Average Error: 39.1 → 23.3
Time: 5.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.30799658357328170507068570933223802163 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -1.566161132000562091721385925067375800416 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le -3.068368918555544922720895417815131992438 \cdot 10^{-192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im - re}\right)}\\ \mathbf{elif}\;re \le 3.880381092944904394836799132431800994872 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot im\right)}\\ \mathbf{elif}\;re \le 9.236024919771600304746451409373163447937 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.30799658357328170507068570933223802163 \cdot 10^{154}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\

\mathbf{elif}\;re \le -1.566161132000562091721385925067375800416 \cdot 10^{-143}:\\
\;\;\;\;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 -3.068368918555544922720895417815131992438 \cdot 10^{-192}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im - re}\right)}\\

\mathbf{elif}\;re \le 3.880381092944904394836799132431800994872 \cdot 10^{-302}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot im\right)}\\

\mathbf{elif}\;re \le 9.236024919771600304746451409373163447937 \cdot 10^{68}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}\right)\\

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

\end{array}
double f(double re, double im) {
        double r342265 = 0.5;
        double r342266 = 2.0;
        double r342267 = re;
        double r342268 = r342267 * r342267;
        double r342269 = im;
        double r342270 = r342269 * r342269;
        double r342271 = r342268 + r342270;
        double r342272 = sqrt(r342271);
        double r342273 = r342272 + r342267;
        double r342274 = r342266 * r342273;
        double r342275 = sqrt(r342274);
        double r342276 = r342265 * r342275;
        return r342276;
}
↓
double f(double re, double im) {
        double r342277 = re;
        double r342278 = -1.3079965835732817e+154;
        bool r342279 = r342277 <= r342278;
        double r342280 = 0.5;
        double r342281 = 2.0;
        double r342282 = im;
        double r342283 = -2.0;
        double r342284 = r342283 * r342277;
        double r342285 = r342282 / r342284;
        double r342286 = r342282 * r342285;
        double r342287 = r342281 * r342286;
        double r342288 = sqrt(r342287);
        double r342289 = r342280 * r342288;
        double r342290 = -1.566161132000562e-143;
        bool r342291 = r342277 <= r342290;
        double r342292 = r342282 * r342282;
        double r342293 = r342281 * r342292;
        double r342294 = sqrt(r342293);
        double r342295 = r342277 * r342277;
        double r342296 = r342295 + r342292;
        double r342297 = sqrt(r342296);
        double r342298 = r342297 - r342277;
        double r342299 = sqrt(r342298);
        double r342300 = r342294 / r342299;
        double r342301 = r342280 * r342300;
        double r342302 = -3.068368918555545e-192;
        bool r342303 = r342277 <= r342302;
        double r342304 = r342282 - r342277;
        double r342305 = r342282 / r342304;
        double r342306 = r342282 * r342305;
        double r342307 = r342281 * r342306;
        double r342308 = sqrt(r342307);
        double r342309 = r342280 * r342308;
        double r342310 = 3.8803810929449044e-302;
        bool r342311 = r342277 <= r342310;
        double r342312 = r342282 / r342298;
        double r342313 = r342312 * r342282;
        double r342314 = r342281 * r342313;
        double r342315 = sqrt(r342314);
        double r342316 = r342280 * r342315;
        double r342317 = 9.2360249197716e+68;
        bool r342318 = r342277 <= r342317;
        double r342319 = sqrt(r342281);
        double r342320 = r342297 + r342277;
        double r342321 = sqrt(r342320);
        double r342322 = r342319 * r342321;
        double r342323 = r342280 * r342322;
        double r342324 = 2.0;
        double r342325 = r342324 * r342277;
        double r342326 = r342281 * r342325;
        double r342327 = sqrt(r342326);
        double r342328 = r342280 * r342327;
        double r342329 = r342318 ? r342323 : r342328;
        double r342330 = r342311 ? r342316 : r342329;
        double r342331 = r342303 ? r342309 : r342330;
        double r342332 = r342291 ? r342301 : r342331;
        double r342333 = r342279 ? r342289 : r342332;
        return r342333;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target34.1
Herbie23.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 6 regimes

  2. if re < -1.3079965835732817e+154

    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. Simplified50.8

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

    6. Applied *-un-lft-identity50.8

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

    7. Applied times-frac50.4

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

    8. Simplified50.4

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

    9. Taylor expanded around -inf 23.9

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

    if -1.3079965835732817e+154 < re < -1.566161132000562e-143

    1. Initial program 45.6

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

    3. Applied flip-+45.6

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

    4. Simplified31.3

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

    6. Applied associate-*r/31.4

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

    7. Applied sqrt-div29.5

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

    if -1.566161132000562e-143 < re < -3.068368918555545e-192

    1. Initial program 30.9

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

    3. Applied flip-+30.5

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

    4. Simplified30.4

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

    6. Applied *-un-lft-identity30.4

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

    7. Applied times-frac27.4

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

    8. Simplified27.4

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

    9. Taylor expanded around 0 38.3

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

    if -3.068368918555545e-192 < re < 3.8803810929449044e-302

    1. Initial program 30.7

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

    3. Applied flip-+30.2

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

    4. Simplified30.2

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

    6. Applied *-un-lft-identity30.2

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

    7. Applied times-frac28.9

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

    8. Simplified28.9

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

    10. Applied *-commutative28.9

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

    if 3.8803810929449044e-302 < re < 9.2360249197716e+68

    1. Initial program 21.4

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

    3. Applied sqrt-prod21.7

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

    if 9.2360249197716e+68 < re

    1. Initial program 48.0

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

    2. Taylor expanded around inf 12.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification23.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.30799658357328170507068570933223802163 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -1.566161132000562091721385925067375800416 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le -3.068368918555544922720895417815131992438 \cdot 10^{-192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{im - re}\right)}\\ \mathbf{elif}\;re \le 3.880381092944904394836799132431800994872 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\sqrt{re \cdot re + im \cdot im} - re} \cdot im\right)}\\ \mathbf{elif}\;re \le 9.236024919771600304746451409373163447937 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(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)))))