Average Error: 38.9 → 19.3
Time: 4.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 6.4768545182660149 \cdot 10^{131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.3474626627347847 \cdot 10^{154}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\

\mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 6.4768545182660149 \cdot 10^{131}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\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 r187325 = 0.5;
        double r187326 = 2.0;
        double r187327 = re;
        double r187328 = r187327 * r187327;
        double r187329 = im;
        double r187330 = r187329 * r187329;
        double r187331 = r187328 + r187330;
        double r187332 = sqrt(r187331);
        double r187333 = r187332 + r187327;
        double r187334 = r187326 * r187333;
        double r187335 = sqrt(r187334);
        double r187336 = r187325 * r187335;
        return r187336;
}
↓
double f(double re, double im) {
        double r187337 = re;
        double r187338 = -1.3474626627347847e+154;
        bool r187339 = r187337 <= r187338;
        double r187340 = 0.5;
        double r187341 = 2.0;
        double r187342 = im;
        double r187343 = -2.0;
        double r187344 = r187343 * r187337;
        double r187345 = r187342 / r187344;
        double r187346 = r187342 * r187345;
        double r187347 = r187341 * r187346;
        double r187348 = sqrt(r187347);
        double r187349 = r187340 * r187348;
        double r187350 = -1.612699741875562e-300;
        bool r187351 = r187337 <= r187350;
        double r187352 = sqrt(r187341);
        double r187353 = fabs(r187342);
        double r187354 = r187352 * r187353;
        double r187355 = r187337 * r187337;
        double r187356 = r187342 * r187342;
        double r187357 = r187355 + r187356;
        double r187358 = sqrt(r187357);
        double r187359 = r187358 - r187337;
        double r187360 = sqrt(r187359);
        double r187361 = r187354 / r187360;
        double r187362 = r187340 * r187361;
        double r187363 = 6.476854518266015e+131;
        bool r187364 = r187337 <= r187363;
        double r187365 = r187358 + r187337;
        double r187366 = r187341 * r187365;
        double r187367 = sqrt(r187366);
        double r187368 = r187340 * r187367;
        double r187369 = 2.0;
        double r187370 = r187369 * r187337;
        double r187371 = r187341 * r187370;
        double r187372 = sqrt(r187371);
        double r187373 = r187340 * r187372;
        double r187374 = r187364 ? r187368 : r187373;
        double r187375 = r187351 ? r187362 : r187374;
        double r187376 = r187339 ? r187349 : r187375;
        return r187376;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.9
Herbie19.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.3474626627347847e+154

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Simplified50.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity50.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}}\]

    7. Applied add-sqr-sqrt58.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)}}^{2}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}\]

    8. Applied unpow-prod-down58.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{\left(\sqrt{im}\right)}^{2} \cdot {\left(\sqrt{im}\right)}^{2}}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}\]

    9. Applied times-frac57.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{\left(\sqrt{im}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}\right)}}\]

    10. Simplified57.9

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

    11. Simplified50.2

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

    12. Taylor expanded around -inf 22.7

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

    if -1.3474626627347847e+154 < re < -1.612699741875562e-300

    1. Initial program 40.6

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

    3. Applied flip-+40.4

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

    4. Simplified31.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-*r/31.6

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

    7. Applied sqrt-div30.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]
    8. Using strategy rm

    9. Applied sqrt-prod30.4

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

    10. Simplified20.5

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

    if -1.612699741875562e-300 < re < 6.476854518266015e+131

    1. Initial program 21.1

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

    if 6.476854518266015e+131 < re

    1. Initial program 57.7

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

    2. Taylor expanded around inf 8.4

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

  4. Final simplification19.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3474626627347847 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 6.4768545182660149 \cdot 10^{131}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))))