Average Error: 39.5 → 24.6
Time: 5.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.4192758919568226 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.4192758919568226 \cdot 10^{153}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{-2 \cdot re}}\\

\mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r137620 = 0.5;
        double r137621 = 2.0;
        double r137622 = re;
        double r137623 = r137622 * r137622;
        double r137624 = im;
        double r137625 = r137624 * r137624;
        double r137626 = r137623 + r137625;
        double r137627 = sqrt(r137626);
        double r137628 = r137627 + r137622;
        double r137629 = r137621 * r137628;
        double r137630 = sqrt(r137629);
        double r137631 = r137620 * r137630;
        return r137631;
}
↓
double f(double re, double im) {
        double r137632 = re;
        double r137633 = -7.419275891956823e+153;
        bool r137634 = r137632 <= r137633;
        double r137635 = 0.5;
        double r137636 = 2.0;
        double r137637 = im;
        double r137638 = 2.0;
        double r137639 = pow(r137637, r137638);
        double r137640 = -2.0;
        double r137641 = r137640 * r137632;
        double r137642 = r137639 / r137641;
        double r137643 = r137636 * r137642;
        double r137644 = sqrt(r137643);
        double r137645 = r137635 * r137644;
        double r137646 = -1.2504367945899628e-181;
        bool r137647 = r137632 <= r137646;
        double r137648 = r137636 * r137639;
        double r137649 = sqrt(r137648);
        double r137650 = r137632 * r137632;
        double r137651 = r137637 * r137637;
        double r137652 = r137650 + r137651;
        double r137653 = sqrt(r137652);
        double r137654 = r137653 - r137632;
        double r137655 = sqrt(r137654);
        double r137656 = r137649 / r137655;
        double r137657 = r137635 * r137656;
        double r137658 = 1.1547189018901299e-253;
        bool r137659 = r137632 <= r137658;
        double r137660 = r137632 + r137637;
        double r137661 = r137636 * r137660;
        double r137662 = sqrt(r137661);
        double r137663 = r137635 * r137662;
        double r137664 = 1.386148847085094e+97;
        bool r137665 = r137632 <= r137664;
        double r137666 = cbrt(r137652);
        double r137667 = r137666 * r137666;
        double r137668 = r137667 * r137666;
        double r137669 = sqrt(r137668);
        double r137670 = r137669 + r137632;
        double r137671 = r137636 * r137670;
        double r137672 = sqrt(r137671);
        double r137673 = r137635 * r137672;
        double r137674 = r137638 * r137632;
        double r137675 = r137636 * r137674;
        double r137676 = sqrt(r137675);
        double r137677 = r137635 * r137676;
        double r137678 = r137665 ? r137673 : r137677;
        double r137679 = r137659 ? r137663 : r137678;
        double r137680 = r137647 ? r137657 : r137679;
        double r137681 = r137634 ? r137645 : r137680;
        return r137681;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
Target34.4
Herbie24.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -7.419275891956823e+153

    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. Simplified49.5

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

    5. Taylor expanded around -inf 30.4

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

    if -7.419275891956823e+153 < re < -1.2504367945899628e-181

    1. Initial program 44.1

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

    3. Applied flip-+44.1

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

    4. Simplified32.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-*r/32.1

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

    7. Applied sqrt-div30.2

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

    if -1.2504367945899628e-181 < re < 1.1547189018901299e-253

    1. Initial program 32.7

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

    3. Applied flip-+32.6

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

    4. Simplified32.6

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

    5. Taylor expanded around inf 33.9

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

    if 1.1547189018901299e-253 < re < 1.386148847085094e+97

    1. Initial program 20.3

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

    3. Applied add-cube-cbrt20.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    if 1.386148847085094e+97 < re

    1. Initial program 51.2

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

    3. Applied flip-+63.2

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

    4. Simplified62.3

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

    5. Taylor expanded around 0 10.7

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

  4. Final simplification24.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.4192758919568226 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))