Average Error: 38.7 → 21.0
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.34046224677095251 \cdot 10^{149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le -1.0150073241706023 \cdot 10^{-273}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.54087667976783686 \cdot 10^{126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.34046224677095251 \cdot 10^{149}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\

\mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\

\mathbf{elif}\;re \le -1.0150073241706023 \cdot 10^{-273}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 1.54087667976783686 \cdot 10^{126}:\\
\;\;\;\;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 r143658 = 0.5;
        double r143659 = 2.0;
        double r143660 = re;
        double r143661 = r143660 * r143660;
        double r143662 = im;
        double r143663 = r143662 * r143662;
        double r143664 = r143661 + r143663;
        double r143665 = sqrt(r143664);
        double r143666 = r143665 + r143660;
        double r143667 = r143659 * r143666;
        double r143668 = sqrt(r143667);
        double r143669 = r143658 * r143668;
        return r143669;
}
↓
double f(double re, double im) {
        double r143670 = re;
        double r143671 = -4.3404622467709525e+149;
        bool r143672 = r143670 <= r143671;
        double r143673 = 0.5;
        double r143674 = 2.0;
        double r143675 = im;
        double r143676 = r143675 * r143675;
        double r143677 = -1.0;
        double r143678 = r143677 * r143670;
        double r143679 = r143678 - r143670;
        double r143680 = r143676 / r143679;
        double r143681 = r143674 * r143680;
        double r143682 = sqrt(r143681);
        double r143683 = r143673 * r143682;
        double r143684 = -2.0278572522938575e-184;
        bool r143685 = r143670 <= r143684;
        double r143686 = sqrt(r143674);
        double r143687 = r143670 * r143670;
        double r143688 = r143687 + r143676;
        double r143689 = sqrt(r143688);
        double r143690 = r143689 - r143670;
        double r143691 = sqrt(r143690);
        double r143692 = r143675 / r143691;
        double r143693 = fabs(r143692);
        double r143694 = r143686 * r143693;
        double r143695 = r143673 * r143694;
        double r143696 = -1.0150073241706023e-273;
        bool r143697 = r143670 <= r143696;
        double r143698 = r143675 + r143670;
        double r143699 = r143674 * r143698;
        double r143700 = sqrt(r143699);
        double r143701 = r143673 * r143700;
        double r143702 = 1.540876679767837e+126;
        bool r143703 = r143670 <= r143702;
        double r143704 = r143689 + r143670;
        double r143705 = r143674 * r143704;
        double r143706 = sqrt(r143705);
        double r143707 = r143673 * r143706;
        double r143708 = 2.0;
        double r143709 = r143708 * r143670;
        double r143710 = r143674 * r143709;
        double r143711 = sqrt(r143710);
        double r143712 = r143673 * r143711;
        double r143713 = r143703 ? r143707 : r143712;
        double r143714 = r143697 ? r143701 : r143713;
        double r143715 = r143685 ? r143695 : r143714;
        double r143716 = r143672 ? r143683 : r143715;
        return r143716;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.7
Target33.5
Herbie21.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -4.3404622467709525e+149

    1. Initial program 63.8

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

    3. Applied flip-+63.8

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

    4. Simplified51.1

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

    5. Taylor expanded around -inf 31.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\color{blue}{-1 \cdot re} - re}}\]

    if -4.3404622467709525e+149 < re < -2.0278572522938575e-184

    1. Initial program 41.4

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

    3. Applied flip-+41.3

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

    4. Simplified28.1

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

    6. Applied add-sqr-sqrt28.2

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

    7. Applied times-frac25.9

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

    9. Applied sqrt-prod25.9

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

    10. Simplified15.2

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

    if -2.0278572522938575e-184 < re < -1.0150073241706023e-273

    1. Initial program 31.2

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

    3. Applied add-sqr-sqrt31.2

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

    4. Applied sqrt-prod31.4

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

    5. Taylor expanded around 0 37.8

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

    if -1.0150073241706023e-273 < re < 1.540876679767837e+126

    1. Initial program 22.3

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

    if 1.540876679767837e+126 < re

    1. Initial program 56.8

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

    2. Taylor expanded around inf 10.0

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

  4. Final simplification21.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.34046224677095251 \cdot 10^{149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{im}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le -1.0150073241706023 \cdot 10^{-273}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 1.54087667976783686 \cdot 10^{126}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))))