Average Error: 37.3 → 23.0
Time: 19.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.129850026901925 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.246514102911935 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 6.902307096639934 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{re + re}}\\ \end{array}\]
double f(double re, double im) {
        double r617578 = 0.5;
        double r617579 = 2.0;
        double r617580 = re;
        double r617581 = r617580 * r617580;
        double r617582 = im;
        double r617583 = r617582 * r617582;
        double r617584 = r617581 + r617583;
        double r617585 = sqrt(r617584);
        double r617586 = r617585 - r617580;
        double r617587 = r617579 * r617586;
        double r617588 = sqrt(r617587);
        double r617589 = r617578 * r617588;
        return r617589;
}
↓
double f(double re, double im) {
        double r617590 = re;
        double r617591 = -4.129850026901925e+87;
        bool r617592 = r617590 <= r617591;
        double r617593 = -2.0;
        double r617594 = r617593 * r617590;
        double r617595 = 2.0;
        double r617596 = r617594 * r617595;
        double r617597 = sqrt(r617596);
        double r617598 = 0.5;
        double r617599 = r617597 * r617598;
        double r617600 = 1.246514102911935e-58;
        bool r617601 = r617590 <= r617600;
        double r617602 = im;
        double r617603 = r617602 * r617602;
        double r617604 = r617590 * r617590;
        double r617605 = r617603 + r617604;
        double r617606 = sqrt(r617605);
        double r617607 = r617606 - r617590;
        double r617608 = r617595 * r617607;
        double r617609 = sqrt(r617608);
        double r617610 = r617609 * r617598;
        double r617611 = 6.902307096639934e+43;
        bool r617612 = r617590 <= r617611;
        double r617613 = sqrt(r617595);
        double r617614 = r617602 * r617613;
        double r617615 = r617606 + r617590;
        double r617616 = sqrt(r617615);
        double r617617 = r617614 / r617616;
        double r617618 = r617598 * r617617;
        double r617619 = r617595 * r617603;
        double r617620 = sqrt(r617619);
        double r617621 = r617590 + r617590;
        double r617622 = sqrt(r617621);
        double r617623 = r617620 / r617622;
        double r617624 = r617598 * r617623;
        double r617625 = r617612 ? r617618 : r617624;
        double r617626 = r617601 ? r617610 : r617625;
        double r617627 = r617592 ? r617599 : r617626;
        return r617627;
}


0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.129850026901925 \cdot 10^{+87}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 1.246514102911935 \cdot 10^{-58}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\

\mathbf{elif}\;re \le 6.902307096639934 \cdot 10^{+43}:\\
\;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{re + re}}\\

\end{array}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 4 regimes

  2. if re < -4.129850026901925e+87

    1. Initial program 47.3

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

    2. Taylor expanded around -inf 10.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

    if -4.129850026901925e+87 < re < 1.246514102911935e-58

    1. Initial program 24.8

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity24.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}\]

    if 1.246514102911935e-58 < re < 6.902307096639934e+43

    1. Initial program 44.0

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--44.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/44.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    5. Applied sqrt-div44.2

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    6. Simplified29.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    7. Taylor expanded around inf 35.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    if 6.902307096639934e+43 < re

    1. Initial program 57.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--57.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/57.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    5. Applied sqrt-div57.1

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    6. Simplified39.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    7. Taylor expanded around inf 23.7

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\color{blue}{re} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification23.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.129850026901925 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.246514102911935 \cdot 10^{-58}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 6.902307096639934 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \frac{im \cdot \sqrt{2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{re + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))