Average Error: 38.8 → 27.4
Time: 4.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.015832581446431561803372347247675693307 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 2.165263052480363094004576098455732339339 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 2.065651901617657621042712121435180714289 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot e^{\log \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
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}{\sqrt{re \cdot re + im \cdot im} - 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 r221366 = 0.5;
        double r221367 = 2.0;
        double r221368 = re;
        double r221369 = r221368 * r221368;
        double r221370 = im;
        double r221371 = r221370 * r221370;
        double r221372 = r221369 + r221371;
        double r221373 = sqrt(r221372);
        double r221374 = r221373 + r221368;
        double r221375 = r221367 * r221374;
        double r221376 = sqrt(r221375);
        double r221377 = r221366 * r221376;
        return r221377;
}
↓
double f(double re, double im) {
        double r221378 = re;
        double r221379 = -1.0158325814464316e-264;
        bool r221380 = r221378 <= r221379;
        double r221381 = 0.5;
        double r221382 = 2.0;
        double r221383 = im;
        double r221384 = r221383 * r221383;
        double r221385 = r221378 * r221378;
        double r221386 = r221385 + r221384;
        double r221387 = sqrt(r221386);
        double r221388 = r221387 - r221378;
        double r221389 = r221384 / r221388;
        double r221390 = r221382 * r221389;
        double r221391 = sqrt(r221390);
        double r221392 = r221381 * r221391;
        double r221393 = 2.165263052480363e-217;
        bool r221394 = r221378 <= r221393;
        double r221395 = r221378 + r221383;
        double r221396 = r221382 * r221395;
        double r221397 = sqrt(r221396);
        double r221398 = r221381 * r221397;
        double r221399 = 2.0656519016176576e+119;
        bool r221400 = r221378 <= r221399;
        double r221401 = sqrt(r221387);
        double r221402 = r221401 * r221401;
        double r221403 = r221402 + r221378;
        double r221404 = log(r221403);
        double r221405 = exp(r221404);
        double r221406 = r221382 * r221405;
        double r221407 = sqrt(r221406);
        double r221408 = r221381 * r221407;
        double r221409 = 2.0;
        double r221410 = r221409 * r221378;
        double r221411 = r221382 * r221410;
        double r221412 = sqrt(r221411);
        double r221413 = r221381 * r221412;
        double r221414 = r221400 ? r221408 : r221413;
        double r221415 = r221394 ? r221398 : r221414;
        double r221416 = r221380 ? r221392 : r221415;
        return r221416;
}

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 flip-+47.2

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

    4. Simplified36.2

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

    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. 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. Using strategy rm

    6. Applied add-sqr-sqrt55.6

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

    7. Applied sqrt-prod55.6

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

    8. Applied sqrt-prod55.6

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

    9. Applied associate-*r*55.6

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

    10. Simplified55.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{3}} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt55.7

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

    13. 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

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

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)))))