Average Error: 38.2 → 26.3
Time: 17.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.896078043343783 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.563508143180615 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.1909114483375085 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re} \cdot 2.0}\\ \mathbf{elif}\;re \le 1.4105722211090752 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 4.4479284942813524 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\sqrt{im \cdot im + re \cdot re} + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.896078043343783 \cdot 10^{-117}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le -2.563508143180615 \cdot 10^{-158}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 1.1909114483375085 \cdot 10^{-307}:\\
\;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re} \cdot 2.0}\\

\mathbf{elif}\;re \le 1.4105722211090752 \cdot 10^{-227}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 4.4479284942813524 \cdot 10^{+142}:\\
\;\;\;\;\sqrt{\left(\sqrt{im \cdot im + re \cdot re} + re\right) \cdot 2.0} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r5910572 = 0.5;
        double r5910573 = 2.0;
        double r5910574 = re;
        double r5910575 = r5910574 * r5910574;
        double r5910576 = im;
        double r5910577 = r5910576 * r5910576;
        double r5910578 = r5910575 + r5910577;
        double r5910579 = sqrt(r5910578);
        double r5910580 = r5910579 + r5910574;
        double r5910581 = r5910573 * r5910580;
        double r5910582 = sqrt(r5910581);
        double r5910583 = r5910572 * r5910582;
        return r5910583;
}
↓
double f(double re, double im) {
        double r5910584 = re;
        double r5910585 = -3.896078043343783e-117;
        bool r5910586 = r5910584 <= r5910585;
        double r5910587 = 0.5;
        double r5910588 = 2.0;
        double r5910589 = im;
        double r5910590 = r5910589 * r5910589;
        double r5910591 = r5910588 * r5910590;
        double r5910592 = sqrt(r5910591);
        double r5910593 = r5910584 * r5910584;
        double r5910594 = r5910590 + r5910593;
        double r5910595 = sqrt(r5910594);
        double r5910596 = r5910595 - r5910584;
        double r5910597 = sqrt(r5910596);
        double r5910598 = r5910592 / r5910597;
        double r5910599 = r5910587 * r5910598;
        double r5910600 = -2.563508143180615e-158;
        bool r5910601 = r5910584 <= r5910600;
        double r5910602 = r5910589 + r5910584;
        double r5910603 = r5910588 * r5910602;
        double r5910604 = sqrt(r5910603);
        double r5910605 = r5910587 * r5910604;
        double r5910606 = 1.1909114483375085e-307;
        bool r5910607 = r5910584 <= r5910606;
        double r5910608 = r5910590 / r5910596;
        double r5910609 = r5910608 * r5910588;
        double r5910610 = sqrt(r5910609);
        double r5910611 = r5910587 * r5910610;
        double r5910612 = 1.4105722211090752e-227;
        bool r5910613 = r5910584 <= r5910612;
        double r5910614 = 4.4479284942813524e+142;
        bool r5910615 = r5910584 <= r5910614;
        double r5910616 = r5910595 + r5910584;
        double r5910617 = r5910616 * r5910588;
        double r5910618 = sqrt(r5910617);
        double r5910619 = r5910618 * r5910587;
        double r5910620 = r5910584 + r5910584;
        double r5910621 = r5910620 * r5910588;
        double r5910622 = sqrt(r5910621);
        double r5910623 = r5910587 * r5910622;
        double r5910624 = r5910615 ? r5910619 : r5910623;
        double r5910625 = r5910613 ? r5910605 : r5910624;
        double r5910626 = r5910607 ? r5910611 : r5910625;
        double r5910627 = r5910601 ? r5910605 : r5910626;
        double r5910628 = r5910586 ? r5910599 : r5910627;
        return r5910628;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.2
Target33.0
Herbie26.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -3.896078043343783e-117

    1. Initial program 50.7

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

    3. Applied flip-+50.7

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

    4. Applied associate-*r/50.7

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

    5. Applied sqrt-div50.8

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

    6. Simplified35.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right) + 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    if -3.896078043343783e-117 < re < -2.563508143180615e-158 or 1.1909114483375085e-307 < re < 1.4105722211090752e-227

    1. Initial program 31.2

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

    3. Applied add-exp-log33.0

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

    5. Applied pow133.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(e^{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}} + re\right)}\]

    6. Applied log-pow33.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(e^{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]

    7. Applied exp-prod33.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} + re\right)}\]

    8. Simplified33.0

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

    9. Taylor expanded around 0 34.9

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

    if -2.563508143180615e-158 < re < 1.1909114483375085e-307

    1. Initial program 30.3

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

    3. Applied add-exp-log32.3

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

    5. Applied flip-+31.9

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

    6. Simplified31.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im + 0}}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re}}\]

    7. Simplified30.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im + 0}{\color{blue}{\sqrt{im \cdot im + re \cdot re} - re}}}\]

    if 1.4105722211090752e-227 < re < 4.4479284942813524e+142

    1. Initial program 18.8

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

    if 4.4479284942813524e+142 < re

    1. Initial program 58.6

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

    3. Applied add-exp-log58.8

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

    5. Applied pow158.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(e^{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}} + re\right)}\]

    6. Applied log-pow58.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(e^{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]

    7. Applied exp-prod58.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} + re\right)}\]

    8. Simplified58.8

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

    9. Taylor expanded around inf 8.8

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.896078043343783 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.563508143180615 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.1909114483375085 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re} \cdot 2.0}\\ \mathbf{elif}\;re \le 1.4105722211090752 \cdot 10^{-227}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 4.4479284942813524 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\sqrt{im \cdot im + re \cdot re} + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(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)))))