Average Error: 38.5 → 22.6
Time: 17.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\\ \mathbf{elif}\;re \le 5.124751274050741168628571362640123162884 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.280297657817536289043603160829670533045 \cdot 10^{-204} \lor \neg \left(re \le 9.727118253535961652403013059453411638468 \cdot 10^{-160}\right) \land re \le 4.202834506095946744840619038062984088453 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{2}} \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\
\;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{-2 \cdot re}}\\

\mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\
\;\;\;\;0.5 \cdot \frac{\frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\\

\mathbf{elif}\;re \le 5.124751274050741168628571362640123162884 \cdot 10^{-246}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 1.280297657817536289043603160829670533045 \cdot 10^{-204} \lor \neg \left(re \le 9.727118253535961652403013059453411638468 \cdot 10^{-160}\right) \land re \le 4.202834506095946744840619038062984088453 \cdot 10^{-94}:\\
\;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{2}} \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r161499 = 0.5;
        double r161500 = 2.0;
        double r161501 = re;
        double r161502 = r161501 * r161501;
        double r161503 = im;
        double r161504 = r161503 * r161503;
        double r161505 = r161502 + r161504;
        double r161506 = sqrt(r161505);
        double r161507 = r161506 + r161501;
        double r161508 = r161500 * r161507;
        double r161509 = sqrt(r161508);
        double r161510 = r161499 * r161509;
        return r161510;
}
↓
double f(double re, double im) {
        double r161511 = re;
        double r161512 = -5.330091552844717e+114;
        bool r161513 = r161511 <= r161512;
        double r161514 = 0.5;
        double r161515 = im;
        double r161516 = fabs(r161515);
        double r161517 = 2.0;
        double r161518 = sqrt(r161517);
        double r161519 = r161516 * r161518;
        double r161520 = -2.0;
        double r161521 = r161520 * r161511;
        double r161522 = sqrt(r161521);
        double r161523 = r161519 / r161522;
        double r161524 = r161514 * r161523;
        double r161525 = -4.2156616274993736e-144;
        bool r161526 = r161511 <= r161525;
        double r161527 = r161511 * r161511;
        double r161528 = r161515 * r161515;
        double r161529 = r161527 + r161528;
        double r161530 = sqrt(r161529);
        double r161531 = r161530 - r161511;
        double r161532 = sqrt(r161531);
        double r161533 = sqrt(r161532);
        double r161534 = r161519 / r161533;
        double r161535 = r161534 / r161533;
        double r161536 = r161514 * r161535;
        double r161537 = 5.124751274050741e-246;
        bool r161538 = r161511 <= r161537;
        double r161539 = r161511 + r161515;
        double r161540 = r161517 * r161539;
        double r161541 = sqrt(r161540);
        double r161542 = r161514 * r161541;
        double r161543 = 1.2802976578175363e-204;
        bool r161544 = r161511 <= r161543;
        double r161545 = 9.727118253535962e-160;
        bool r161546 = r161511 <= r161545;
        double r161547 = !r161546;
        double r161548 = 4.202834506095947e-94;
        bool r161549 = r161511 <= r161548;
        bool r161550 = r161547 && r161549;
        bool r161551 = r161544 || r161550;
        double r161552 = r161531 / r161517;
        double r161553 = r161515 / r161552;
        double r161554 = r161553 * r161515;
        double r161555 = sqrt(r161554);
        double r161556 = r161514 * r161555;
        double r161557 = r161511 + r161511;
        double r161558 = r161517 * r161557;
        double r161559 = sqrt(r161558);
        double r161560 = r161514 * r161559;
        double r161561 = r161551 ? r161556 : r161560;
        double r161562 = r161538 ? r161542 : r161561;
        double r161563 = r161526 ? r161536 : r161562;
        double r161564 = r161513 ? r161524 : r161563;
        return r161564;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.5
Target33.3
Herbie22.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -5.330091552844717e+114

    1. Initial program 61.8

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

    3. Applied flip-+61.8

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

    4. Applied associate-*r/61.9

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

    5. Applied sqrt-div61.9

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

    6. Simplified45.4

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

    8. Applied sqrt-prod45.3

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

    9. Simplified43.1

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

    10. Taylor expanded around -inf 8.9

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

    if -5.330091552844717e+114 < re < -4.2156616274993736e-144

    1. Initial program 43.2

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

    3. Applied flip-+43.2

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

    4. Applied associate-*r/43.4

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

    5. Applied sqrt-div43.5

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

    6. Simplified28.4

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

    8. Applied sqrt-prod28.3

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

    9. Simplified15.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\left|im\right|} \cdot \sqrt{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    10. Using strategy rm

    11. Applied add-sqr-sqrt15.5

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

    12. Applied sqrt-prod15.7

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

    13. Applied associate-/r*15.7

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

    if -4.2156616274993736e-144 < re < 5.124751274050741e-246

    1. Initial program 31.6

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

    2. Taylor expanded around 0 36.2

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

    if 5.124751274050741e-246 < re < 1.2802976578175363e-204 or 9.727118253535962e-160 < re < 4.202834506095947e-94

    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 flip-+33.1

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

    4. Applied associate-*r/33.2

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

    5. Applied sqrt-div33.6

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

    6. Simplified33.6

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

    8. Applied sqrt-undiv33.2

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

    9. Simplified32.8

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

    if 1.2802976578175363e-204 < re < 9.727118253535962e-160 or 4.202834506095947e-94 < re

    1. Initial program 33.3

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

    2. Taylor expanded around inf 23.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification22.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \frac{\frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\\ \mathbf{elif}\;re \le 5.124751274050741168628571362640123162884 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.280297657817536289043603160829670533045 \cdot 10^{-204} \lor \neg \left(re \le 9.727118253535961652403013059453411638468 \cdot 10^{-160}\right) \land re \le 4.202834506095946744840619038062984088453 \cdot 10^{-94}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{2}} \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))))