Average Error: 38.8 → 27.4
Time: 3.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.015832581446431561803372347247675693307 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{\sqrt{re \cdot re + im \cdot im} + -1 \cdot re}}\\ \mathbf{elif}\;re \le 2.165263052480363094004576098455732339339 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 2.065651901617657621042712121435180714289 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot e^{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.015832581446431561803372347247675693307 \cdot 10^{-264}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{\sqrt{re \cdot re + im \cdot im} + -1 \cdot re}}\\

\mathbf{elif}\;re \le 2.165263052480363094004576098455732339339 \cdot 10^{-217}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 2.065651901617657621042712121435180714289 \cdot 10^{119}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot e^{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \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 r248254 = 0.5;
        double r248255 = 2.0;
        double r248256 = re;
        double r248257 = r248256 * r248256;
        double r248258 = im;
        double r248259 = r248258 * r248258;
        double r248260 = r248257 + r248259;
        double r248261 = sqrt(r248260);
        double r248262 = r248261 + r248256;
        double r248263 = r248255 * r248262;
        double r248264 = sqrt(r248263);
        double r248265 = r248254 * r248264;
        return r248265;
}
↓
double f(double re, double im) {
        double r248266 = re;
        double r248267 = -1.0158325814464316e-264;
        bool r248268 = r248266 <= r248267;
        double r248269 = 0.5;
        double r248270 = 2.0;
        double r248271 = im;
        double r248272 = r248271 * r248271;
        double r248273 = 0.0;
        double r248274 = r248272 + r248273;
        double r248275 = r248266 * r248266;
        double r248276 = r248275 + r248272;
        double r248277 = sqrt(r248276);
        double r248278 = -1.0;
        double r248279 = r248278 * r248266;
        double r248280 = r248277 + r248279;
        double r248281 = r248274 / r248280;
        double r248282 = r248270 * r248281;
        double r248283 = sqrt(r248282);
        double r248284 = r248269 * r248283;
        double r248285 = 2.165263052480363e-217;
        bool r248286 = r248266 <= r248285;
        double r248287 = r248266 + r248271;
        double r248288 = r248270 * r248287;
        double r248289 = sqrt(r248288);
        double r248290 = r248269 * r248289;
        double r248291 = 2.0656519016176576e+119;
        bool r248292 = r248266 <= r248291;
        double r248293 = sqrt(r248277);
        double r248294 = r248293 * r248293;
        double r248295 = r248294 + r248266;
        double r248296 = log(r248295);
        double r248297 = exp(r248296);
        double r248298 = r248270 * r248297;
        double r248299 = sqrt(r248298);
        double r248300 = r248269 * r248299;
        double r248301 = 2.0;
        double r248302 = r248301 * r248266;
        double r248303 = r248270 * r248302;
        double r248304 = sqrt(r248303);
        double r248305 = r248269 * r248304;
        double r248306 = r248292 ? r248300 : r248305;
        double r248307 = r248286 ? r248290 : r248306;
        double r248308 = r248268 ? r248284 : r248307;
        return r248308;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target33.8
Herbie27.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.0158325814464316e-264

    1. Initial program 47.3

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

    3. Applied add-sqr-sqrt47.3

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

    4. Applied sqrt-prod48.0

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

    6. Applied flip-+47.9

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

    7. Simplified36.3

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

    8. Simplified36.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{\color{blue}{\sqrt{re \cdot re + im \cdot im} + -1 \cdot re}}}\]

    if -1.0158325814464316e-264 < re < 2.165263052480363e-217

    1. Initial program 30.5

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

    3. Applied add-sqr-sqrt30.5

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

    4. Applied sqrt-prod30.7

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

    5. Taylor expanded around 0 33.0

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

    if 2.165263052480363e-217 < re < 2.0656519016176576e+119

    1. Initial program 18.0

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

    3. Applied add-sqr-sqrt18.0

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

    4. Applied sqrt-prod18.0

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

    6. Applied add-exp-log20.6

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

    if 2.0656519016176576e+119 < re

    1. Initial program 55.6

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

    3. Applied add-sqr-sqrt55.6

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

    4. Applied sqrt-prod55.6

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

    5. Taylor expanded around inf 9.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.015832581446431561803372347247675693307 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{\sqrt{re \cdot re + im \cdot im} + -1 \cdot re}}\\ \mathbf{elif}\;re \le 2.165263052480363094004576098455732339339 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 2.065651901617657621042712121435180714289 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot e^{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))))