Average Error: 37.2 → 25.7
Time: 15.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.940844341770915 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 2.1282394422164936 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.940844341770915 \cdot 10^{-249}:\\
\;\;\;\;\sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\

\mathbf{elif}\;re \le 2.1282394422164936 \cdot 10^{+111}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\

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

\end{array}
double f(double re, double im) {
        double r6235375 = 0.5;
        double r6235376 = 2.0;
        double r6235377 = re;
        double r6235378 = r6235377 * r6235377;
        double r6235379 = im;
        double r6235380 = r6235379 * r6235379;
        double r6235381 = r6235378 + r6235380;
        double r6235382 = sqrt(r6235381);
        double r6235383 = r6235382 + r6235377;
        double r6235384 = r6235376 * r6235383;
        double r6235385 = sqrt(r6235384);
        double r6235386 = r6235375 * r6235385;
        return r6235386;
}
↓
double f(double re, double im) {
        double r6235387 = re;
        double r6235388 = -1.940844341770915e-249;
        bool r6235389 = r6235387 <= r6235388;
        double r6235390 = 2.0;
        double r6235391 = im;
        double r6235392 = r6235391 * r6235391;
        double r6235393 = r6235387 * r6235387;
        double r6235394 = r6235392 + r6235393;
        double r6235395 = sqrt(r6235394);
        double r6235396 = r6235395 - r6235387;
        double r6235397 = r6235392 / r6235396;
        double r6235398 = r6235390 * r6235397;
        double r6235399 = sqrt(r6235398);
        double r6235400 = 0.5;
        double r6235401 = r6235399 * r6235400;
        double r6235402 = 1.0730314248253915e-288;
        bool r6235403 = r6235387 <= r6235402;
        double r6235404 = r6235391 + r6235387;
        double r6235405 = r6235390 * r6235404;
        double r6235406 = sqrt(r6235405);
        double r6235407 = r6235406 * r6235400;
        double r6235408 = 2.1282394422164936e+111;
        bool r6235409 = r6235387 <= r6235408;
        double r6235410 = r6235387 + r6235395;
        double r6235411 = r6235390 * r6235410;
        double r6235412 = sqrt(r6235411);
        double r6235413 = r6235400 * r6235412;
        double r6235414 = r6235387 + r6235387;
        double r6235415 = r6235414 * r6235390;
        double r6235416 = sqrt(r6235415);
        double r6235417 = r6235400 * r6235416;
        double r6235418 = r6235409 ? r6235413 : r6235417;
        double r6235419 = r6235403 ? r6235407 : r6235418;
        double r6235420 = r6235389 ? r6235401 : r6235419;
        return r6235420;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target32.4
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.940844341770915e-249

    1. Initial program 46.4

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

    3. Applied add-sqr-sqrt47.0

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

    5. Applied flip-+47.0

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

    6. Simplified35.6

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

    7. Simplified35.6

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

    if -1.940844341770915e-249 < re < 1.0730314248253915e-288

    1. Initial program 30.2

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

    3. Applied add-sqr-sqrt30.3

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

    4. Taylor expanded around 0 29.6

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

    if 1.0730314248253915e-288 < re < 2.1282394422164936e+111

    1. Initial program 19.4

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

    if 2.1282394422164936e+111 < re

    1. Initial program 50.2

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

    2. Taylor expanded around inf 9.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.940844341770915 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 2.1282394422164936 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(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)))))