Average Error: 38.6 → 12.1
Time: 3.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 122037.3472471400746144354343414306640625 \lor \neg \left(re \le 9.353483842517877248816988325405588452621 \cdot 10^{131} \lor \neg \left(re \le 3.360843720164907787265527774503692456559 \cdot 10^{248}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 122037.3472471400746144354343414306640625 \lor \neg \left(re \le 9.353483842517877248816988325405588452621 \cdot 10^{131} \lor \neg \left(re \le 3.360843720164907787265527774503692456559 \cdot 10^{248}\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{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r10824 = 0.5;
        double r10825 = 2.0;
        double r10826 = re;
        double r10827 = r10826 * r10826;
        double r10828 = im;
        double r10829 = r10828 * r10828;
        double r10830 = r10827 + r10829;
        double r10831 = sqrt(r10830);
        double r10832 = r10831 - r10826;
        double r10833 = r10825 * r10832;
        double r10834 = sqrt(r10833);
        double r10835 = r10824 * r10834;
        return r10835;
}
↓
double f(double re, double im) {
        double r10836 = re;
        double r10837 = 122037.34724714007;
        bool r10838 = r10836 <= r10837;
        double r10839 = 9.353483842517877e+131;
        bool r10840 = r10836 <= r10839;
        double r10841 = 3.360843720164908e+248;
        bool r10842 = r10836 <= r10841;
        double r10843 = !r10842;
        bool r10844 = r10840 || r10843;
        double r10845 = !r10844;
        bool r10846 = r10838 || r10845;
        double r10847 = 0.5;
        double r10848 = 2.0;
        double r10849 = im;
        double r10850 = hypot(r10836, r10849);
        double r10851 = r10850 - r10836;
        double r10852 = 0.0;
        double r10853 = r10851 + r10852;
        double r10854 = r10848 * r10853;
        double r10855 = sqrt(r10854);
        double r10856 = r10847 * r10855;
        double r10857 = fma(r10849, r10849, r10852);
        double r10858 = r10836 + r10850;
        double r10859 = r10857 / r10858;
        double r10860 = r10848 * r10859;
        double r10861 = sqrt(r10860);
        double r10862 = r10847 * r10861;
        double r10863 = r10846 ? r10856 : r10862;
        return r10863;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 2 regimes

  2. if re < 122037.34724714007 or 9.353483842517877e+131 < re < 3.360843720164908e+248

    1. Initial program 35.9

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

    3. Applied add-cube-cbrt36.1

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

    4. Applied add-sqr-sqrt36.1

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

    5. Applied sqrt-prod36.2

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

    6. Applied prod-diff36.2

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

    7. Simplified10.2

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

    8. Simplified8.9

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

    if 122037.34724714007 < re < 9.353483842517877e+131 or 3.360843720164908e+248 < re

    1. Initial program 54.3

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

    3. Applied flip--54.3

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

    4. Simplified36.2

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

    5. Simplified30.4

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 122037.3472471400746144354343414306640625 \lor \neg \left(re \le 9.353483842517877248816988325405588452621 \cdot 10^{131} \lor \neg \left(re \le 3.360843720164907787265527774503692456559 \cdot 10^{248}\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))))