Average Error: 38.9 → 11.2
Time: 3.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 98254381358.475433:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 98254381358.475433:\\
\;\;\;\;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{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r11492 = 0.5;
        double r11493 = 2.0;
        double r11494 = re;
        double r11495 = r11494 * r11494;
        double r11496 = im;
        double r11497 = r11496 * r11496;
        double r11498 = r11495 + r11497;
        double r11499 = sqrt(r11498);
        double r11500 = r11499 - r11494;
        double r11501 = r11493 * r11500;
        double r11502 = sqrt(r11501);
        double r11503 = r11492 * r11502;
        return r11503;
}
↓
double f(double re, double im) {
        double r11504 = re;
        double r11505 = 98254381358.47543;
        bool r11506 = r11504 <= r11505;
        double r11507 = 0.5;
        double r11508 = 2.0;
        double r11509 = 1.0;
        double r11510 = im;
        double r11511 = hypot(r11504, r11510);
        double r11512 = r11511 - r11504;
        double r11513 = r11509 * r11512;
        double r11514 = r11508 * r11513;
        double r11515 = sqrt(r11514);
        double r11516 = r11507 * r11515;
        double r11517 = 0.0;
        double r11518 = fma(r11510, r11510, r11517);
        double r11519 = r11504 + r11511;
        double r11520 = r11518 / r11519;
        double r11521 = r11508 * r11520;
        double r11522 = sqrt(r11521);
        double r11523 = r11507 * r11522;
        double r11524 = r11506 ? r11516 : r11523;
        return r11524;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 2 regimes

  2. if re < 98254381358.47543

    1. Initial program 32.9

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

    3. Applied *-un-lft-identity32.9

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

    4. Applied *-un-lft-identity32.9

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

    5. Applied distribute-lft-out--32.9

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

    6. Simplified4.9

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

    if 98254381358.47543 < re

    1. Initial program 57.0

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

    3. Applied flip--57.0

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

    4. Simplified40.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{\mathsf{fma}\left(im, im, 0\right)}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Simplified30.3

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 98254381358.475433:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{\mathsf{fma}\left(im, im, 0\right)}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))))