Average Error: 38.6 → 29.5
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \cdot im \le 3.409052956304601154818324670790727469319 \cdot 10^{-322}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.862091955235903216101784317539283248128 \cdot 10^{293}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \cdot im \le 3.409052956304601154818324670790727469319 \cdot 10^{-322}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;im \cdot im \le 1.862091955235903216101784317539283248128 \cdot 10^{293}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r14419 = 0.5;
        double r14420 = 2.0;
        double r14421 = re;
        double r14422 = r14421 * r14421;
        double r14423 = im;
        double r14424 = r14423 * r14423;
        double r14425 = r14422 + r14424;
        double r14426 = sqrt(r14425);
        double r14427 = r14426 - r14421;
        double r14428 = r14420 * r14427;
        double r14429 = sqrt(r14428);
        double r14430 = r14419 * r14429;
        return r14430;
}
↓
double f(double re, double im) {
        double r14431 = im;
        double r14432 = r14431 * r14431;
        double r14433 = 3.4090529563046e-322;
        bool r14434 = r14432 <= r14433;
        double r14435 = 0.5;
        double r14436 = 2.0;
        double r14437 = -2.0;
        double r14438 = re;
        double r14439 = r14437 * r14438;
        double r14440 = r14436 * r14439;
        double r14441 = sqrt(r14440);
        double r14442 = r14435 * r14441;
        double r14443 = 1.8620919552359032e+293;
        bool r14444 = r14432 <= r14443;
        double r14445 = r14438 * r14438;
        double r14446 = r14445 + r14432;
        double r14447 = sqrt(r14446);
        double r14448 = sqrt(r14447);
        double r14449 = r14448 * r14448;
        double r14450 = r14449 - r14438;
        double r14451 = r14436 * r14450;
        double r14452 = sqrt(r14451);
        double r14453 = r14435 * r14452;
        double r14454 = r14431 - r14438;
        double r14455 = r14436 * r14454;
        double r14456 = sqrt(r14455);
        double r14457 = r14435 * r14456;
        double r14458 = r14444 ? r14453 : r14457;
        double r14459 = r14434 ? r14442 : r14458;
        return r14459;
}

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 (* im im) < 3.4090529563046e-322

    1. Initial program 44.3

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

    2. Taylor expanded around -inf 36.4

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

    if 3.4090529563046e-322 < (* im im) < 1.8620919552359032e+293

    1. Initial program 23.4

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

    3. Applied add-sqr-sqrt23.4

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

    4. Applied sqrt-prod23.5

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

    if 1.8620919552359032e+293 < (* im im)

    1. Initial program 61.8

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

    2. Taylor expanded around 0 34.4

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

  4. Final simplification29.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \cdot im \le 3.409052956304601154818324670790727469319 \cdot 10^{-322}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;im \cdot im \le 1.862091955235903216101784317539283248128 \cdot 10^{293}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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)))))