Average Error: 38.9 → 26.8
Time: 4.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.83207374814195553 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.7097317442318103 \cdot 10^{134}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.83207374814195553 \cdot 10^{-305}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 1.7097317442318103 \cdot 10^{134}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \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 r121474 = 0.5;
        double r121475 = 2.0;
        double r121476 = re;
        double r121477 = r121476 * r121476;
        double r121478 = im;
        double r121479 = r121478 * r121478;
        double r121480 = r121477 + r121479;
        double r121481 = sqrt(r121480);
        double r121482 = r121481 + r121476;
        double r121483 = r121475 * r121482;
        double r121484 = sqrt(r121483);
        double r121485 = r121474 * r121484;
        return r121485;
}
↓
double f(double re, double im) {
        double r121486 = re;
        double r121487 = -3.8320737481419555e-305;
        bool r121488 = r121486 <= r121487;
        double r121489 = 0.5;
        double r121490 = 2.0;
        double r121491 = im;
        double r121492 = r121491 * r121491;
        double r121493 = r121486 * r121486;
        double r121494 = r121493 + r121492;
        double r121495 = sqrt(r121494);
        double r121496 = r121495 - r121486;
        double r121497 = r121492 / r121496;
        double r121498 = r121490 * r121497;
        double r121499 = sqrt(r121498);
        double r121500 = r121489 * r121499;
        double r121501 = 1.7097317442318103e+134;
        bool r121502 = r121486 <= r121501;
        double r121503 = cbrt(r121494);
        double r121504 = fabs(r121503);
        double r121505 = sqrt(r121503);
        double r121506 = r121504 * r121505;
        double r121507 = r121506 + r121486;
        double r121508 = r121490 * r121507;
        double r121509 = sqrt(r121508);
        double r121510 = r121489 * r121509;
        double r121511 = r121486 + r121486;
        double r121512 = r121490 * r121511;
        double r121513 = sqrt(r121512);
        double r121514 = r121489 * r121513;
        double r121515 = r121502 ? r121510 : r121514;
        double r121516 = r121488 ? r121500 : r121515;
        return r121516;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.8
Herbie26.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -3.8320737481419555e-305

    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-+46.3

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

    4. Simplified36.1

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

    if -3.8320737481419555e-305 < re < 1.7097317442318103e+134

    1. Initial program 20.8

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

    3. Applied add-cube-cbrt21.0

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

    4. Applied sqrt-prod21.0

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

    5. Simplified21.0

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

    if 1.7097317442318103e+134 < re

    1. Initial program 58.7

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

    2. Taylor expanded around inf 9.3

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

  4. Final simplification26.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.83207374814195553 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.7097317442318103 \cdot 10^{134}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))))