Average Error: 39.2 → 23.8
Time: 4.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.150193189996774185649058855667814599131 \cdot 10^{145}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -8.463201405765729115322317591242469230422 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le -2.584350837219368294692435261917815944642 \cdot 10^{-192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le -3.228290100181764909249096677380323651935 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 2.52693442221263941553214481169152971763 \cdot 10^{70}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{2 \cdot re}\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.150193189996774185649058855667814599131 \cdot 10^{145}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -8.463201405765729115322317591242469230422 \cdot 10^{-144}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \le -2.584350837219368294692435261917815944642 \cdot 10^{-192}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le -3.228290100181764909249096677380323651935 \cdot 10^{-270}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\

\mathbf{elif}\;re \le 2.52693442221263941553214481169152971763 \cdot 10^{70}:\\
\;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right)\\

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

\end{array}
double f(double re, double im) {
        double r18616 = 0.5;
        double r18617 = 2.0;
        double r18618 = re;
        double r18619 = r18618 * r18618;
        double r18620 = im;
        double r18621 = r18620 * r18620;
        double r18622 = r18619 + r18621;
        double r18623 = sqrt(r18622);
        double r18624 = r18623 - r18618;
        double r18625 = r18617 * r18624;
        double r18626 = sqrt(r18625);
        double r18627 = r18616 * r18626;
        return r18627;
}
↓
double f(double re, double im) {
        double r18628 = re;
        double r18629 = -2.1501931899967742e+145;
        bool r18630 = r18628 <= r18629;
        double r18631 = 0.5;
        double r18632 = 2.0;
        double r18633 = -2.0;
        double r18634 = r18633 * r18628;
        double r18635 = r18632 * r18634;
        double r18636 = sqrt(r18635);
        double r18637 = r18631 * r18636;
        double r18638 = -8.463201405765729e-144;
        bool r18639 = r18628 <= r18638;
        double r18640 = r18628 * r18628;
        double r18641 = im;
        double r18642 = r18641 * r18641;
        double r18643 = r18640 + r18642;
        double r18644 = sqrt(r18643);
        double r18645 = r18644 - r18628;
        double r18646 = r18632 * r18645;
        double r18647 = sqrt(r18646);
        double r18648 = r18631 * r18647;
        double r18649 = -2.5843508372193683e-192;
        bool r18650 = r18628 <= r18649;
        double r18651 = r18641 - r18628;
        double r18652 = r18632 * r18651;
        double r18653 = sqrt(r18652);
        double r18654 = r18631 * r18653;
        double r18655 = -3.228290100181765e-270;
        bool r18656 = r18628 <= r18655;
        double r18657 = r18628 + r18641;
        double r18658 = -r18657;
        double r18659 = r18632 * r18658;
        double r18660 = sqrt(r18659);
        double r18661 = r18631 * r18660;
        double r18662 = 2.5269344222126394e+70;
        bool r18663 = r18628 <= r18662;
        double r18664 = r18644 + r18628;
        double r18665 = r18641 / r18664;
        double r18666 = r18641 * r18665;
        double r18667 = r18632 * r18666;
        double r18668 = sqrt(r18667);
        double r18669 = cbrt(r18668);
        double r18670 = r18669 * r18669;
        double r18671 = r18670 * r18669;
        double r18672 = r18631 * r18671;
        double r18673 = 2.0;
        double r18674 = r18673 * r18628;
        double r18675 = r18641 / r18674;
        double r18676 = r18641 * r18675;
        double r18677 = r18632 * r18676;
        double r18678 = sqrt(r18677);
        double r18679 = r18631 * r18678;
        double r18680 = r18663 ? r18672 : r18679;
        double r18681 = r18656 ? r18661 : r18680;
        double r18682 = r18650 ? r18654 : r18681;
        double r18683 = r18639 ? r18648 : r18682;
        double r18684 = r18630 ? r18637 : r18683;
        return r18684;
}

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 6 regimes

  2. if re < -2.1501931899967742e+145

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 8.4

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

    if -2.1501931899967742e+145 < re < -8.463201405765729e-144

    1. Initial program 17.0

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

    if -8.463201405765729e-144 < re < -2.5843508372193683e-192

    1. Initial program 25.1

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

    2. Taylor expanded around 0 37.4

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

    if -2.5843508372193683e-192 < re < -3.228290100181765e-270

    1. Initial program 29.3

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

    3. Applied flip--30.9

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

    4. Simplified30.9

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

    5. Taylor expanded around -inf 34.1

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

    if -3.228290100181765e-270 < re < 2.5269344222126394e+70

    1. Initial program 36.7

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

    3. Applied flip--36.6

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

    4. Simplified30.2

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

    6. Applied *-un-lft-identity30.2

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

    7. Applied times-frac28.1

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

    8. Simplified28.1

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

    10. Applied add-cube-cbrt28.8

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

    if 2.5269344222126394e+70 < re

    1. Initial program 60.0

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

    3. Applied flip--60.0

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

    4. Simplified44.2

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

    6. Applied *-un-lft-identity44.2

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

    7. Applied times-frac43.8

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

    8. Simplified43.8

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

    9. Taylor expanded around inf 27.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\color{blue}{2 \cdot re}}\right)}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification23.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.150193189996774185649058855667814599131 \cdot 10^{145}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -8.463201405765729115322317591242469230422 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le -2.584350837219368294692435261917815944642 \cdot 10^{-192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le -3.228290100181764909249096677380323651935 \cdot 10^{-270}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 2.52693442221263941553214481169152971763 \cdot 10^{70}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{2 \cdot re}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(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)))))