Average Error: 37.8 → 27.2
Time: 16.7s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.709153750764979 \cdot 10^{-176}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2.0}\\ \mathbf{elif}\;re \le 2.539134336777917 \cdot 10^{-238}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 7.000752032910662 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.709153750764979 \cdot 10^{-176}:\\
\;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2.0}\\

\mathbf{elif}\;re \le 2.539134336777917 \cdot 10^{-238}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 7.000752032910662 \cdot 10^{+122}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)\right) \cdot 2.0}\\

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

\end{array}
double f(double re, double im) {
        double r2784579 = 0.5;
        double r2784580 = 2.0;
        double r2784581 = re;
        double r2784582 = r2784581 * r2784581;
        double r2784583 = im;
        double r2784584 = r2784583 * r2784583;
        double r2784585 = r2784582 + r2784584;
        double r2784586 = sqrt(r2784585);
        double r2784587 = r2784586 + r2784581;
        double r2784588 = r2784580 * r2784587;
        double r2784589 = sqrt(r2784588);
        double r2784590 = r2784579 * r2784589;
        return r2784590;
}
↓
double f(double re, double im) {
        double r2784591 = re;
        double r2784592 = -2.709153750764979e-176;
        bool r2784593 = r2784591 <= r2784592;
        double r2784594 = 0.5;
        double r2784595 = im;
        double r2784596 = r2784595 * r2784595;
        double r2784597 = r2784591 * r2784591;
        double r2784598 = r2784597 + r2784596;
        double r2784599 = sqrt(r2784598);
        double r2784600 = r2784599 - r2784591;
        double r2784601 = r2784596 / r2784600;
        double r2784602 = 2.0;
        double r2784603 = r2784601 * r2784602;
        double r2784604 = sqrt(r2784603);
        double r2784605 = r2784594 * r2784604;
        double r2784606 = 2.539134336777917e-238;
        bool r2784607 = r2784591 <= r2784606;
        double r2784608 = r2784595 + r2784591;
        double r2784609 = r2784602 * r2784608;
        double r2784610 = sqrt(r2784609);
        double r2784611 = r2784594 * r2784610;
        double r2784612 = 7.000752032910662e+122;
        bool r2784613 = r2784591 <= r2784612;
        double r2784614 = sqrt(r2784599);
        double r2784615 = sqrt(r2784614);
        double r2784616 = r2784615 * r2784614;
        double r2784617 = r2784615 * r2784616;
        double r2784618 = r2784591 + r2784617;
        double r2784619 = r2784618 * r2784602;
        double r2784620 = sqrt(r2784619);
        double r2784621 = r2784594 * r2784620;
        double r2784622 = r2784591 + r2784591;
        double r2784623 = r2784602 * r2784622;
        double r2784624 = sqrt(r2784623);
        double r2784625 = r2784594 * r2784624;
        double r2784626 = r2784613 ? r2784621 : r2784625;
        double r2784627 = r2784607 ? r2784611 : r2784626;
        double r2784628 = r2784593 ? r2784605 : r2784627;
        return r2784628;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.8
Target33.0
Herbie27.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -2.709153750764979e-176

    1. Initial program 49.5

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

    3. Applied flip-+49.5

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

    4. Simplified37.1

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

    if -2.709153750764979e-176 < re < 2.539134336777917e-238

    1. Initial program 29.3

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

    3. Applied add-sqr-sqrt29.3

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

    4. Applied sqrt-prod29.5

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

    5. Taylor expanded around 0 33.5

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

    if 2.539134336777917e-238 < re < 7.000752032910662e+122

    1. Initial program 19.0

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

    3. Applied add-sqr-sqrt19.0

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

    4. Applied sqrt-prod19.1

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

    6. Applied add-sqr-sqrt19.1

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

    7. Applied sqrt-prod19.1

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

    8. Applied sqrt-prod19.1

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

    9. Applied associate-*r*19.1

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

    if 7.000752032910662e+122 < re

    1. Initial program 52.6

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

    2. Taylor expanded around inf 9.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.709153750764979 \cdot 10^{-176}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2.0}\\ \mathbf{elif}\;re \le 2.539134336777917 \cdot 10^{-238}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 7.000752032910662 \cdot 10^{+122}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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)))))