Average Error: 38.6 → 26.3
Time: 16.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le 5.609857205188480997814633429622826314871 \cdot 10^{85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le 5.609857205188480997814633429622826314871 \cdot 10^{85}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r6332785 = 0.5;
        double r6332786 = 2.0;
        double r6332787 = re;
        double r6332788 = r6332787 * r6332787;
        double r6332789 = im;
        double r6332790 = r6332789 * r6332789;
        double r6332791 = r6332788 + r6332790;
        double r6332792 = sqrt(r6332791);
        double r6332793 = r6332792 + r6332787;
        double r6332794 = r6332786 * r6332793;
        double r6332795 = sqrt(r6332794);
        double r6332796 = r6332785 * r6332795;
        return r6332796;
}
↓
double f(double re, double im) {
        double r6332797 = re;
        double r6332798 = -9.064518896973367e-262;
        bool r6332799 = r6332797 <= r6332798;
        double r6332800 = 0.5;
        double r6332801 = 2.0;
        double r6332802 = im;
        double r6332803 = r6332802 * r6332802;
        double r6332804 = r6332801 * r6332803;
        double r6332805 = sqrt(r6332804);
        double r6332806 = r6332797 * r6332797;
        double r6332807 = r6332803 + r6332806;
        double r6332808 = sqrt(r6332807);
        double r6332809 = r6332808 - r6332797;
        double r6332810 = sqrt(r6332809);
        double r6332811 = r6332805 / r6332810;
        double r6332812 = r6332800 * r6332811;
        double r6332813 = 5.609857205188481e+85;
        bool r6332814 = r6332797 <= r6332813;
        double r6332815 = r6332808 + r6332797;
        double r6332816 = r6332801 * r6332815;
        double r6332817 = sqrt(r6332816);
        double r6332818 = r6332800 * r6332817;
        double r6332819 = r6332797 + r6332797;
        double r6332820 = r6332801 * r6332819;
        double r6332821 = sqrt(r6332820);
        double r6332822 = r6332800 * r6332821;
        double r6332823 = r6332814 ? r6332818 : r6332822;
        double r6332824 = r6332799 ? r6332812 : r6332823;
        return r6332824;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.8
Herbie26.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -9.064518896973367e-262

    1. Initial program 47.2

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

    3. Applied flip-+47.1

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

    4. Applied associate-*r/47.2

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

    5. Applied sqrt-div47.2

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

    6. Simplified36.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    if -9.064518896973367e-262 < re < 5.609857205188481e+85

    1. Initial program 21.5

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

    if 5.609857205188481e+85 < re

    1. Initial program 48.8

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

    2. Taylor expanded around inf 10.6

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le 5.609857205188480997814633429622826314871 \cdot 10^{85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))