Average Error: 38.8 → 27.0
Time: 1.4m
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right) \cdot 2}\\ \mathbf{elif}\;re \le 5.712445954823859518117956639119879892462 \cdot 10^{-130}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right) \cdot 2}\\

\mathbf{elif}\;re \le 5.712445954823859518117956639119879892462 \cdot 10^{-130}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r1030783 = 0.5;
        double r1030784 = 2.0;
        double r1030785 = re;
        double r1030786 = r1030785 * r1030785;
        double r1030787 = im;
        double r1030788 = r1030787 * r1030787;
        double r1030789 = r1030786 + r1030788;
        double r1030790 = sqrt(r1030789);
        double r1030791 = r1030790 - r1030785;
        double r1030792 = r1030784 * r1030791;
        double r1030793 = sqrt(r1030792);
        double r1030794 = r1030783 * r1030793;
        return r1030794;
}
↓
double f(double re, double im) {
        double r1030795 = re;
        double r1030796 = -8.039547558546272e+104;
        bool r1030797 = r1030795 <= r1030796;
        double r1030798 = -2.0;
        double r1030799 = r1030798 * r1030795;
        double r1030800 = 2.0;
        double r1030801 = r1030799 * r1030800;
        double r1030802 = sqrt(r1030801);
        double r1030803 = 0.5;
        double r1030804 = r1030802 * r1030803;
        double r1030805 = -3.240034389533804e-262;
        bool r1030806 = r1030795 <= r1030805;
        double r1030807 = r1030795 * r1030795;
        double r1030808 = im;
        double r1030809 = r1030808 * r1030808;
        double r1030810 = r1030807 + r1030809;
        double r1030811 = sqrt(r1030810);
        double r1030812 = sqrt(r1030811);
        double r1030813 = r1030812 * r1030812;
        double r1030814 = r1030813 - r1030795;
        double r1030815 = r1030814 * r1030800;
        double r1030816 = sqrt(r1030815);
        double r1030817 = r1030803 * r1030816;
        double r1030818 = 5.7124459548238595e-130;
        bool r1030819 = r1030795 <= r1030818;
        double r1030820 = r1030808 - r1030795;
        double r1030821 = r1030800 * r1030820;
        double r1030822 = sqrt(r1030821);
        double r1030823 = r1030803 * r1030822;
        double r1030824 = r1030809 * r1030800;
        double r1030825 = sqrt(r1030824);
        double r1030826 = r1030795 + r1030811;
        double r1030827 = sqrt(r1030826);
        double r1030828 = r1030825 / r1030827;
        double r1030829 = r1030828 * r1030803;
        double r1030830 = r1030819 ? r1030823 : r1030829;
        double r1030831 = r1030806 ? r1030817 : r1030830;
        double r1030832 = r1030797 ? r1030804 : r1030831;
        return r1030832;
}

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 4 regimes

  2. if re < -8.039547558546272e+104

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 9.4

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

    if -8.039547558546272e+104 < re < -3.240034389533804e-262

    1. Initial program 19.5

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

    3. Applied add-sqr-sqrt19.5

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

    4. Applied sqrt-prod19.6

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

    6. Applied *-un-lft-identity19.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \color{blue}{\left(1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)} - re\right)}\]

    7. Applied associate-*r*19.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot 1\right) \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

    if -3.240034389533804e-262 < re < 5.7124459548238595e-130

    1. Initial program 31.7

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

    2. Taylor expanded around 0 35.0

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

    if 5.7124459548238595e-130 < re

    1. Initial program 52.0

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

    3. Applied flip--52.0

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

    4. Applied associate-*r/52.0

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

    5. Applied sqrt-div52.0

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

    6. Simplified37.1

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(0 + im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.039547558546271631628934502111036051442 \cdot 10^{104}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -3.240034389533803828723034610715031267776 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right) \cdot 2}\\ \mathbf{elif}\;re \le 5.712445954823859518117956639119879892462 \cdot 10^{-130}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))))