Average Error: 37.1 → 25.8
Time: 18.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.674549697368973 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 7057901.046632865:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]
double f(double re, double im) {
        double r37946246 = 0.5;
        double r37946247 = 2.0;
        double r37946248 = re;
        double r37946249 = r37946248 * r37946248;
        double r37946250 = im;
        double r37946251 = r37946250 * r37946250;
        double r37946252 = r37946249 + r37946251;
        double r37946253 = sqrt(r37946252);
        double r37946254 = r37946253 + r37946248;
        double r37946255 = r37946247 * r37946254;
        double r37946256 = sqrt(r37946255);
        double r37946257 = r37946246 * r37946256;
        return r37946257;
}
↓
double f(double re, double im) {
        double r37946258 = re;
        double r37946259 = -7.674549697368973e-232;
        bool r37946260 = r37946258 <= r37946259;
        double r37946261 = im;
        double r37946262 = r37946261 * r37946261;
        double r37946263 = 2.0;
        double r37946264 = r37946262 * r37946263;
        double r37946265 = sqrt(r37946264);
        double r37946266 = r37946258 * r37946258;
        double r37946267 = r37946262 + r37946266;
        double r37946268 = sqrt(r37946267);
        double r37946269 = r37946268 - r37946258;
        double r37946270 = sqrt(r37946269);
        double r37946271 = r37946265 / r37946270;
        double r37946272 = 0.5;
        double r37946273 = r37946271 * r37946272;
        double r37946274 = 7057901.046632865;
        bool r37946275 = r37946258 <= r37946274;
        double r37946276 = r37946258 + r37946268;
        double r37946277 = r37946263 * r37946276;
        double r37946278 = sqrt(r37946277);
        double r37946279 = r37946278 * r37946272;
        double r37946280 = r37946258 + r37946258;
        double r37946281 = r37946263 * r37946280;
        double r37946282 = sqrt(r37946281);
        double r37946283 = r37946282 * r37946272;
        double r37946284 = r37946275 ? r37946279 : r37946283;
        double r37946285 = r37946260 ? r37946273 : r37946284;
        return r37946285;
}


0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.674549697368973 \cdot 10^{-232}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 7057901.046632865:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\

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

\end{array}

Error

Bits error versus re

Bits error versus im

Target

Original37.1
Target32.0
Herbie25.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -7.674549697368973e-232

    1. Initial program 47.3

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

    3. Applied flip-+47.3

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

    4. Applied associate-*r/47.3

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

    5. Applied sqrt-div47.4

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

    6. Simplified34.6

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

    if -7.674549697368973e-232 < re < 7057901.046632865

    1. Initial program 22.6

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

    if 7057901.046632865 < re

    1. Initial program 38.0

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

    2. Taylor expanded around inf 14.6

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

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.674549697368973 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 7057901.046632865:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 
(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)))))