Average Error: 38.3 → 12.1
Time: 4.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.09350039928565661 \cdot 10^{227}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 1.09350039928565661 \cdot 10^{227}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\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 r20856 = 0.5;
        double r20857 = 2.0;
        double r20858 = re;
        double r20859 = r20858 * r20858;
        double r20860 = im;
        double r20861 = r20860 * r20860;
        double r20862 = r20859 + r20861;
        double r20863 = sqrt(r20862);
        double r20864 = r20863 - r20858;
        double r20865 = r20857 * r20864;
        double r20866 = sqrt(r20865);
        double r20867 = r20856 * r20866;
        return r20867;
}
↓
double f(double re, double im) {
        double r20868 = re;
        double r20869 = 1.0935003992856566e+227;
        bool r20870 = r20868 <= r20869;
        double r20871 = 0.5;
        double r20872 = 2.0;
        double r20873 = im;
        double r20874 = hypot(r20868, r20873);
        double r20875 = r20874 - r20868;
        double r20876 = r20872 * r20875;
        double r20877 = sqrt(r20876);
        double r20878 = r20871 * r20877;
        double r20879 = 2.0;
        double r20880 = pow(r20873, r20879);
        double r20881 = 0.0;
        double r20882 = r20880 + r20881;
        double r20883 = r20868 + r20874;
        double r20884 = r20882 / r20883;
        double r20885 = r20872 * r20884;
        double r20886 = sqrt(r20885);
        double r20887 = r20871 * r20886;
        double r20888 = r20870 ? r20878 : r20887;
        return r20888;
}

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 < 1.0935003992856566e+227

    1. Initial program 36.5

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

    3. Applied hypot-def10.7

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

    if 1.0935003992856566e+227 < 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. Simplified48.6

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

    5. Simplified31.6

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.09350039928565661 \cdot 10^{227}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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)))))