Average Error: 38.7 → 13.7
Time: 3.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \cdot im \le 2.23420910232322685 \cdot 10^{-163} \lor \neg \left(im \cdot im \le 6.8486120940779959 \cdot 10^{-98} \lor \neg \left(im \cdot im \le 1123211316354332.8 \lor \neg \left(im \cdot im \le 1.7186702316517893 \cdot 10^{70}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \cdot im \le 2.23420910232322685 \cdot 10^{-163} \lor \neg \left(im \cdot im \le 6.8486120940779959 \cdot 10^{-98} \lor \neg \left(im \cdot im \le 1123211316354332.8 \lor \neg \left(im \cdot im \le 1.7186702316517893 \cdot 10^{70}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r11869 = 0.5;
        double r11870 = 2.0;
        double r11871 = re;
        double r11872 = r11871 * r11871;
        double r11873 = im;
        double r11874 = r11873 * r11873;
        double r11875 = r11872 + r11874;
        double r11876 = sqrt(r11875);
        double r11877 = r11876 - r11871;
        double r11878 = r11870 * r11877;
        double r11879 = sqrt(r11878);
        double r11880 = r11869 * r11879;
        return r11880;
}
↓
double f(double re, double im) {
        double r11881 = im;
        double r11882 = r11881 * r11881;
        double r11883 = 2.234209102323227e-163;
        bool r11884 = r11882 <= r11883;
        double r11885 = 6.848612094077996e-98;
        bool r11886 = r11882 <= r11885;
        double r11887 = 1123211316354332.8;
        bool r11888 = r11882 <= r11887;
        double r11889 = 1.7186702316517893e+70;
        bool r11890 = r11882 <= r11889;
        double r11891 = !r11890;
        bool r11892 = r11888 || r11891;
        double r11893 = !r11892;
        bool r11894 = r11886 || r11893;
        double r11895 = !r11894;
        bool r11896 = r11884 || r11895;
        double r11897 = 0.5;
        double r11898 = 2.0;
        double r11899 = re;
        double r11900 = hypot(r11899, r11881);
        double r11901 = r11900 - r11899;
        double r11902 = 0.0;
        double r11903 = r11901 + r11902;
        double r11904 = r11898 * r11903;
        double r11905 = sqrt(r11904);
        double r11906 = r11897 * r11905;
        double r11907 = 2.0;
        double r11908 = pow(r11881, r11907);
        double r11909 = r11908 + r11902;
        double r11910 = r11899 + r11900;
        double r11911 = r11909 / r11910;
        double r11912 = r11898 * r11911;
        double r11913 = sqrt(r11912);
        double r11914 = r11897 * r11913;
        double r11915 = r11896 ? r11906 : r11914;
        return r11915;
}

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 2 regimes

  2. if (* im im) < 2.234209102323227e-163 or 6.848612094077996e-98 < (* im im) < 1123211316354332.8 or 1.7186702316517893e+70 < (* im im)

    1. Initial program 40.2

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

    3. Applied add-cube-cbrt40.8

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

    4. Applied add-sqr-sqrt40.8

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

    5. Applied sqrt-prod40.9

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

    6. Applied prod-diff41.0

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

    7. Simplified15.1

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

    8. Simplified12.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + \color{blue}{0}\right)}\]

    if 2.234209102323227e-163 < (* im im) < 6.848612094077996e-98 or 1123211316354332.8 < (* im im) < 1.7186702316517893e+70

    1. Initial program 24.9

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

    3. Applied flip--34.4

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

    4. Simplified25.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Simplified21.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{re + \mathsf{hypot}\left(re, im\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \le 2.23420910232322685 \cdot 10^{-163} \lor \neg \left(im \cdot im \le 6.8486120940779959 \cdot 10^{-98} \lor \neg \left(im \cdot im \le 1123211316354332.8 \lor \neg \left(im \cdot im \le 1.7186702316517893 \cdot 10^{70}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(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)))))