Average Error: 38.8 → 26.1
Time: 9.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.1094940511471951 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.3723278512607863 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{1}} \cdot \sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.1094940511471951 \cdot 10^{119}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -1.3723278512607863 \cdot 10^{-307}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{1}} \cdot \sqrt{re \cdot re + im \cdot im} - re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r17588 = 0.5;
        double r17589 = 2.0;
        double r17590 = re;
        double r17591 = r17590 * r17590;
        double r17592 = im;
        double r17593 = r17592 * r17592;
        double r17594 = r17591 + r17593;
        double r17595 = sqrt(r17594);
        double r17596 = r17595 - r17590;
        double r17597 = r17589 * r17596;
        double r17598 = sqrt(r17597);
        double r17599 = r17588 * r17598;
        return r17599;
}
↓
double f(double re, double im) {
        double r17600 = re;
        double r17601 = -4.109494051147195e+119;
        bool r17602 = r17600 <= r17601;
        double r17603 = 0.5;
        double r17604 = 2.0;
        double r17605 = -2.0;
        double r17606 = r17605 * r17600;
        double r17607 = r17604 * r17606;
        double r17608 = sqrt(r17607);
        double r17609 = r17603 * r17608;
        double r17610 = -1.3723278512607863e-307;
        bool r17611 = r17600 <= r17610;
        double r17612 = 1.0;
        double r17613 = sqrt(r17612);
        double r17614 = sqrt(r17613);
        double r17615 = r17600 * r17600;
        double r17616 = im;
        double r17617 = r17616 * r17616;
        double r17618 = r17615 + r17617;
        double r17619 = sqrt(r17618);
        double r17620 = r17614 * r17619;
        double r17621 = r17620 - r17600;
        double r17622 = r17604 * r17621;
        double r17623 = sqrt(r17622);
        double r17624 = r17603 * r17623;
        double r17625 = 2.0;
        double r17626 = pow(r17616, r17625);
        double r17627 = r17626 * r17604;
        double r17628 = sqrt(r17627);
        double r17629 = r17619 + r17600;
        double r17630 = sqrt(r17629);
        double r17631 = r17628 / r17630;
        double r17632 = r17631 * r17603;
        double r17633 = r17611 ? r17624 : r17632;
        double r17634 = r17602 ? r17609 : r17633;
        return r17634;
}

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 3 regimes

  2. if re < -4.109494051147195e+119

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 9.8

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

    if -4.109494051147195e+119 < re < -1.3723278512607863e-307

    1. Initial program 20.1

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

    3. Applied add-sqr-sqrt20.1

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

    4. Applied sqrt-prod20.2

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

    6. Applied *-un-lft-identity20.2

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

    7. Applied sqrt-prod20.2

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

    8. Applied sqrt-prod20.2

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

    9. Applied associate-*l*20.2

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

    10. Simplified20.1

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

    if -1.3723278512607863e-307 < re

    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--46.1

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

    4. Applied associate-*r/46.1

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

    5. Applied sqrt-div46.2

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

    6. Simplified35.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left({im}^{2} + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1094940511471951 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.3723278512607863 \cdot 10^{-307}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{1}} \cdot \sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}} \cdot 0.5\\ \end{array}\]

Reproduce

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