Average Error: 38.5 → 12.3
Time: 4.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.53980607688204982 \cdot 10^{-62} \lor \neg \left(re \le 84660.474554424669 \lor \neg \left(re \le 2.1063886874167957 \cdot 10^{226}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 1.53980607688204982 \cdot 10^{-62} \lor \neg \left(re \le 84660.474554424669 \lor \neg \left(re \le 2.1063886874167957 \cdot 10^{226}\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 r20859 = 0.5;
        double r20860 = 2.0;
        double r20861 = re;
        double r20862 = r20861 * r20861;
        double r20863 = im;
        double r20864 = r20863 * r20863;
        double r20865 = r20862 + r20864;
        double r20866 = sqrt(r20865);
        double r20867 = r20866 - r20861;
        double r20868 = r20860 * r20867;
        double r20869 = sqrt(r20868);
        double r20870 = r20859 * r20869;
        return r20870;
}
↓
double f(double re, double im) {
        double r20871 = re;
        double r20872 = 1.5398060768820498e-62;
        bool r20873 = r20871 <= r20872;
        double r20874 = 84660.47455442467;
        bool r20875 = r20871 <= r20874;
        double r20876 = 2.1063886874167957e+226;
        bool r20877 = r20871 <= r20876;
        double r20878 = !r20877;
        bool r20879 = r20875 || r20878;
        double r20880 = !r20879;
        bool r20881 = r20873 || r20880;
        double r20882 = 0.5;
        double r20883 = 2.0;
        double r20884 = im;
        double r20885 = hypot(r20871, r20884);
        double r20886 = r20885 - r20871;
        double r20887 = 0.0;
        double r20888 = r20886 + r20887;
        double r20889 = r20883 * r20888;
        double r20890 = sqrt(r20889);
        double r20891 = r20882 * r20890;
        double r20892 = 2.0;
        double r20893 = pow(r20884, r20892);
        double r20894 = r20893 + r20887;
        double r20895 = r20871 + r20885;
        double r20896 = r20894 / r20895;
        double r20897 = r20883 * r20896;
        double r20898 = sqrt(r20897);
        double r20899 = r20882 * r20898;
        double r20900 = r20881 ? r20891 : r20899;
        return r20900;
}

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 re < 1.5398060768820498e-62 or 84660.47455442467 < re < 2.1063886874167957e+226

    1. Initial program 36.3

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

    3. Applied add-cube-cbrt36.9

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

    4. Applied add-sqr-sqrt36.9

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

    5. Applied sqrt-prod36.9

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

    6. Applied prod-diff37.1

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

    7. Simplified10.8

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

    8. Simplified9.7

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

    if 1.5398060768820498e-62 < re < 84660.47455442467 or 2.1063886874167957e+226 < re

    1. Initial program 54.4

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

    3. Applied flip--54.4

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

    4. Simplified40.9

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

    5. Simplified31.0

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.53980607688204982 \cdot 10^{-62} \lor \neg \left(re \le 84660.474554424669 \lor \neg \left(re \le 2.1063886874167957 \cdot 10^{226}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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)))))