Average Error: 37.3 → 26.9
Time: 19.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\
\;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\
\;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\
\;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\

\end{array}
double f(double re, double im) {
        double r7714269 = 0.5;
        double r7714270 = 2.0;
        double r7714271 = re;
        double r7714272 = r7714271 * r7714271;
        double r7714273 = im;
        double r7714274 = r7714273 * r7714273;
        double r7714275 = r7714272 + r7714274;
        double r7714276 = sqrt(r7714275);
        double r7714277 = r7714276 + r7714271;
        double r7714278 = r7714270 * r7714277;
        double r7714279 = sqrt(r7714278);
        double r7714280 = r7714269 * r7714279;
        return r7714280;
}
↓
double f(double re, double im) {
        double r7714281 = re;
        double r7714282 = -1.0974932438808633e+26;
        bool r7714283 = r7714281 <= r7714282;
        double r7714284 = im;
        double r7714285 = r7714284 * r7714284;
        double r7714286 = 2.0;
        double r7714287 = r7714285 * r7714286;
        double r7714288 = sqrt(r7714287);
        double r7714289 = r7714281 * r7714281;
        double r7714290 = r7714285 + r7714289;
        double r7714291 = sqrt(r7714290);
        double r7714292 = r7714291 - r7714281;
        double r7714293 = sqrt(r7714292);
        double r7714294 = r7714288 / r7714293;
        double r7714295 = 0.5;
        double r7714296 = r7714294 * r7714295;
        double r7714297 = -4.4945327826415316e-20;
        bool r7714298 = r7714281 <= r7714297;
        double r7714299 = r7714281 + r7714284;
        double r7714300 = r7714299 * r7714286;
        double r7714301 = sqrt(r7714300);
        double r7714302 = r7714301 * r7714295;
        double r7714303 = -7.961223836723572e-96;
        bool r7714304 = r7714281 <= r7714303;
        double r7714305 = r7714285 / r7714292;
        double r7714306 = r7714286 * r7714305;
        double r7714307 = sqrt(r7714306);
        double r7714308 = r7714295 * r7714307;
        double r7714309 = -2.538815066158378e-267;
        bool r7714310 = r7714281 <= r7714309;
        double r7714311 = 1.8791426213625292e+66;
        bool r7714312 = r7714281 <= r7714311;
        double r7714313 = r7714281 + r7714291;
        double r7714314 = r7714313 * r7714286;
        double r7714315 = sqrt(r7714314);
        double r7714316 = r7714315 * r7714295;
        double r7714317 = r7714281 + r7714281;
        double r7714318 = r7714317 * r7714286;
        double r7714319 = sqrt(r7714318);
        double r7714320 = r7714319 * r7714295;
        double r7714321 = r7714312 ? r7714316 : r7714320;
        double r7714322 = r7714310 ? r7714302 : r7714321;
        double r7714323 = r7714304 ? r7714308 : r7714322;
        double r7714324 = r7714298 ? r7714302 : r7714323;
        double r7714325 = r7714283 ? r7714296 : r7714324;
        return r7714325;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target32.5
Herbie26.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.0974932438808633e+26

    1. Initial program 56.7

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-exp-log59.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]
    4. Using strategy rm

    5. Applied flip-+59.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re \cdot re}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re}}}\]

    6. Applied associate-*r/59.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re \cdot re\right)}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re}}}\]

    7. Applied sqrt-div59.1

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} \cdot e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re \cdot re\right)}}{\sqrt{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re}}}\]

    8. Simplified39.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im\right) \cdot 2.0}}}{\sqrt{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re}}\]

    9. Simplified38.8

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    if -1.0974932438808633e+26 < re < -4.4945327826415316e-20 or -7.961223836723572e-96 < re < -2.538815066158378e-267

    1. Initial program 33.8

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-exp-log35.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}} + re\right)}\]

    4. Taylor expanded around 0 38.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} + re\right)}\]

    if -4.4945327826415316e-20 < re < -7.961223836723572e-96

    1. Initial program 38.9

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+38.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Simplified28.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

    if -2.538815066158378e-267 < re < 1.8791426213625292e+66

    1. Initial program 21.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    if 1.8791426213625292e+66 < re

    1. Initial program 45.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around inf 11.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))