Average Error: 38.6 → 26.4
Time: 14.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.258700308413734263198473092360990096615 \cdot 10^{-288}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.015933577747527344666564222799068286537 \cdot 10^{138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.258700308413734263198473092360990096615 \cdot 10^{-288}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 1.015933577747527344666564222799068286537 \cdot 10^{138}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r95495 = 0.5;
        double r95496 = 2.0;
        double r95497 = re;
        double r95498 = r95497 * r95497;
        double r95499 = im;
        double r95500 = r95499 * r95499;
        double r95501 = r95498 + r95500;
        double r95502 = sqrt(r95501);
        double r95503 = r95502 + r95497;
        double r95504 = r95496 * r95503;
        double r95505 = sqrt(r95504);
        double r95506 = r95495 * r95505;
        return r95506;
}
↓
double f(double re, double im) {
        double r95507 = re;
        double r95508 = -3.2587003084137343e-288;
        bool r95509 = r95507 <= r95508;
        double r95510 = 0.5;
        double r95511 = im;
        double r95512 = r95511 * r95511;
        double r95513 = 2.0;
        double r95514 = r95512 * r95513;
        double r95515 = sqrt(r95514);
        double r95516 = r95507 * r95507;
        double r95517 = r95516 + r95512;
        double r95518 = sqrt(r95517);
        double r95519 = r95518 - r95507;
        double r95520 = sqrt(r95519);
        double r95521 = r95515 / r95520;
        double r95522 = r95510 * r95521;
        double r95523 = 1.0159335777475273e+138;
        bool r95524 = r95507 <= r95523;
        double r95525 = sqrt(r95518);
        double r95526 = cbrt(r95517);
        double r95527 = fabs(r95526);
        double r95528 = sqrt(r95527);
        double r95529 = r95525 * r95528;
        double r95530 = sqrt(r95526);
        double r95531 = sqrt(r95530);
        double r95532 = r95529 * r95531;
        double r95533 = r95532 + r95507;
        double r95534 = r95513 * r95533;
        double r95535 = sqrt(r95534);
        double r95536 = r95510 * r95535;
        double r95537 = r95507 + r95507;
        double r95538 = r95513 * r95537;
        double r95539 = sqrt(r95538);
        double r95540 = r95510 * r95539;
        double r95541 = r95524 ? r95536 : r95540;
        double r95542 = r95509 ? r95522 : r95541;
        return r95542;
}

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
Target33.7
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -3.2587003084137343e-288

    1. Initial program 46.5

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

    3. Applied flip-+46.4

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

    4. Applied associate-*r/46.4

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

    5. Applied sqrt-div46.5

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

    6. Simplified35.4

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

    if -3.2587003084137343e-288 < re < 1.0159335777475273e+138

    1. Initial program 20.9

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

    3. Applied add-sqr-sqrt20.9

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

    4. Applied sqrt-prod21.0

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

    6. Applied add-cube-cbrt21.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    7. Applied sqrt-prod21.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    8. Applied sqrt-prod21.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \color{blue}{\left(\sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right)} + re\right)}\]

    9. Applied associate-*r*21.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    10. Simplified21.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|}\right)} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    if 1.0159335777475273e+138 < re

    1. Initial program 58.8

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.258700308413734263198473092360990096615 \cdot 10^{-288}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.015933577747527344666564222799068286537 \cdot 10^{138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))