Average Error: 38.7 → 24.3
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -4.31226287577943336 \cdot 10^{124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -1.31798426233677 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{elif}\;im \le -5.84508579610625006 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le 3.10359725716777497 \cdot 10^{-130}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;im \le 7.78847036828617129 \cdot 10^{89}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -4.31226287577943336 \cdot 10^{124}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\

\mathbf{elif}\;im \le -1.31798426233677 \cdot 10^{-105}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\

\mathbf{elif}\;im \le -5.84508579610625006 \cdot 10^{-260}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\

\mathbf{elif}\;im \le 3.10359725716777497 \cdot 10^{-130}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;im \le 7.78847036828617129 \cdot 10^{89}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\

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

\end{array}
double f(double re, double im) {
        double r18566 = 0.5;
        double r18567 = 2.0;
        double r18568 = re;
        double r18569 = r18568 * r18568;
        double r18570 = im;
        double r18571 = r18570 * r18570;
        double r18572 = r18569 + r18571;
        double r18573 = sqrt(r18572);
        double r18574 = r18573 - r18568;
        double r18575 = r18567 * r18574;
        double r18576 = sqrt(r18575);
        double r18577 = r18566 * r18576;
        return r18577;
}
↓
double f(double re, double im) {
        double r18578 = im;
        double r18579 = -4.312262875779433e+124;
        bool r18580 = r18578 <= r18579;
        double r18581 = 0.5;
        double r18582 = 2.0;
        double r18583 = re;
        double r18584 = r18583 + r18578;
        double r18585 = -r18584;
        double r18586 = r18582 * r18585;
        double r18587 = sqrt(r18586);
        double r18588 = r18581 * r18587;
        double r18589 = -1.3179842623367683e-105;
        bool r18590 = r18578 <= r18589;
        double r18591 = r18583 * r18583;
        double r18592 = r18578 * r18578;
        double r18593 = r18591 + r18592;
        double r18594 = sqrt(r18593);
        double r18595 = r18594 + r18583;
        double r18596 = r18595 / r18578;
        double r18597 = r18578 / r18596;
        double r18598 = r18582 * r18597;
        double r18599 = sqrt(r18598);
        double r18600 = r18581 * r18599;
        double r18601 = -5.84508579610625e-260;
        bool r18602 = r18578 <= r18601;
        double r18603 = 3.103597257167775e-130;
        bool r18604 = r18578 <= r18603;
        double r18605 = -2.0;
        double r18606 = r18605 * r18583;
        double r18607 = r18582 * r18606;
        double r18608 = sqrt(r18607);
        double r18609 = r18581 * r18608;
        double r18610 = 7.788470368286171e+89;
        bool r18611 = r18578 <= r18610;
        double r18612 = r18578 - r18583;
        double r18613 = r18582 * r18612;
        double r18614 = sqrt(r18613);
        double r18615 = r18581 * r18614;
        double r18616 = r18611 ? r18600 : r18615;
        double r18617 = r18604 ? r18609 : r18616;
        double r18618 = r18602 ? r18588 : r18617;
        double r18619 = r18590 ? r18600 : r18618;
        double r18620 = r18580 ? r18588 : r18619;
        return r18620;
}

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 4 regimes

  2. if im < -4.312262875779433e+124 or -1.3179842623367683e-105 < im < -5.84508579610625e-260

    1. Initial program 49.8

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

    3. Applied flip--56.9

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

    4. Simplified53.6

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

    5. Taylor expanded around -inf 25.3

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

    if -4.312262875779433e+124 < im < -1.3179842623367683e-105 or 3.103597257167775e-130 < im < 7.788470368286171e+89

    1. Initial program 23.7

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

    3. Applied flip--32.0

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

    4. Simplified24.2

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

    6. Applied unpow224.2

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

    7. Applied associate-/l*24.2

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

    if -5.84508579610625e-260 < im < 3.103597257167775e-130

    1. Initial program 41.2

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

    2. Taylor expanded around -inf 36.6

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

    if 7.788470368286171e+89 < im

    1. Initial program 49.7

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

    2. Taylor expanded around 0 10.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification24.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.31226287577943336 \cdot 10^{124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -1.31798426233677 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{elif}\;im \le -5.84508579610625006 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le 3.10359725716777497 \cdot 10^{-130}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;im \le 7.78847036828617129 \cdot 10^{89}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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)))))