Average Error: 38.4 → 26.4
Time: 18.5s
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 r983766 = 0.5;
        double r983767 = 2.0;
        double r983768 = re;
        double r983769 = r983768 * r983768;
        double r983770 = im;
        double r983771 = r983770 * r983770;
        double r983772 = r983769 + r983771;
        double r983773 = sqrt(r983772);
        double r983774 = r983773 - r983768;
        double r983775 = r983767 * r983774;
        double r983776 = sqrt(r983775);
        double r983777 = r983766 * r983776;
        return r983777;
}
↓
double f(double re, double im) {
        double r983778 = re;
        double r983779 = -2.222006465724332e+103;
        bool r983780 = r983778 <= r983779;
        double r983781 = -2.0;
        double r983782 = r983781 * r983778;
        double r983783 = 2.0;
        double r983784 = r983782 * r983783;
        double r983785 = sqrt(r983784);
        double r983786 = 0.5;
        double r983787 = r983785 * r983786;
        double r983788 = -1.5110809447304374e-305;
        bool r983789 = r983778 <= r983788;
        double r983790 = r983778 * r983778;
        double r983791 = im;
        double r983792 = r983791 * r983791;
        double r983793 = r983790 + r983792;
        double r983794 = sqrt(r983793);
        double r983795 = cbrt(r983794);
        double r983796 = sqrt(r983795);
        double r983797 = cbrt(r983793);
        double r983798 = fabs(r983797);
        double r983799 = r983796 * r983798;
        double r983800 = r983796 * r983799;
        double r983801 = r983800 - r983778;
        double r983802 = r983801 * r983783;
        double r983803 = sqrt(r983802);
        double r983804 = r983786 * r983803;
        double r983805 = r983794 + r983778;
        double r983806 = r983792 / r983805;
        double r983807 = r983806 * r983783;
        double r983808 = sqrt(r983807);
        double r983809 = r983786 * r983808;
        double r983810 = r983789 ? r983804 : r983809;
        double r983811 = r983780 ? r983787 : r983810;
        return r983811;
}

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)))))