Average Error: 38.2 → 22.1
Time: 10.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.32924863909638652 \cdot 10^{154}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 9.9898197352700734 \cdot 10^{-90}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.32924863909638652 \cdot 10^{154}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{-2 \cdot re}} \cdot 0.5\\

\mathbf{elif}\;re \le 9.9898197352700734 \cdot 10^{-90}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\

\end{array}
double f(double re, double im) {
        double r372287 = 0.5;
        double r372288 = 2.0;
        double r372289 = re;
        double r372290 = r372289 * r372289;
        double r372291 = im;
        double r372292 = r372291 * r372291;
        double r372293 = r372290 + r372292;
        double r372294 = sqrt(r372293);
        double r372295 = r372294 + r372289;
        double r372296 = r372288 * r372295;
        double r372297 = sqrt(r372296);
        double r372298 = r372287 * r372297;
        return r372298;
}
↓
double f(double re, double im) {
        double r372299 = re;
        double r372300 = -1.3292486390963865e+154;
        bool r372301 = r372299 <= r372300;
        double r372302 = im;
        double r372303 = r372302 * r372302;
        double r372304 = 2.0;
        double r372305 = r372303 * r372304;
        double r372306 = sqrt(r372305);
        double r372307 = -2.0;
        double r372308 = r372307 * r372299;
        double r372309 = sqrt(r372308);
        double r372310 = r372306 / r372309;
        double r372311 = 0.5;
        double r372312 = r372310 * r372311;
        double r372313 = 9.989819735270073e-90;
        bool r372314 = r372299 <= r372313;
        double r372315 = sqrt(r372304);
        double r372316 = fabs(r372302);
        double r372317 = r372315 * r372316;
        double r372318 = r372299 * r372299;
        double r372319 = r372318 + r372303;
        double r372320 = sqrt(r372319);
        double r372321 = r372320 - r372299;
        double r372322 = sqrt(r372321);
        double r372323 = r372317 / r372322;
        double r372324 = r372311 * r372323;
        double r372325 = 2.0;
        double r372326 = r372325 * r372299;
        double r372327 = r372304 * r372326;
        double r372328 = sqrt(r372327);
        double r372329 = r372311 * r372328;
        double r372330 = r372314 ? r372324 : r372329;
        double r372331 = r372301 ? r372312 : r372330;
        return r372331;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.2
Target33.2
Herbie22.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -1.3292486390963865e+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. Applied associate-*r/64.0

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

    5. Applied sqrt-div64.0

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

    6. Simplified51.3

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

    7. Taylor expanded around -inf 21.3

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

    if -1.3292486390963865e+154 < re < 9.989819735270073e-90

    1. Initial program 35.2

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

    3. Applied flip-+37.3

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

    4. Applied associate-*r/37.3

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

    5. Applied sqrt-div37.6

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

    6. Simplified30.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod30.5

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

    9. Simplified23.8

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

    if 9.989819735270073e-90 < re

    1. Initial program 33.8

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

    2. Taylor expanded around inf 19.7

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.32924863909638652 \cdot 10^{154}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 9.9898197352700734 \cdot 10^{-90}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))))