Average Error: 39.2 → 24.1
Time: 5.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.792801863201988590980342546956552051615 \cdot 10^{155}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\ \mathbf{elif}\;re \le -4.201891031489358046275693965712763863255 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le -3.886845242418946786634156336833997677928 \cdot 10^{-283}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 8.854091567426593596102456745468139063682 \cdot 10^{133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.792801863201988590980342546956552051615 \cdot 10^{155}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\

\mathbf{elif}\;re \le -4.201891031489358046275693965712763863255 \cdot 10^{-217}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le -3.886845242418946786634156336833997677928 \cdot 10^{-283}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 8.854091567426593596102456745468139063682 \cdot 10^{133}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\

\end{array}
double f(double re, double im) {
        double r169310 = 0.5;
        double r169311 = 2.0;
        double r169312 = re;
        double r169313 = r169312 * r169312;
        double r169314 = im;
        double r169315 = r169314 * r169314;
        double r169316 = r169313 + r169315;
        double r169317 = sqrt(r169316);
        double r169318 = r169317 + r169312;
        double r169319 = r169311 * r169318;
        double r169320 = sqrt(r169319);
        double r169321 = r169310 * r169320;
        return r169321;
}
↓
double f(double re, double im) {
        double r169322 = re;
        double r169323 = -6.792801863201989e+155;
        bool r169324 = r169322 <= r169323;
        double r169325 = 0.5;
        double r169326 = 2.0;
        double r169327 = im;
        double r169328 = r169327 * r169327;
        double r169329 = -1.0;
        double r169330 = r169329 * r169322;
        double r169331 = r169330 - r169322;
        double r169332 = r169328 / r169331;
        double r169333 = r169326 * r169332;
        double r169334 = sqrt(r169333);
        double r169335 = r169325 * r169334;
        double r169336 = -4.201891031489358e-217;
        bool r169337 = r169322 <= r169336;
        double r169338 = r169326 * r169328;
        double r169339 = sqrt(r169338);
        double r169340 = r169322 * r169322;
        double r169341 = r169340 + r169328;
        double r169342 = sqrt(r169341);
        double r169343 = r169342 - r169322;
        double r169344 = sqrt(r169343);
        double r169345 = r169339 / r169344;
        double r169346 = r169325 * r169345;
        double r169347 = -3.886845242418947e-283;
        bool r169348 = r169322 <= r169347;
        double r169349 = r169327 + r169322;
        double r169350 = r169326 * r169349;
        double r169351 = sqrt(r169350);
        double r169352 = r169325 * r169351;
        double r169353 = 8.854091567426594e+133;
        bool r169354 = r169322 <= r169353;
        double r169355 = cbrt(r169341);
        double r169356 = fabs(r169355);
        double r169357 = sqrt(r169355);
        double r169358 = r169356 * r169357;
        double r169359 = r169358 + r169322;
        double r169360 = r169326 * r169359;
        double r169361 = sqrt(r169360);
        double r169362 = r169325 * r169361;
        double r169363 = r169322 + r169322;
        double r169364 = r169326 * r169363;
        double r169365 = sqrt(r169364);
        double r169366 = r169325 * r169365;
        double r169367 = r169354 ? r169362 : r169366;
        double r169368 = r169348 ? r169352 : r169367;
        double r169369 = r169337 ? r169346 : r169368;
        double r169370 = r169324 ? r169335 : r169369;
        return r169370;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.2
Target34.0
Herbie24.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -6.792801863201989e+155

    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. Simplified50.1

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

    5. Taylor expanded around -inf 31.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\color{blue}{-1 \cdot re} - re}}\]

    if -6.792801863201989e+155 < re < -4.201891031489358e-217

    1. Initial program 41.9

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

    3. Applied flip-+41.8

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

    4. Simplified30.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-*r/30.6

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

    7. Applied sqrt-div29.3

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

    if -4.201891031489358e-217 < re < -3.886845242418947e-283

    1. Initial program 34.6

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around 0 35.0

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

    if -3.886845242418947e-283 < re < 8.854091567426594e+133

    1. Initial program 21.7

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

    3. Applied add-cube-cbrt21.9

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

    4. Applied sqrt-prod21.9

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

    5. Simplified21.9

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

    if 8.854091567426594e+133 < re

    1. Initial program 58.8

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around inf 8.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification24.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.792801863201988590980342546956552051615 \cdot 10^{155}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{-1 \cdot re - re}}\\ \mathbf{elif}\;re \le -4.201891031489358046275693965712763863255 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le -3.886845242418946786634156336833997677928 \cdot 10^{-283}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 8.854091567426593596102456745468139063682 \cdot 10^{133}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))