathbf{elif}\;im \le -4.0771067353042325 \cdot 10^{-295}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;im \le 1.1321272697325898 \cdot 10^{-70}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;im \le 2.19133113958616996 \cdot 10^{-60}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;im \le 2.9805355872812398 \cdot 10^{118}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

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

\end{array}
double f(double re, double im) {
        double r20894 = 0.5;
        double r20895 = 2.0;
        double r20896 = re;
        double r20897 = r20896 * r20896;
        double r20898 = im;
        double r20899 = r20898 * r20898;
        double r20900 = r20897 + r20899;
        double r20901 = sqrt(r20900);
        double r20902 = r20901 - r20896;
        double r20903 = r20895 * r20902;
        double r20904 = sqrt(r20903);
        double r20905 = r20894 * r20904;
        return r20905;
}
↓
double f(double re, double im) {
        double r20906 = im;
        double r20907 = -2.5585399092543814e-148;
        bool r20908 = r20906 <= r20907;
        double r20909 = 0.5;
        double r20910 = 2.0;
        double r20911 = re;
        double r20912 = r20911 * r20911;
        double r20913 = r20906 * r20906;
        double r20914 = r20912 + r20913;
        double r20915 = cbrt(r20914);
        double r20916 = fabs(r20915);
        double r20917 = sqrt(r20916);
        double r20918 = sqrt(r20915);
        double r20919 = sqrt(r20918);
        double r20920 = r20917 * r20919;
        double r20921 = r20920 * r20920;
        double r20922 = r20921 - r20911;
        double r20923 = r20910 * r20922;
        double r20924 = sqrt(r20923);
        double r20925 = r20909 * r20924;
        double r20926 = -4.0771067353042325e-295;
        bool r20927 = r20906 <= r20926;
        double r20928 = -1.0;
        double r20929 = r20928 * r20911;
        double r20930 = r20929 - r20911;
        double r20931 = r20910 * r20930;
        double r20932 = sqrt(r20931);
        double r20933 = r20909 * r20932;
        double r20934 = 1.1321272697325898e-70;
        bool r20935 = r20906 <= r20934;
        double r20936 = 2.0;
        double r20937 = pow(r20906, r20936);
        double r20938 = sqrt(r20914);
        double r20939 = r20938 + r20911;
        double r20940 = r20937 / r20939;
        double r20941 = r20910 * r20940;
        double r20942 = sqrt(r20941);
        double r20943 = r20909 * r20942;
        double r20944 = 2.19133113958617e-60;
        bool r20945 = r20906 <= r20944;
        double r20946 = 2.98053558728124e+118;
        bool r20947 = r20906 <= r20946;
        double r20948 = r20906 - r20911;
        double r20949 = r20910 * r20948;
        double r20950 = sqrt(r20949);
        double r20951 = r20909 * r20950;
        double r20952 = r20947 ? r20943 : r20951;
        double r20953 = r20945 ? r20933 : r20952;
        double r20954 = r20935 ? r20943 : r20953;
        double r20955 = r20927 ? r20933 : r20954;
        double r20956 = r20908 ? r20925 : r20955;
        return r20956;
}

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 im < -2.5585399092543814e-148

    1. Initial program 35.4

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

    3. Applied add-cube-cbrt35.6

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

    4. Applied sqrt-prod35.6

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

    5. Simplified35.6

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

    7. Applied add-sqr-sqrt35.6

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

    8. Applied sqrt-prod35.6

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

    9. Applied add-sqr-sqrt35.7

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

    10. Applied unswap-sqr35.7

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

    if -2.5585399092543814e-148 < im < -4.0771067353042325e-295 or 1.1321272697325898e-70 < im < 2.19133113958617e-60

    1. Initial program 42.1

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

    2. Taylor expanded around -inf 35.6

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

    if -4.0771067353042325e-295 < im < 1.1321272697325898e-70 or 2.19133113958617e-60 < im < 2.98053558728124e+118

    1. Initial program 31.6

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

    3. Applied flip--43.3

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

    4. Simplified35.3

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

    if 2.98053558728124e+118 < im

    1. Initial program 55.5

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

    2. Taylor expanded around 0 10.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification31.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.5585399092543814 \cdot 10^{-148}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \left(\sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) - re\right)}\\ \mathbf{elif}\;im \le -4.0771067353042325 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.1321272697325898 \cdot 10^{-70}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 2.19133113958616996 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 2.9805355872812398 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))))