Average Error: 39.0 → 20.0
Time: 17.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.966965058059024834951651891485124795183 \cdot 10^{106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -7.953827581321288477217614759533074653297 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{e^{\log \left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot 3}}\\ \mathbf{elif}\;re \le 1.299753781433129405351729142907841247123 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.966965058059024834951651891485124795183 \cdot 10^{106}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -7.953827581321288477217614759533074653297 \cdot 10^{-141}:\\
\;\;\;\;0.5 \cdot \sqrt[3]{e^{\log \left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot 3}}\\

\mathbf{elif}\;re \le 1.299753781433129405351729142907841247123 \cdot 10^{154}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)\\

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

\end{array}
double f(double re, double im) {
        double r31907 = 0.5;
        double r31908 = 2.0;
        double r31909 = re;
        double r31910 = r31909 * r31909;
        double r31911 = im;
        double r31912 = r31911 * r31911;
        double r31913 = r31910 + r31912;
        double r31914 = sqrt(r31913);
        double r31915 = r31914 - r31909;
        double r31916 = r31908 * r31915;
        double r31917 = sqrt(r31916);
        double r31918 = r31907 * r31917;
        return r31918;
}
↓
double f(double re, double im) {
        double r31919 = re;
        double r31920 = -8.966965058059025e+106;
        bool r31921 = r31919 <= r31920;
        double r31922 = 0.5;
        double r31923 = 2.0;
        double r31924 = -2.0;
        double r31925 = r31924 * r31919;
        double r31926 = r31923 * r31925;
        double r31927 = sqrt(r31926);
        double r31928 = r31922 * r31927;
        double r31929 = -7.953827581321288e-141;
        bool r31930 = r31919 <= r31929;
        double r31931 = r31919 * r31919;
        double r31932 = im;
        double r31933 = r31932 * r31932;
        double r31934 = r31931 + r31933;
        double r31935 = sqrt(r31934);
        double r31936 = r31935 - r31919;
        double r31937 = r31923 * r31936;
        double r31938 = sqrt(r31937);
        double r31939 = log(r31938);
        double r31940 = 3.0;
        double r31941 = r31939 * r31940;
        double r31942 = exp(r31941);
        double r31943 = cbrt(r31942);
        double r31944 = r31922 * r31943;
        double r31945 = 1.2997537814331294e+154;
        bool r31946 = r31919 <= r31945;
        double r31947 = sqrt(r31923);
        double r31948 = fabs(r31932);
        double r31949 = r31935 + r31919;
        double r31950 = sqrt(r31949);
        double r31951 = r31948 / r31950;
        double r31952 = r31947 * r31951;
        double r31953 = r31922 * r31952;
        double r31954 = r31923 * r31933;
        double r31955 = sqrt(r31954);
        double r31956 = r31919 + r31919;
        double r31957 = sqrt(r31956);
        double r31958 = r31955 / r31957;
        double r31959 = r31922 * r31958;
        double r31960 = r31946 ? r31953 : r31959;
        double r31961 = r31930 ? r31944 : r31960;
        double r31962 = r31921 ? r31928 : r31961;
        return r31962;
}

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 < -8.966965058059025e+106

    1. Initial program 53.4

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

    3. Applied add-sqr-sqrt53.4

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

    4. Applied sqrt-prod53.4

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

    5. Taylor expanded around -inf 10.7

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

    if -8.966965058059025e+106 < re < -7.953827581321288e-141

    1. Initial program 16.3

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

    3. Applied add-cbrt-cube16.7

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

    4. Simplified16.7

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

    6. Applied add-exp-log19.0

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

    7. Applied pow-exp19.3

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

    if -7.953827581321288e-141 < re < 1.2997537814331294e+154

    1. Initial program 36.9

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

    3. Applied flip--37.6

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

    4. Applied associate-*r/37.6

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

    5. Applied sqrt-div37.9

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

    6. Simplified30.5

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

    8. Applied *-un-lft-identity30.5

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

    9. Applied sqrt-prod30.5

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

    10. Applied sqrt-prod30.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    11. Applied times-frac30.6

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

    12. Simplified30.6

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

    13. Simplified23.2

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

    if 1.2997537814331294e+154 < re

    1. Initial program 64.0

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

    3. Applied flip--64.0

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

    4. Applied associate-*r/64.0

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

    5. Applied sqrt-div64.0

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

    6. Simplified51.7

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

    7. Taylor expanded around inf 20.0

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

  4. Final simplification20.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.966965058059024834951651891485124795183 \cdot 10^{106}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -7.953827581321288477217614759533074653297 \cdot 10^{-141}:\\ \;\;\;\;0.5 \cdot \sqrt[3]{e^{\log \left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot 3}}\\ \mathbf{elif}\;re \le 1.299753781433129405351729142907841247123 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)))))