Average Error: 38.4 → 27.2
Time: 14.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.756706552252228103614804055367357436769 \cdot 10^{164}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \le -1.016798043080117215536296671868401307267 \cdot 10^{113}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + im\right)} \cdot 0.5\\ \mathbf{elif}\;re \le -5.204735522622992940875512007096881566611 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.756706552252228103614804055367357436769 \cdot 10^{164}:\\
\;\;\;\;0\\

\mathbf{elif}\;re \le -1.016798043080117215536296671868401307267 \cdot 10^{113}:\\
\;\;\;\;\sqrt{2 \cdot \left(re + im\right)} \cdot 0.5\\

\mathbf{elif}\;re \le -5.204735522622992940875512007096881566611 \cdot 10^{-274}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r6450622 = 0.5;
        double r6450623 = 2.0;
        double r6450624 = re;
        double r6450625 = r6450624 * r6450624;
        double r6450626 = im;
        double r6450627 = r6450626 * r6450626;
        double r6450628 = r6450625 + r6450627;
        double r6450629 = sqrt(r6450628);
        double r6450630 = r6450629 + r6450624;
        double r6450631 = r6450623 * r6450630;
        double r6450632 = sqrt(r6450631);
        double r6450633 = r6450622 * r6450632;
        return r6450633;
}
↓
double f(double re, double im) {
        double r6450634 = re;
        double r6450635 = -1.756706552252228e+164;
        bool r6450636 = r6450634 <= r6450635;
        double r6450637 = 0.0;
        double r6450638 = -1.0167980430801172e+113;
        bool r6450639 = r6450634 <= r6450638;
        double r6450640 = 2.0;
        double r6450641 = im;
        double r6450642 = r6450634 + r6450641;
        double r6450643 = r6450640 * r6450642;
        double r6450644 = sqrt(r6450643);
        double r6450645 = 0.5;
        double r6450646 = r6450644 * r6450645;
        double r6450647 = -5.204735522622993e-274;
        bool r6450648 = r6450634 <= r6450647;
        double r6450649 = r6450641 * r6450641;
        double r6450650 = r6450640 * r6450649;
        double r6450651 = sqrt(r6450650);
        double r6450652 = r6450634 * r6450634;
        double r6450653 = r6450649 + r6450652;
        double r6450654 = sqrt(r6450653);
        double r6450655 = r6450654 - r6450634;
        double r6450656 = sqrt(r6450655);
        double r6450657 = r6450651 / r6450656;
        double r6450658 = r6450645 * r6450657;
        double r6450659 = 7.747777771049568e+94;
        bool r6450660 = r6450634 <= r6450659;
        double r6450661 = r6450654 + r6450634;
        double r6450662 = r6450640 * r6450661;
        double r6450663 = sqrt(r6450662);
        double r6450664 = r6450645 * r6450663;
        double r6450665 = r6450634 + r6450634;
        double r6450666 = r6450640 * r6450665;
        double r6450667 = sqrt(r6450666);
        double r6450668 = r6450645 * r6450667;
        double r6450669 = r6450660 ? r6450664 : r6450668;
        double r6450670 = r6450648 ? r6450658 : r6450669;
        double r6450671 = r6450639 ? r6450646 : r6450670;
        double r6450672 = r6450636 ? r6450637 : r6450671;
        return r6450672;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.3
Herbie27.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.756706552252228e+164

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 49.8

      \[\leadsto 0.5 \cdot \color{blue}{0}\]

    if -1.756706552252228e+164 < re < -1.0167980430801172e+113

    1. Initial program 55.2

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

    3. Applied add-sqr-sqrt55.2

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

    4. Applied sqrt-prod58.4

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

    5. Taylor expanded around 0 53.9

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

    if -1.0167980430801172e+113 < re < -5.204735522622993e-274

    1. Initial program 38.3

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

    3. Applied flip-+38.1

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

    4. Applied associate-*r/38.1

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

    5. Applied sqrt-div38.3

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

    6. Simplified29.5

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

    if -5.204735522622993e-274 < re < 7.747777771049568e+94

    1. Initial program 22.1

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

    if 7.747777771049568e+94 < re

    1. Initial program 49.0

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

    2. Taylor expanded around inf 11.2

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

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.756706552252228103614804055367357436769 \cdot 10^{164}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \le -1.016798043080117215536296671868401307267 \cdot 10^{113}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + im\right)} \cdot 0.5\\ \mathbf{elif}\;re \le -5.204735522622992940875512007096881566611 \cdot 10^{-274}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le 7.747777771049567852122186762181106639836 \cdot 10^{94}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))))