Average Error: 37.3 → 17.8
Time: 18.8s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1639866401777603 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left|im\right| \cdot \sqrt{2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\ \;\;\;\;\left(\left(\left|im\right| \cdot \sqrt{2.0}\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 3.4426323618367234 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 3.165636632302708 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.1639866401777603 \cdot 10^{+111}:\\
\;\;\;\;\frac{\left|im\right| \cdot \sqrt{2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\

\mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\
\;\;\;\;\left(\left(\left|im\right| \cdot \sqrt{2.0}\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot 0.5\\

\mathbf{elif}\;re \le 3.4426323618367234 \cdot 10^{-252}:\\
\;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 3.165636632302708 \cdot 10^{+48}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r4402253 = 0.5;
        double r4402254 = 2.0;
        double r4402255 = re;
        double r4402256 = r4402255 * r4402255;
        double r4402257 = im;
        double r4402258 = r4402257 * r4402257;
        double r4402259 = r4402256 + r4402258;
        double r4402260 = sqrt(r4402259);
        double r4402261 = r4402260 + r4402255;
        double r4402262 = r4402254 * r4402261;
        double r4402263 = sqrt(r4402262);
        double r4402264 = r4402253 * r4402263;
        return r4402264;
}
↓
double f(double re, double im) {
        double r4402265 = re;
        double r4402266 = -1.1639866401777603e+111;
        bool r4402267 = r4402265 <= r4402266;
        double r4402268 = im;
        double r4402269 = fabs(r4402268);
        double r4402270 = 2.0;
        double r4402271 = sqrt(r4402270);
        double r4402272 = r4402269 * r4402271;
        double r4402273 = -2.0;
        double r4402274 = r4402273 * r4402265;
        double r4402275 = sqrt(r4402274);
        double r4402276 = r4402272 / r4402275;
        double r4402277 = 0.5;
        double r4402278 = r4402276 * r4402277;
        double r4402279 = -6.3441075981154e-203;
        bool r4402280 = r4402265 <= r4402279;
        double r4402281 = 1.0;
        double r4402282 = r4402265 * r4402265;
        double r4402283 = r4402268 * r4402268;
        double r4402284 = r4402282 + r4402283;
        double r4402285 = sqrt(r4402284);
        double r4402286 = r4402285 - r4402265;
        double r4402287 = sqrt(r4402286);
        double r4402288 = r4402281 / r4402287;
        double r4402289 = r4402272 * r4402288;
        double r4402290 = r4402289 * r4402277;
        double r4402291 = 3.4426323618367234e-252;
        bool r4402292 = r4402265 <= r4402291;
        double r4402293 = r4402265 + r4402268;
        double r4402294 = r4402293 * r4402270;
        double r4402295 = sqrt(r4402294);
        double r4402296 = r4402295 * r4402277;
        double r4402297 = 3.165636632302708e+48;
        bool r4402298 = r4402265 <= r4402297;
        double r4402299 = r4402285 + r4402265;
        double r4402300 = r4402270 * r4402299;
        double r4402301 = sqrt(r4402300);
        double r4402302 = r4402277 * r4402301;
        double r4402303 = r4402265 + r4402265;
        double r4402304 = r4402303 * r4402270;
        double r4402305 = sqrt(r4402304);
        double r4402306 = r4402305 * r4402277;
        double r4402307 = r4402298 ? r4402302 : r4402306;
        double r4402308 = r4402292 ? r4402296 : r4402307;
        double r4402309 = r4402280 ? r4402290 : r4402308;
        double r4402310 = r4402267 ? r4402278 : r4402309;
        return r4402310;
}

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
Target32.6
Herbie17.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.1639866401777603e+111

    1. Initial program 60.7

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

    3. Applied flip-+60.7

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

    4. Applied associate-*r/60.7

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

    5. Applied sqrt-div60.7

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

    6. Simplified46.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im - 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod46.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{im \cdot im - 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    9. Simplified44.1

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

    10. Taylor expanded around -inf 10.0

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \left|im\right|}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -1.1639866401777603e+111 < re < -6.3441075981154e-203

    1. Initial program 40.7

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

    3. Applied flip-+40.7

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

    4. Applied associate-*r/40.7

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

    5. Applied sqrt-div40.8

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

    6. Simplified29.8

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im - 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod29.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{im \cdot im - 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    9. Simplified18.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \color{blue}{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    10. Using strategy rm

    11. Applied div-inv18.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2.0} \cdot \left|im\right|\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

    if -6.3441075981154e-203 < re < 3.4426323618367234e-252

    1. Initial program 27.7

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

    2. Taylor expanded around 0 31.8

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

    if 3.4426323618367234e-252 < re < 3.165636632302708e+48

    1. Initial program 19.3

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

    if 3.165636632302708e+48 < re

    1. Initial program 41.9

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

    2. Taylor expanded around inf 13.0

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1639866401777603 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left|im\right| \cdot \sqrt{2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\ \;\;\;\;\left(\left(\left|im\right| \cdot \sqrt{2.0}\right) \cdot \frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 3.4426323618367234 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 3.165636632302708 \cdot 10^{+48}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(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)))))