Average Error: 37.6 → 22.9
Time: 21.8s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.92754656328539 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.0335100130965686 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{\frac{\left(im \cdot im\right) \cdot 2.0}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 1.7588975328913653 \cdot 10^{-163}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 3.0746636010417827 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.92754656328539 \cdot 10^{+93}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\

\mathbf{elif}\;re \le -2.0335100130965686 \cdot 10^{-243}:\\
\;\;\;\;\sqrt{\frac{\left(im \cdot im\right) \cdot 2.0}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 1.7588975328913653 \cdot 10^{-163}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 3.0746636010417827 \cdot 10^{+98}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\

\end{array}
double f(double re, double im) {
        double r38786566 = 0.5;
        double r38786567 = 2.0;
        double r38786568 = re;
        double r38786569 = r38786568 * r38786568;
        double r38786570 = im;
        double r38786571 = r38786570 * r38786570;
        double r38786572 = r38786569 + r38786571;
        double r38786573 = sqrt(r38786572);
        double r38786574 = r38786573 + r38786568;
        double r38786575 = r38786567 * r38786574;
        double r38786576 = sqrt(r38786575);
        double r38786577 = r38786566 * r38786576;
        return r38786577;
}
↓
double f(double re, double im) {
        double r38786578 = re;
        double r38786579 = -8.92754656328539e+93;
        bool r38786580 = r38786578 <= r38786579;
        double r38786581 = im;
        double r38786582 = r38786581 * r38786581;
        double r38786583 = 2.0;
        double r38786584 = r38786582 * r38786583;
        double r38786585 = sqrt(r38786584);
        double r38786586 = -2.0;
        double r38786587 = r38786586 * r38786578;
        double r38786588 = sqrt(r38786587);
        double r38786589 = r38786585 / r38786588;
        double r38786590 = 0.5;
        double r38786591 = r38786589 * r38786590;
        double r38786592 = -2.0335100130965686e-243;
        bool r38786593 = r38786578 <= r38786592;
        double r38786594 = r38786578 * r38786578;
        double r38786595 = r38786582 + r38786594;
        double r38786596 = sqrt(r38786595);
        double r38786597 = r38786596 - r38786578;
        double r38786598 = r38786584 / r38786597;
        double r38786599 = sqrt(r38786598);
        double r38786600 = r38786599 * r38786590;
        double r38786601 = 1.7588975328913653e-163;
        bool r38786602 = r38786578 <= r38786601;
        double r38786603 = r38786581 + r38786578;
        double r38786604 = r38786583 * r38786603;
        double r38786605 = sqrt(r38786604);
        double r38786606 = r38786590 * r38786605;
        double r38786607 = 3.0746636010417827e+98;
        bool r38786608 = r38786578 <= r38786607;
        double r38786609 = r38786596 + r38786578;
        double r38786610 = r38786583 * r38786609;
        double r38786611 = sqrt(r38786610);
        double r38786612 = r38786590 * r38786611;
        double r38786613 = r38786578 + r38786578;
        double r38786614 = r38786613 * r38786583;
        double r38786615 = sqrt(r38786614);
        double r38786616 = r38786590 * r38786615;
        double r38786617 = r38786608 ? r38786612 : r38786616;
        double r38786618 = r38786602 ? r38786606 : r38786617;
        double r38786619 = r38786593 ? r38786600 : r38786618;
        double r38786620 = r38786580 ? r38786591 : r38786619;
        return r38786620;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.8
Herbie22.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -8.92754656328539e+93

    1. Initial program 59.5

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+59.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Applied associate-*r/59.5

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    5. Applied sqrt-div59.5

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    6. Simplified42.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    7. Taylor expanded around -inf 22.4

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -8.92754656328539e+93 < re < -2.0335100130965686e-243

    1. Initial program 39.2

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+39.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Applied associate-*r/39.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    5. Applied sqrt-div39.3

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    6. Simplified30.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-undiv30.6

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\frac{2.0 \cdot \left(im \cdot im\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    if -2.0335100130965686e-243 < re < 1.7588975328913653e-163

    1. Initial program 28.9

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around 0 34.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(re + im\right)}}\]

    if 1.7588975328913653e-163 < re < 3.0746636010417827e+98

    1. Initial program 15.0

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    if 3.0746636010417827e+98 < re

    1. Initial program 49.3

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-exp-log50.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]

    4. Taylor expanded around inf 9.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification22.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.92754656328539 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.0335100130965686 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{\frac{\left(im \cdot im\right) \cdot 2.0}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 1.7588975328913653 \cdot 10^{-163}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 3.0746636010417827 \cdot 10^{+98}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (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)))))