Average Error: 38.9 → 31.1
Time: 4.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -4.4013028877896469 \cdot 10^{-95}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 7.9416639037502523 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;im \le 1.25420999700278674 \cdot 10^{58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -4.4013028877896469 \cdot 10^{-95}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;im \le 7.9416639037502523 \cdot 10^{-159}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;im \le 1.25420999700278674 \cdot 10^{58}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

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

\end{array}
double f(double re, double im) {
        double r18820 = 0.5;
        double r18821 = 2.0;
        double r18822 = re;
        double r18823 = r18822 * r18822;
        double r18824 = im;
        double r18825 = r18824 * r18824;
        double r18826 = r18823 + r18825;
        double r18827 = sqrt(r18826);
        double r18828 = r18827 - r18822;
        double r18829 = r18821 * r18828;
        double r18830 = sqrt(r18829);
        double r18831 = r18820 * r18830;
        return r18831;
}
↓
double f(double re, double im) {
        double r18832 = im;
        double r18833 = -4.401302887789647e-95;
        bool r18834 = r18832 <= r18833;
        double r18835 = 0.5;
        double r18836 = 2.0;
        double r18837 = 2.0;
        double r18838 = pow(r18832, r18837);
        double r18839 = re;
        double r18840 = r18839 * r18839;
        double r18841 = r18832 * r18832;
        double r18842 = r18840 + r18841;
        double r18843 = sqrt(r18842);
        double r18844 = r18843 + r18839;
        double r18845 = r18838 / r18844;
        double r18846 = r18836 * r18845;
        double r18847 = sqrt(r18846);
        double r18848 = r18835 * r18847;
        double r18849 = 7.941663903750252e-159;
        bool r18850 = r18832 <= r18849;
        double r18851 = -2.0;
        double r18852 = r18851 * r18839;
        double r18853 = r18836 * r18852;
        double r18854 = sqrt(r18853);
        double r18855 = r18835 * r18854;
        double r18856 = 1.2542099970027867e+58;
        bool r18857 = r18832 <= r18856;
        double r18858 = r18832 - r18839;
        double r18859 = r18836 * r18858;
        double r18860 = sqrt(r18859);
        double r18861 = r18835 * r18860;
        double r18862 = r18857 ? r18848 : r18861;
        double r18863 = r18850 ? r18855 : r18862;
        double r18864 = r18834 ? r18848 : r18863;
        return r18864;
}

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 3 regimes

  2. if im < -4.401302887789647e-95 or 7.941663903750252e-159 < im < 1.2542099970027867e+58

    1. Initial program 33.8

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

    3. Applied flip--40.0

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

    4. Simplified34.8

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

    if -4.401302887789647e-95 < im < 7.941663903750252e-159

    1. Initial program 42.0

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

    2. Taylor expanded around -inf 37.9

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

    if 1.2542099970027867e+58 < im

    1. Initial program 46.9

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

    2. Taylor expanded around 0 12.5

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

  4. Final simplification31.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.4013028877896469 \cdot 10^{-95}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 7.9416639037502523 \cdot 10^{-159}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;im \le 1.25420999700278674 \cdot 10^{58}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(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)))))