Average Error: 38.7 → 25.5
Time: 3.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.9191202825942023 \cdot 10^{46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.6134652433551429 \cdot 10^{-224}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 2.24796075526010884 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\ \mathbf{elif}\;re \le 4.1429610238547999 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.915911405075526 \cdot 10^{72}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;re \le 9.90103109639506297 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 4.9135228890152114 \cdot 10^{139}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.9191202825942023 \cdot 10^{46}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\

\mathbf{elif}\;re \le 1.6134652433551429 \cdot 10^{-224}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le 2.24796075526010884 \cdot 10^{-102}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\

\mathbf{elif}\;re \le 4.1429610238547999 \cdot 10^{-51}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\

\mathbf{elif}\;re \le 5.915911405075526 \cdot 10^{72}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;re \le 9.90103109639506297 \cdot 10^{118}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\

\mathbf{elif}\;re \le 4.9135228890152114 \cdot 10^{139}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + re}}\\

\end{array}
double f(double re, double im) {
        double r16626 = 0.5;
        double r16627 = 2.0;
        double r16628 = re;
        double r16629 = r16628 * r16628;
        double r16630 = im;
        double r16631 = r16630 * r16630;
        double r16632 = r16629 + r16631;
        double r16633 = sqrt(r16632);
        double r16634 = r16633 - r16628;
        double r16635 = r16627 * r16634;
        double r16636 = sqrt(r16635);
        double r16637 = r16626 * r16636;
        return r16637;
}
↓
double f(double re, double im) {
        double r16638 = re;
        double r16639 = -5.919120282594202e+46;
        bool r16640 = r16638 <= r16639;
        double r16641 = 0.5;
        double r16642 = 2.0;
        double r16643 = -1.0;
        double r16644 = r16643 * r16638;
        double r16645 = r16644 - r16638;
        double r16646 = r16642 * r16645;
        double r16647 = sqrt(r16646);
        double r16648 = r16641 * r16647;
        double r16649 = 7.942397244706197e-271;
        bool r16650 = r16638 <= r16649;
        double r16651 = r16638 * r16638;
        double r16652 = im;
        double r16653 = r16652 * r16652;
        double r16654 = r16651 + r16653;
        double r16655 = cbrt(r16654);
        double r16656 = fabs(r16655);
        double r16657 = sqrt(r16655);
        double r16658 = r16656 * r16657;
        double r16659 = r16658 - r16638;
        double r16660 = r16642 * r16659;
        double r16661 = sqrt(r16660);
        double r16662 = r16641 * r16661;
        double r16663 = 1.613465243355143e-224;
        bool r16664 = r16638 <= r16663;
        double r16665 = r16652 - r16638;
        double r16666 = r16642 * r16665;
        double r16667 = sqrt(r16666);
        double r16668 = r16641 * r16667;
        double r16669 = 2.247960755260109e-102;
        bool r16670 = r16638 <= r16669;
        double r16671 = sqrt(r16654);
        double r16672 = r16671 + r16638;
        double r16673 = r16652 / r16672;
        double r16674 = r16652 * r16673;
        double r16675 = r16642 * r16674;
        double r16676 = sqrt(r16675);
        double r16677 = r16641 * r16676;
        double r16678 = 4.1429610238548e-51;
        bool r16679 = r16638 <= r16678;
        double r16680 = r16638 + r16652;
        double r16681 = -r16680;
        double r16682 = r16642 * r16681;
        double r16683 = sqrt(r16682);
        double r16684 = r16641 * r16683;
        double r16685 = 5.9159114050755255e+72;
        bool r16686 = r16638 <= r16685;
        double r16687 = 2.0;
        double r16688 = pow(r16652, r16687);
        double r16689 = r16642 * r16688;
        double r16690 = sqrt(r16689);
        double r16691 = sqrt(r16672);
        double r16692 = r16690 / r16691;
        double r16693 = r16641 * r16692;
        double r16694 = 9.901031096395063e+118;
        bool r16695 = r16638 <= r16694;
        double r16696 = 4.913522889015211e+139;
        bool r16697 = r16638 <= r16696;
        double r16698 = r16638 + r16638;
        double r16699 = r16688 / r16698;
        double r16700 = r16642 * r16699;
        double r16701 = sqrt(r16700);
        double r16702 = r16641 * r16701;
        double r16703 = r16697 ? r16693 : r16702;
        double r16704 = r16695 ? r16684 : r16703;
        double r16705 = r16686 ? r16693 : r16704;
        double r16706 = r16679 ? r16684 : r16705;
        double r16707 = r16670 ? r16677 : r16706;
        double r16708 = r16664 ? r16668 : r16707;
        double r16709 = r16650 ? r16662 : r16708;
        double r16710 = r16640 ? r16648 : r16709;
        return r16710;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 7 regimes

  2. if re < -5.919120282594202e+46

    1. Initial program 43.7

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

    2. Taylor expanded around -inf 14.0

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

    if -5.919120282594202e+46 < re < 7.942397244706197e-271

    1. Initial program 22.8

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

    3. Applied add-cube-cbrt23.0

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

    4. Applied sqrt-prod23.0

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

    5. Simplified23.0

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

    if 7.942397244706197e-271 < re < 1.613465243355143e-224

    1. Initial program 30.0

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

    2. Taylor expanded around 0 32.7

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

    if 1.613465243355143e-224 < re < 2.247960755260109e-102

    1. Initial program 34.3

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

    3. Applied flip--33.9

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

    4. Simplified33.0

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

    6. Applied *-un-lft-identity33.0

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

    7. Applied add-sqr-sqrt49.1

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

    8. Applied unpow-prod-down49.1

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

    9. Applied times-frac47.0

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

    10. Simplified46.9

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

    11. Simplified28.6

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

    if 2.247960755260109e-102 < re < 4.1429610238548e-51 or 5.9159114050755255e+72 < re < 9.901031096395063e+118

    1. Initial program 45.1

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

    3. Applied flip--45.1

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

    4. Simplified30.7

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

    5. Taylor expanded around -inf 46.8

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

    if 4.1429610238548e-51 < re < 5.9159114050755255e+72 or 9.901031096395063e+118 < re < 4.913522889015211e+139

    1. Initial program 46.2

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

    3. Applied flip--46.2

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

    4. Simplified30.0

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

    6. Applied associate-*r/30.0

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

    7. Applied sqrt-div27.9

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

    if 4.913522889015211e+139 < re

    1. Initial program 63.3

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

    3. Applied flip--63.3

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

    4. Simplified48.8

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

    5. Taylor expanded around inf 30.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{re} + re}}\]
  3. Recombined 7 regimes into one program.

  4. Final simplification25.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.9191202825942023 \cdot 10^{46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.6134652433551429 \cdot 10^{-224}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 2.24796075526010884 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\ \mathbf{elif}\;re \le 4.1429610238547999 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.915911405075526 \cdot 10^{72}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;re \le 9.90103109639506297 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 4.9135228890152114 \cdot 10^{139}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))