Average Error: 38.6 → 26.4
Time: 8.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.5808863745987776 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 6.6745168043242478 \cdot 10^{142}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.5808863745987776 \cdot 10^{-301}:\\
\;\;\;\;\sqrt{\frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le 6.6745168043242478 \cdot 10^{142}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{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 r250647 = 0.5;
        double r250648 = 2.0;
        double r250649 = re;
        double r250650 = r250649 * r250649;
        double r250651 = im;
        double r250652 = r250651 * r250651;
        double r250653 = r250650 + r250652;
        double r250654 = sqrt(r250653);
        double r250655 = r250654 + r250649;
        double r250656 = r250648 * r250655;
        double r250657 = sqrt(r250656);
        double r250658 = r250647 * r250657;
        return r250658;
}
↓
double f(double re, double im) {
        double r250659 = re;
        double r250660 = -6.580886374598778e-301;
        bool r250661 = r250659 <= r250660;
        double r250662 = im;
        double r250663 = 2.0;
        double r250664 = pow(r250662, r250663);
        double r250665 = r250659 * r250659;
        double r250666 = r250662 * r250662;
        double r250667 = r250665 + r250666;
        double r250668 = sqrt(r250667);
        double r250669 = r250668 - r250659;
        double r250670 = r250664 / r250669;
        double r250671 = 2.0;
        double r250672 = r250670 * r250671;
        double r250673 = sqrt(r250672);
        double r250674 = 0.5;
        double r250675 = r250673 * r250674;
        double r250676 = 6.674516804324248e+142;
        bool r250677 = r250659 <= r250676;
        double r250678 = r250668 + r250659;
        double r250679 = r250671 * r250678;
        double r250680 = sqrt(r250679);
        double r250681 = r250674 * r250680;
        double r250682 = r250663 * r250659;
        double r250683 = r250671 * r250682;
        double r250684 = sqrt(r250683);
        double r250685 = r250674 * r250684;
        double r250686 = r250677 ? r250681 : r250685;
        double r250687 = r250661 ? r250675 : r250686;
        return r250687;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.2
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -6.580886374598778e-301

    1. Initial program 46.4

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

    3. Applied add-exp-log48.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]
    4. Using strategy rm

    5. Applied flip-+48.5

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

    6. Simplified36.7

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

    7. Simplified35.4

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

    if -6.580886374598778e-301 < re < 6.674516804324248e+142

    1. Initial program 20.5

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

    if 6.674516804324248e+142 < re

    1. Initial program 60.6

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.5808863745987776 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re} \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 6.6745168043242478 \cdot 10^{142}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))))