Average Error: 38.6 → 26.3
Time: 19.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 5.609857205188480997814633429622826314871 \cdot 10^{85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 5.609857205188480997814633429622826314871 \cdot 10^{85}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\

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

\end{array}
double f(double re, double im) {
        double r6601448 = 0.5;
        double r6601449 = 2.0;
        double r6601450 = re;
        double r6601451 = r6601450 * r6601450;
        double r6601452 = im;
        double r6601453 = r6601452 * r6601452;
        double r6601454 = r6601451 + r6601453;
        double r6601455 = sqrt(r6601454);
        double r6601456 = r6601455 + r6601450;
        double r6601457 = r6601449 * r6601456;
        double r6601458 = sqrt(r6601457);
        double r6601459 = r6601448 * r6601458;
        return r6601459;
}
↓
double f(double re, double im) {
        double r6601460 = re;
        double r6601461 = -9.064518896973367e-262;
        bool r6601462 = r6601460 <= r6601461;
        double r6601463 = im;
        double r6601464 = r6601463 * r6601463;
        double r6601465 = 2.0;
        double r6601466 = r6601464 * r6601465;
        double r6601467 = sqrt(r6601466);
        double r6601468 = r6601460 * r6601460;
        double r6601469 = r6601464 + r6601468;
        double r6601470 = sqrt(r6601469);
        double r6601471 = r6601470 - r6601460;
        double r6601472 = sqrt(r6601471);
        double r6601473 = r6601467 / r6601472;
        double r6601474 = 0.5;
        double r6601475 = r6601473 * r6601474;
        double r6601476 = 5.609857205188481e+85;
        bool r6601477 = r6601460 <= r6601476;
        double r6601478 = r6601460 + r6601470;
        double r6601479 = r6601465 * r6601478;
        double r6601480 = sqrt(r6601479);
        double r6601481 = r6601474 * r6601480;
        double r6601482 = r6601460 + r6601460;
        double r6601483 = r6601465 * r6601482;
        double r6601484 = sqrt(r6601483);
        double r6601485 = r6601474 * r6601484;
        double r6601486 = r6601477 ? r6601481 : r6601485;
        double r6601487 = r6601462 ? r6601475 : r6601486;
        return r6601487;
}

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
Target33.8
Herbie26.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -9.064518896973367e-262

    1. Initial program 47.2

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

    3. Applied add-exp-log49.4

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

    5. Applied flip-+49.3

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

    6. Applied associate-*r/49.3

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

    7. Applied sqrt-div49.3

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

    8. Simplified37.3

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

    9. Simplified36.2

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

    if -9.064518896973367e-262 < re < 5.609857205188481e+85

    1. Initial program 21.5

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

    if 5.609857205188481e+85 < re

    1. Initial program 48.8

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

    2. Taylor expanded around inf 10.6

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 5.609857205188480997814633429622826314871 \cdot 10^{85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))