Average Error: 38.3 → 11.2
Time: 4.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 7.84862278851990408 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 7.84862278851990408 \cdot 10^{68}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r20438 = 0.5;
        double r20439 = 2.0;
        double r20440 = re;
        double r20441 = r20440 * r20440;
        double r20442 = im;
        double r20443 = r20442 * r20442;
        double r20444 = r20441 + r20443;
        double r20445 = sqrt(r20444);
        double r20446 = r20445 - r20440;
        double r20447 = r20439 * r20446;
        double r20448 = sqrt(r20447);
        double r20449 = r20438 * r20448;
        return r20449;
}
↓
double f(double re, double im) {
        double r20450 = re;
        double r20451 = 7.848622788519904e+68;
        bool r20452 = r20450 <= r20451;
        double r20453 = 0.5;
        double r20454 = 2.0;
        double r20455 = im;
        double r20456 = hypot(r20450, r20455);
        double r20457 = r20456 - r20450;
        double r20458 = r20454 * r20457;
        double r20459 = sqrt(r20458);
        double r20460 = r20453 * r20459;
        double r20461 = 2.0;
        double r20462 = pow(r20455, r20461);
        double r20463 = 0.0;
        double r20464 = r20462 + r20463;
        double r20465 = r20450 + r20456;
        double r20466 = r20464 / r20465;
        double r20467 = r20454 * r20466;
        double r20468 = sqrt(r20467);
        double r20469 = r20453 * r20468;
        double r20470 = r20452 ? r20460 : r20469;
        return r20470;
}

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 2 regimes

  2. if re < 7.848622788519904e+68

    1. Initial program 33.3

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

    3. Applied hypot-def6.5

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

    if 7.848622788519904e+68 < re

    1. Initial program 59.6

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

    3. Applied flip--59.6

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

    4. Simplified43.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Simplified30.9

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 7.84862278851990408 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))))