Average Error: 38.7 → 25.5
Time: 3.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.9191202825942023 \cdot 10^{46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 7.9423972447061974 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.6134652433551429 \cdot 10^{-224}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 2.24796075526010884 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\ \mathbf{elif}\;re \le 4.1429610238547999 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.915911405075526 \cdot 10^{72}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;re \le 9.90103109639506297 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 4.9135228890152114 \cdot 10^{139}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + re}}\\ \end{array}\]
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(-2 \cdot 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 r16603 = 0.5;
        double r16604 = 2.0;
        double r16605 = re;
        double r16606 = r16605 * r16605;
        double r16607 = im;
        double r16608 = r16607 * r16607;
        double r16609 = r16606 + r16608;
        double r16610 = sqrt(r16609);
        double r16611 = r16610 - r16605;
        double r16612 = r16604 * r16611;
        double r16613 = sqrt(r16612);
        double r16614 = r16603 * r16613;
        return r16614;
}
↓
double f(double re, double im) {
        double r16615 = re;
        double r16616 = -5.919120282594202e+46;
        bool r16617 = r16615 <= r16616;
        double r16618 = 0.5;
        double r16619 = 2.0;
        double r16620 = -2.0;
        double r16621 = r16620 * r16615;
        double r16622 = r16619 * r16621;
        double r16623 = sqrt(r16622);
        double r16624 = r16618 * r16623;
        double r16625 = 7.942397244706197e-271;
        bool r16626 = r16615 <= r16625;
        double r16627 = r16615 * r16615;
        double r16628 = im;
        double r16629 = r16628 * r16628;
        double r16630 = r16627 + r16629;
        double r16631 = cbrt(r16630);
        double r16632 = fabs(r16631);
        double r16633 = sqrt(r16631);
        double r16634 = r16632 * r16633;
        double r16635 = r16634 - r16615;
        double r16636 = r16619 * r16635;
        double r16637 = sqrt(r16636);
        double r16638 = r16618 * r16637;
        double r16639 = 1.613465243355143e-224;
        bool r16640 = r16615 <= r16639;
        double r16641 = r16628 - r16615;
        double r16642 = r16619 * r16641;
        double r16643 = sqrt(r16642);
        double r16644 = r16618 * r16643;
        double r16645 = 2.247960755260109e-102;
        bool r16646 = r16615 <= r16645;
        double r16647 = sqrt(r16630);
        double r16648 = r16647 + r16615;
        double r16649 = r16628 / r16648;
        double r16650 = r16628 * r16649;
        double r16651 = r16619 * r16650;
        double r16652 = sqrt(r16651);
        double r16653 = r16618 * r16652;
        double r16654 = 4.1429610238548e-51;
        bool r16655 = r16615 <= r16654;
        double r16656 = r16615 + r16628;
        double r16657 = -r16656;
        double r16658 = r16619 * r16657;
        double r16659 = sqrt(r16658);
        double r16660 = r16618 * r16659;
        double r16661 = 5.9159114050755255e+72;
        bool r16662 = r16615 <= r16661;
        double r16663 = 2.0;
        double r16664 = pow(r16628, r16663);
        double r16665 = r16619 * r16664;
        double r16666 = sqrt(r16665);
        double r16667 = sqrt(r16648);
        double r16668 = r16666 / r16667;
        double r16669 = r16618 * r16668;
        double r16670 = 9.901031096395063e+118;
        bool r16671 = r16615 <= r16670;
        double r16672 = 4.913522889015211e+139;
        bool r16673 = r16615 <= r16672;
        double r16674 = r16615 + r16615;
        double r16675 = r16664 / r16674;
        double r16676 = r16619 * r16675;
        double r16677 = sqrt(r16676);
        double r16678 = r16618 * r16677;
        double r16679 = r16673 ? r16669 : r16678;
        double r16680 = r16671 ? r16660 : r16679;
        double r16681 = r16662 ? r16669 : r16680;
        double r16682 = r16655 ? r16660 : r16681;
        double r16683 = r16646 ? r16653 : r16682;
        double r16684 = r16640 ? r16644 : r16683;
        double r16685 = r16626 ? r16638 : r16684;
        double r16686 = r16617 ? r16624 : r16685;
        return r16686;
}

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

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

    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

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

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)))))