Average Error: 38.4 → 26.4
Time: 18.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.222006465724332039862348815623896484961 \cdot 10^{103}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -1.511080944730437448708717796208898738523 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right) - re\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re} \cdot 2}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.222006465724332039862348815623896484961 \cdot 10^{103}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le -1.511080944730437448708717796208898738523 \cdot 10^{-305}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right) - re\right) \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re} \cdot 2}\\

\end{array}
double f(double re, double im) {
        double r983768 = 0.5;
        double r983769 = 2.0;
        double r983770 = re;
        double r983771 = r983770 * r983770;
        double r983772 = im;
        double r983773 = r983772 * r983772;
        double r983774 = r983771 + r983773;
        double r983775 = sqrt(r983774);
        double r983776 = r983775 - r983770;
        double r983777 = r983769 * r983776;
        double r983778 = sqrt(r983777);
        double r983779 = r983768 * r983778;
        return r983779;
}
↓
double f(double re, double im) {
        double r983780 = re;
        double r983781 = -2.222006465724332e+103;
        bool r983782 = r983780 <= r983781;
        double r983783 = -2.0;
        double r983784 = r983783 * r983780;
        double r983785 = 2.0;
        double r983786 = r983784 * r983785;
        double r983787 = sqrt(r983786);
        double r983788 = 0.5;
        double r983789 = r983787 * r983788;
        double r983790 = -1.5110809447304374e-305;
        bool r983791 = r983780 <= r983790;
        double r983792 = r983780 * r983780;
        double r983793 = im;
        double r983794 = r983793 * r983793;
        double r983795 = r983792 + r983794;
        double r983796 = sqrt(r983795);
        double r983797 = cbrt(r983796);
        double r983798 = sqrt(r983797);
        double r983799 = cbrt(r983795);
        double r983800 = fabs(r983799);
        double r983801 = r983798 * r983800;
        double r983802 = r983798 * r983801;
        double r983803 = r983802 - r983780;
        double r983804 = r983803 * r983785;
        double r983805 = sqrt(r983804);
        double r983806 = r983788 * r983805;
        double r983807 = r983796 + r983780;
        double r983808 = r983794 / r983807;
        double r983809 = r983808 * r983785;
        double r983810 = sqrt(r983809);
        double r983811 = r983788 * r983810;
        double r983812 = r983791 ? r983806 : r983811;
        double r983813 = r983782 ? r983789 : r983812;
        return r983813;
}

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 re < -2.222006465724332e+103

    1. Initial program 52.0

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

    2. Taylor expanded around -inf 10.1

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

    if -2.222006465724332e+103 < re < -1.5110809447304374e-305

    1. Initial program 20.4

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

    3. Applied add-cube-cbrt20.6

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

    4. Applied sqrt-prod20.7

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

    5. Simplified20.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt20.7

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

    8. Applied cbrt-prod20.6

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

    9. Applied sqrt-prod20.7

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

    10. Applied associate-*r*20.7

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

    if -1.5110809447304374e-305 < re

    1. Initial program 45.4

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

    3. Applied flip--45.3

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

    4. Simplified35.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im + 0}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.222006465724332039862348815623896484961 \cdot 10^{103}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -1.511080944730437448708717796208898738523 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt{\sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} \cdot \left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right) - re\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re} \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))