Average Error: 38.0 → 11.2
Time: 3.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 3.226330986754208067950915824673981172987 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 3.226330986754208067950915824673981172987 \cdot 10^{-16}:\\
\;\;\;\;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 r13505 = 0.5;
        double r13506 = 2.0;
        double r13507 = re;
        double r13508 = r13507 * r13507;
        double r13509 = im;
        double r13510 = r13509 * r13509;
        double r13511 = r13508 + r13510;
        double r13512 = sqrt(r13511);
        double r13513 = r13512 - r13507;
        double r13514 = r13506 * r13513;
        double r13515 = sqrt(r13514);
        double r13516 = r13505 * r13515;
        return r13516;
}
↓
double f(double re, double im) {
        double r13517 = re;
        double r13518 = 3.226330986754208e-16;
        bool r13519 = r13517 <= r13518;
        double r13520 = 0.5;
        double r13521 = 2.0;
        double r13522 = 1.0;
        double r13523 = im;
        double r13524 = hypot(r13517, r13523);
        double r13525 = r13524 - r13517;
        double r13526 = r13522 * r13525;
        double r13527 = r13521 * r13526;
        double r13528 = sqrt(r13527);
        double r13529 = r13520 * r13528;
        double r13530 = 2.0;
        double r13531 = pow(r13523, r13530);
        double r13532 = 0.0;
        double r13533 = r13531 + r13532;
        double r13534 = r13517 + r13524;
        double r13535 = r13533 / r13534;
        double r13536 = r13521 * r13535;
        double r13537 = sqrt(r13536);
        double r13538 = r13520 * r13537;
        double r13539 = r13519 ? r13529 : r13538;
        return r13539;
}

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 < 3.226330986754208e-16

    1. Initial program 31.4

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

    3. Applied *-un-lft-identity31.4

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

    4. Applied *-un-lft-identity31.4

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

    5. Applied distribute-lft-out--31.4

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

    6. Simplified4.4

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

    if 3.226330986754208e-16 < re

    1. Initial program 56.4

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

    3. Applied flip--56.4

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

    4. Simplified39.5

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

    5. Simplified30.1

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 3.226330986754208067950915824673981172987 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))))