Average Error: 39.1 → 14.2
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -4.17890779846015664 \cdot 10^{114} \lor \neg \left(im \le -8.3405669767788097 \cdot 10^{54} \lor \neg \left(im \le 8.5755005426590888 \cdot 10^{-157} \lor \neg \left(im \le 3.33713048032984944 \cdot 10^{120}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -4.17890779846015664 \cdot 10^{114} \lor \neg \left(im \le -8.3405669767788097 \cdot 10^{54} \lor \neg \left(im \le 8.5755005426590888 \cdot 10^{-157} \lor \neg \left(im \le 3.33713048032984944 \cdot 10^{120}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\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 r24385 = 0.5;
        double r24386 = 2.0;
        double r24387 = re;
        double r24388 = r24387 * r24387;
        double r24389 = im;
        double r24390 = r24389 * r24389;
        double r24391 = r24388 + r24390;
        double r24392 = sqrt(r24391);
        double r24393 = r24392 - r24387;
        double r24394 = r24386 * r24393;
        double r24395 = sqrt(r24394);
        double r24396 = r24385 * r24395;
        return r24396;
}
↓
double f(double re, double im) {
        double r24397 = im;
        double r24398 = -4.1789077984601566e+114;
        bool r24399 = r24397 <= r24398;
        double r24400 = -8.34056697677881e+54;
        bool r24401 = r24397 <= r24400;
        double r24402 = 8.575500542659089e-157;
        bool r24403 = r24397 <= r24402;
        double r24404 = 3.3371304803298494e+120;
        bool r24405 = r24397 <= r24404;
        double r24406 = !r24405;
        bool r24407 = r24403 || r24406;
        double r24408 = !r24407;
        bool r24409 = r24401 || r24408;
        double r24410 = !r24409;
        bool r24411 = r24399 || r24410;
        double r24412 = 0.5;
        double r24413 = 2.0;
        double r24414 = re;
        double r24415 = hypot(r24414, r24397);
        double r24416 = r24415 - r24414;
        double r24417 = 0.0;
        double r24418 = r24416 + r24417;
        double r24419 = r24413 * r24418;
        double r24420 = sqrt(r24419);
        double r24421 = r24412 * r24420;
        double r24422 = 2.0;
        double r24423 = pow(r24397, r24422);
        double r24424 = r24423 + r24417;
        double r24425 = r24414 + r24415;
        double r24426 = r24424 / r24425;
        double r24427 = r24413 * r24426;
        double r24428 = sqrt(r24427);
        double r24429 = r24412 * r24428;
        double r24430 = r24411 ? r24421 : r24429;
        return r24430;
}

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 im < -4.1789077984601566e+114 or -8.34056697677881e+54 < im < 8.575500542659089e-157 or 3.3371304803298494e+120 < im

    1. Initial program 44.5

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

    3. Applied add-cube-cbrt45.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - \color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}\right)}\]

    4. Applied add-sqr-sqrt45.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    5. Applied sqrt-prod45.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    6. Applied prod-diff45.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}, \sqrt{\sqrt{re \cdot re + im \cdot im}}, -\sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}}\]

    7. Simplified15.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right)} + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}\]

    8. Simplified12.5

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

    if -4.1789077984601566e+114 < im < -8.34056697677881e+54 or 8.575500542659089e-157 < im < 3.3371304803298494e+120

    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 flip--31.6

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

    4. Simplified24.1

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

    5. Simplified19.3

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

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.17890779846015664 \cdot 10^{114} \lor \neg \left(im \le -8.3405669767788097 \cdot 10^{54} \lor \neg \left(im \le 8.5755005426590888 \cdot 10^{-157} \lor \neg \left(im \le 3.33713048032984944 \cdot 10^{120}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))))