Average Error: 37.5 → 26.1
Time: 17.7s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 3.2277128143621017 \cdot 10^{-299}:\\ \;\;\;\;\left(\frac{\sqrt{im \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2.0}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 3.920451788983101 \cdot 10^{+83}:\\ \;\;\;\;\left(\sqrt{2.0} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 3.2277128143621017 \cdot 10^{-299}:\\
\;\;\;\;\left(\frac{\sqrt{im \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2.0}\right) \cdot 0.5\\

\mathbf{elif}\;re \le 3.920451788983101 \cdot 10^{+83}:\\
\;\;\;\;\left(\sqrt{2.0} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}\right) \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r6808270 = 0.5;
        double r6808271 = 2.0;
        double r6808272 = re;
        double r6808273 = r6808272 * r6808272;
        double r6808274 = im;
        double r6808275 = r6808274 * r6808274;
        double r6808276 = r6808273 + r6808275;
        double r6808277 = sqrt(r6808276);
        double r6808278 = r6808277 + r6808272;
        double r6808279 = r6808271 * r6808278;
        double r6808280 = sqrt(r6808279);
        double r6808281 = r6808270 * r6808280;
        return r6808281;
}
↓
double f(double re, double im) {
        double r6808282 = re;
        double r6808283 = 3.2277128143621017e-299;
        bool r6808284 = r6808282 <= r6808283;
        double r6808285 = im;
        double r6808286 = r6808285 * r6808285;
        double r6808287 = sqrt(r6808286);
        double r6808288 = r6808282 * r6808282;
        double r6808289 = r6808288 + r6808286;
        double r6808290 = sqrt(r6808289);
        double r6808291 = r6808290 - r6808282;
        double r6808292 = sqrt(r6808291);
        double r6808293 = r6808287 / r6808292;
        double r6808294 = 2.0;
        double r6808295 = sqrt(r6808294);
        double r6808296 = r6808293 * r6808295;
        double r6808297 = 0.5;
        double r6808298 = r6808296 * r6808297;
        double r6808299 = 3.920451788983101e+83;
        bool r6808300 = r6808282 <= r6808299;
        double r6808301 = r6808282 + r6808290;
        double r6808302 = sqrt(r6808301);
        double r6808303 = r6808295 * r6808302;
        double r6808304 = r6808303 * r6808297;
        double r6808305 = r6808282 + r6808282;
        double r6808306 = r6808305 * r6808294;
        double r6808307 = sqrt(r6808306);
        double r6808308 = r6808297 * r6808307;
        double r6808309 = r6808300 ? r6808304 : r6808308;
        double r6808310 = r6808284 ? r6808298 : r6808309;
        return r6808310;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target32.9
Herbie26.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < 3.2277128143621017e-299

    1. Initial program 45.1

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

    3. Applied sqrt-prod45.2

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

    5. Applied flip-+45.2

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

    6. Applied sqrt-div45.2

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

    7. Simplified35.1

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

    if 3.2277128143621017e-299 < re < 3.920451788983101e+83

    1. Initial program 19.7

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

    3. Applied sqrt-prod20.0

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

    if 3.920451788983101e+83 < re

    1. Initial program 45.9

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

    2. Taylor expanded around inf 11.5

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

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 3.2277128143621017 \cdot 10^{-299}:\\ \;\;\;\;\left(\frac{\sqrt{im \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2.0}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 3.920451788983101 \cdot 10^{+83}:\\ \;\;\;\;\left(\sqrt{2.0} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))