Average Error: 38.8 → 19.8
Time: 11.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.3738667209186418 \cdot 10^{156}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\left(-re\right) - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.3650813229174731 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}\\ \mathbf{elif}\;re \le 3.3295675416621997 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 5.202316813475107 \cdot 10^{92}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.3738667209186418 \cdot 10^{156}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\left(-re\right) - re}} \cdot 0.5\\

\mathbf{elif}\;re \le -2.3650813229174731 \cdot 10^{-264}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}\\

\mathbf{elif}\;re \le 3.3295675416621997 \cdot 10^{-138}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 5.202316813475107 \cdot 10^{92}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{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 r232615 = 0.5;
        double r232616 = 2.0;
        double r232617 = re;
        double r232618 = r232617 * r232617;
        double r232619 = im;
        double r232620 = r232619 * r232619;
        double r232621 = r232618 + r232620;
        double r232622 = sqrt(r232621);
        double r232623 = r232622 + r232617;
        double r232624 = r232616 * r232623;
        double r232625 = sqrt(r232624);
        double r232626 = r232615 * r232625;
        return r232626;
}
↓
double f(double re, double im) {
        double r232627 = re;
        double r232628 = -3.373866720918642e+156;
        bool r232629 = r232627 <= r232628;
        double r232630 = im;
        double r232631 = r232630 * r232630;
        double r232632 = 2.0;
        double r232633 = r232631 * r232632;
        double r232634 = sqrt(r232633);
        double r232635 = -r232627;
        double r232636 = r232635 - r232627;
        double r232637 = sqrt(r232636);
        double r232638 = r232634 / r232637;
        double r232639 = 0.5;
        double r232640 = r232638 * r232639;
        double r232641 = -2.365081322917473e-264;
        bool r232642 = r232627 <= r232641;
        double r232643 = sqrt(r232632);
        double r232644 = r232627 * r232627;
        double r232645 = r232644 + r232631;
        double r232646 = sqrt(r232645);
        double r232647 = r232646 - r232627;
        double r232648 = sqrt(r232647);
        double r232649 = fabs(r232630);
        double r232650 = r232648 / r232649;
        double r232651 = r232643 / r232650;
        double r232652 = r232639 * r232651;
        double r232653 = 3.3295675416622e-138;
        bool r232654 = r232627 <= r232653;
        double r232655 = r232627 + r232630;
        double r232656 = r232632 * r232655;
        double r232657 = sqrt(r232656);
        double r232658 = r232639 * r232657;
        double r232659 = 5.202316813475107e+92;
        bool r232660 = r232627 <= r232659;
        double r232661 = sqrt(r232646);
        double r232662 = r232661 * r232661;
        double r232663 = r232662 + r232627;
        double r232664 = r232632 * r232663;
        double r232665 = sqrt(r232664);
        double r232666 = r232639 * r232665;
        double r232667 = r232627 + r232627;
        double r232668 = r232632 * r232667;
        double r232669 = sqrt(r232668);
        double r232670 = r232639 * r232669;
        double r232671 = r232660 ? r232666 : r232670;
        double r232672 = r232654 ? r232658 : r232671;
        double r232673 = r232642 ? r232652 : r232672;
        double r232674 = r232629 ? r232640 : r232673;
        return r232674;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target33.8
Herbie19.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -3.373866720918642e+156

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Applied associate-*r/64.0

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

    5. Applied sqrt-div64.0

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

    6. Simplified50.4

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

    7. Taylor expanded around -inf 18.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(0 + im \cdot im\right)}}{\sqrt{\color{blue}{-1 \cdot re} - re}}\]

    8. Simplified18.6

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

    if -3.373866720918642e+156 < re < -2.365081322917473e-264

    1. Initial program 40.8

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

    3. Applied flip-+40.7

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

    4. Applied associate-*r/40.7

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

    5. Applied sqrt-div40.8

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

    6. Simplified29.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(0 + im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod30.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{0 + im \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    9. Applied associate-/l*30.0

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{0 + im \cdot im}}}}\]

    10. Simplified19.5

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

    if -2.365081322917473e-264 < re < 3.3295675416622e-138

    1. Initial program 29.0

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

    3. Applied add-sqr-sqrt29.0

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

    4. Applied sqrt-prod29.2

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

    5. Taylor expanded around 0 34.3

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

    if 3.3295675416622e-138 < re < 5.202316813475107e+92

    1. Initial program 16.6

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

    3. Applied add-sqr-sqrt16.6

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

    4. Applied sqrt-prod16.6

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

    if 5.202316813475107e+92 < re

    1. Initial program 49.9

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

    3. Applied add-sqr-sqrt49.9

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

    4. Applied sqrt-prod49.9

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

    5. Taylor expanded around inf 10.3

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

  4. Final simplification19.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.3738667209186418 \cdot 10^{156}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\left(-re\right) - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.3650813229174731 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\left|im\right|}}\\ \mathbf{elif}\;re \le 3.3295675416621997 \cdot 10^{-138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 5.202316813475107 \cdot 10^{92}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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)))))