Average Error: 38.6 → 11.9
Time: 4.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 6.234036773388662816925137796587363022033 \cdot 10^{189}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 6.234036773388662816925137796587363022033 \cdot 10^{189}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\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 r30406 = 0.5;
        double r30407 = 2.0;
        double r30408 = re;
        double r30409 = r30408 * r30408;
        double r30410 = im;
        double r30411 = r30410 * r30410;
        double r30412 = r30409 + r30411;
        double r30413 = sqrt(r30412);
        double r30414 = r30413 - r30408;
        double r30415 = r30407 * r30414;
        double r30416 = sqrt(r30415);
        double r30417 = r30406 * r30416;
        return r30417;
}
↓
double f(double re, double im) {
        double r30418 = re;
        double r30419 = 6.234036773388663e+189;
        bool r30420 = r30418 <= r30419;
        double r30421 = 0.5;
        double r30422 = 2.0;
        double r30423 = 1.0;
        double r30424 = im;
        double r30425 = hypot(r30418, r30424);
        double r30426 = r30425 - r30418;
        double r30427 = r30423 * r30426;
        double r30428 = r30422 * r30427;
        double r30429 = sqrt(r30428);
        double r30430 = r30421 * r30429;
        double r30431 = 2.0;
        double r30432 = pow(r30424, r30431);
        double r30433 = 0.0;
        double r30434 = r30432 + r30433;
        double r30435 = r30418 + r30425;
        double r30436 = r30434 / r30435;
        double r30437 = r30422 * r30436;
        double r30438 = sqrt(r30437);
        double r30439 = r30421 * r30438;
        double r30440 = r30420 ? r30430 : r30439;
        return r30440;
}

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 < 6.234036773388663e+189

    1. Initial program 35.9

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

    3. Applied *-un-lft-identity35.9

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

    4. Applied *-un-lft-identity35.9

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

    5. Applied distribute-lft-out--35.9

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

    6. Simplified9.7

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

    if 6.234036773388663e+189 < re

    1. Initial program 64.0

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

    3. Applied flip--64.0

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

    4. Simplified50.6

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

    5. Simplified32.6

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 6.234036773388662816925137796587363022033 \cdot 10^{189}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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)))))