Average Error: 39.3 → 26.8
Time: 4.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\ \mathbf{elif}\;re \le 3.29386860837205705 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\

\mathbf{elif}\;re \le 3.29386860837205705 \cdot 10^{-252}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r23417 = 0.5;
        double r23418 = 2.0;
        double r23419 = re;
        double r23420 = r23419 * r23419;
        double r23421 = im;
        double r23422 = r23421 * r23421;
        double r23423 = r23420 + r23422;
        double r23424 = sqrt(r23423);
        double r23425 = r23424 - r23419;
        double r23426 = r23418 * r23425;
        double r23427 = sqrt(r23426);
        double r23428 = r23417 * r23427;
        return r23428;
}
↓
double f(double re, double im) {
        double r23429 = re;
        double r23430 = -6.485144449769119e+83;
        bool r23431 = r23429 <= r23430;
        double r23432 = 0.5;
        double r23433 = 2.0;
        double r23434 = -1.0;
        double r23435 = r23434 * r23429;
        double r23436 = r23435 - r23429;
        double r23437 = r23433 * r23436;
        double r23438 = sqrt(r23437);
        double r23439 = r23432 * r23438;
        double r23440 = 9.194803090293717e-296;
        bool r23441 = r23429 <= r23440;
        double r23442 = r23429 * r23429;
        double r23443 = im;
        double r23444 = r23443 * r23443;
        double r23445 = r23442 + r23444;
        double r23446 = sqrt(r23445);
        double r23447 = r23446 - r23429;
        double r23448 = sqrt(r23447);
        double r23449 = r23448 * r23448;
        double r23450 = r23433 * r23449;
        double r23451 = sqrt(r23450);
        double r23452 = r23432 * r23451;
        double r23453 = 3.293868608372057e-252;
        bool r23454 = r23429 <= r23453;
        double r23455 = r23443 - r23429;
        double r23456 = r23433 * r23455;
        double r23457 = sqrt(r23456);
        double r23458 = r23432 * r23457;
        double r23459 = 2.0;
        double r23460 = pow(r23443, r23459);
        double r23461 = r23446 + r23429;
        double r23462 = r23460 / r23461;
        double r23463 = r23433 * r23462;
        double r23464 = sqrt(r23463);
        double r23465 = r23432 * r23464;
        double r23466 = r23454 ? r23458 : r23465;
        double r23467 = r23441 ? r23452 : r23466;
        double r23468 = r23431 ? r23439 : r23467;
        return r23468;
}

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 re < -6.485144449769119e+83

    1. Initial program 49.8

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

    2. Taylor expanded around -inf 11.3

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

    if -6.485144449769119e+83 < re < 9.194803090293717e-296

    1. Initial program 21.2

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

    3. Applied add-sqr-sqrt21.4

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

    if 9.194803090293717e-296 < re < 3.293868608372057e-252

    1. Initial program 34.0

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

    2. Taylor expanded around 0 32.7

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

    if 3.293868608372057e-252 < re

    1. Initial program 48.4

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

    3. Applied flip--48.3

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

    4. Simplified36.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification26.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\ \mathbf{elif}\;re \le 3.29386860837205705 \cdot 10^{-252}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))))