Average Error: 38.3 → 18.9
Time: 17.1s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.084312799804569 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 4.5556084918187003 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2.0}\right)\\ \mathbf{elif}\;re \le 6.306382616635563 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 1.5059919692642864 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.084312799804569 \cdot 10^{+102}:\\
\;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\

\mathbf{elif}\;re \le 4.5556084918187003 \cdot 10^{-234}:\\
\;\;\;\;0.5 \cdot \left(\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2.0}\right)\\

\mathbf{elif}\;re \le 6.306382616635563 \cdot 10^{-190}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2.0}\\

\mathbf{elif}\;re \le 1.5059919692642864 \cdot 10^{+126}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}\\

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

\end{array}
double f(double re, double im) {
        double r3290459 = 0.5;
        double r3290460 = 2.0;
        double r3290461 = re;
        double r3290462 = r3290461 * r3290461;
        double r3290463 = im;
        double r3290464 = r3290463 * r3290463;
        double r3290465 = r3290462 + r3290464;
        double r3290466 = sqrt(r3290465);
        double r3290467 = r3290466 + r3290461;
        double r3290468 = r3290460 * r3290467;
        double r3290469 = sqrt(r3290468);
        double r3290470 = r3290459 * r3290469;
        return r3290470;
}
↓
double f(double re, double im) {
        double r3290471 = re;
        double r3290472 = -6.084312799804569e+102;
        bool r3290473 = r3290471 <= r3290472;
        double r3290474 = 2.0;
        double r3290475 = im;
        double r3290476 = r3290475 * r3290475;
        double r3290477 = r3290474 * r3290476;
        double r3290478 = sqrt(r3290477);
        double r3290479 = -2.0;
        double r3290480 = r3290479 * r3290471;
        double r3290481 = sqrt(r3290480);
        double r3290482 = r3290478 / r3290481;
        double r3290483 = 0.5;
        double r3290484 = r3290482 * r3290483;
        double r3290485 = 4.5556084918187003e-234;
        bool r3290486 = r3290471 <= r3290485;
        double r3290487 = fabs(r3290475);
        double r3290488 = r3290471 * r3290471;
        double r3290489 = r3290488 + r3290476;
        double r3290490 = sqrt(r3290489);
        double r3290491 = r3290490 - r3290471;
        double r3290492 = sqrt(r3290491);
        double r3290493 = r3290487 / r3290492;
        double r3290494 = sqrt(r3290474);
        double r3290495 = r3290493 * r3290494;
        double r3290496 = r3290483 * r3290495;
        double r3290497 = 6.306382616635563e-190;
        bool r3290498 = r3290471 <= r3290497;
        double r3290499 = r3290475 + r3290471;
        double r3290500 = r3290499 * r3290474;
        double r3290501 = sqrt(r3290500);
        double r3290502 = r3290483 * r3290501;
        double r3290503 = 1.5059919692642864e+126;
        bool r3290504 = r3290471 <= r3290503;
        double r3290505 = sqrt(r3290490);
        double r3290506 = r3290505 * r3290505;
        double r3290507 = r3290471 + r3290506;
        double r3290508 = r3290474 * r3290507;
        double r3290509 = sqrt(r3290508);
        double r3290510 = r3290483 * r3290509;
        double r3290511 = r3290471 + r3290471;
        double r3290512 = r3290474 * r3290511;
        double r3290513 = sqrt(r3290512);
        double r3290514 = r3290483 * r3290513;
        double r3290515 = r3290504 ? r3290510 : r3290514;
        double r3290516 = r3290498 ? r3290502 : r3290515;
        double r3290517 = r3290486 ? r3290496 : r3290516;
        double r3290518 = r3290473 ? r3290484 : r3290517;
        return r3290518;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.2
Herbie18.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -6.084312799804569e+102

    1. Initial program 60.0

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

    3. Applied flip-+60.0

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

    4. Applied associate-*r/60.0

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

    5. Applied sqrt-div60.0

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

    6. Simplified44.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    7. Taylor expanded around -inf 20.2

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -6.084312799804569e+102 < re < 4.5556084918187003e-234

    1. Initial program 36.9

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

    3. Applied flip-+37.0

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

    4. Applied associate-*r/37.0

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

    5. Applied sqrt-div37.2

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

    6. Simplified30.1

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity30.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

    9. Applied sqrt-prod30.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    10. Applied sqrt-prod30.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    11. Applied times-frac30.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2.0}}{\sqrt{1}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

    12. Simplified30.2

      \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{2.0}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

    13. Simplified22.1

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

    if 4.5556084918187003e-234 < re < 6.306382616635563e-190

    1. Initial program 31.4

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

    2. Taylor expanded around 0 30.9

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

    if 6.306382616635563e-190 < re < 1.5059919692642864e+126

    1. Initial program 16.4

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

    3. Applied add-sqr-sqrt16.5

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

    if 1.5059919692642864e+126 < re

    1. Initial program 55.3

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

    3. Applied add-sqr-sqrt55.3

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

    4. Taylor expanded around inf 9.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.084312799804569 \cdot 10^{+102}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 4.5556084918187003 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2.0}\right)\\ \mathbf{elif}\;re \le 6.306382616635563 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im + re\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 1.5059919692642864 \cdot 10^{+126}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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)))))