Average Error: 38.3 → 18.9
Time: 14.2s
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 r3238606 = 0.5;
        double r3238607 = 2.0;
        double r3238608 = re;
        double r3238609 = r3238608 * r3238608;
        double r3238610 = im;
        double r3238611 = r3238610 * r3238610;
        double r3238612 = r3238609 + r3238611;
        double r3238613 = sqrt(r3238612);
        double r3238614 = r3238613 + r3238608;
        double r3238615 = r3238607 * r3238614;
        double r3238616 = sqrt(r3238615);
        double r3238617 = r3238606 * r3238616;
        return r3238617;
}
↓
double f(double re, double im) {
        double r3238618 = re;
        double r3238619 = -6.084312799804569e+102;
        bool r3238620 = r3238618 <= r3238619;
        double r3238621 = 2.0;
        double r3238622 = im;
        double r3238623 = r3238622 * r3238622;
        double r3238624 = r3238621 * r3238623;
        double r3238625 = sqrt(r3238624);
        double r3238626 = -2.0;
        double r3238627 = r3238626 * r3238618;
        double r3238628 = sqrt(r3238627);
        double r3238629 = r3238625 / r3238628;
        double r3238630 = 0.5;
        double r3238631 = r3238629 * r3238630;
        double r3238632 = 4.5556084918187003e-234;
        bool r3238633 = r3238618 <= r3238632;
        double r3238634 = fabs(r3238622);
        double r3238635 = r3238618 * r3238618;
        double r3238636 = r3238635 + r3238623;
        double r3238637 = sqrt(r3238636);
        double r3238638 = r3238637 - r3238618;
        double r3238639 = sqrt(r3238638);
        double r3238640 = r3238634 / r3238639;
        double r3238641 = sqrt(r3238621);
        double r3238642 = r3238640 * r3238641;
        double r3238643 = r3238630 * r3238642;
        double r3238644 = 6.306382616635563e-190;
        bool r3238645 = r3238618 <= r3238644;
        double r3238646 = r3238622 + r3238618;
        double r3238647 = r3238646 * r3238621;
        double r3238648 = sqrt(r3238647);
        double r3238649 = r3238630 * r3238648;
        double r3238650 = 1.5059919692642864e+126;
        bool r3238651 = r3238618 <= r3238650;
        double r3238652 = sqrt(r3238637);
        double r3238653 = r3238652 * r3238652;
        double r3238654 = r3238618 + r3238653;
        double r3238655 = r3238621 * r3238654;
        double r3238656 = sqrt(r3238655);
        double r3238657 = r3238630 * r3238656;
        double r3238658 = r3238618 + r3238618;
        double r3238659 = r3238621 * r3238658;
        double r3238660 = sqrt(r3238659);
        double r3238661 = r3238630 * r3238660;
        double r3238662 = r3238651 ? r3238657 : r3238661;
        double r3238663 = r3238645 ? r3238649 : r3238662;
        double r3238664 = r3238633 ? r3238643 : r3238663;
        double r3238665 = r3238620 ? r3238631 : r3238664;
        return r3238665;
}

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. Using strategy rm

    3. Applied add-sqr-sqrt31.4

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

    4. Applied sqrt-prod31.5

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

    5. 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-sqrt16.4

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

    4. Applied sqrt-prod16.5

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

    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. 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)))))